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Sampling bias corrections for discrete and Gaussian partial information decompositions
Open accessAbstract
Partial information decomposition (PID) has emerged as a principled way to decompose the information carried by neural activity into components identifying whether interactions among neurons or brain areas generate synergistic or redundant information. Here, we demonstrate that empirical measures of synergy and redundancy based on either Gaussian or discrete probability estimators suffer from a substantial limited-sampling estimation bias. This bias is much larger for synergy than for redundancy. The gap between them increases with the number of parameters specifying the probability distributions. We develop procedures that effectively correct for the bias and provide rules of thumb for the sample sizes required to obtain unbiased estimates. We show that, when used on empirical brain datasets, they successfully remove large synergy biases across species, recording modalities, and experimental designs. Our bias corrections extend the range of neuroscience questions and experimental designs addressable with PID and allow accurate comparisons between synergy and redundancy.
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TL;DR
Partial information decomposition (PID) estimates of synergy and redundancy from discrete or Gaussian probability estimators carry a substantial limited-sampling bias, and this bias is much larger for synergy than for redundancy. The authors develop bias-correction procedures and sample-size rules of thumb, then validate them on simulations and five empirical neural datasets.
Key findings
- Plugin estimates of joint information, synergy, and redundancy are upward biased; for tens of samples the bias can be as large as or larger than the target quantities. Discrete synergy bias scales approximately as \((S \times R^2)/N\), whereas redundancy bias grows at most as \((S \times R)/N\); Gaussian synergy bias grows faster with source/target dimension \(d\) than redundancy bias.
- The recommended merged correction combines two corrections with opposite residual-error tendencies. The sufficient-sampling regimes are \(N > 4 \times S \times R^2\) for discrete PID and \(N > 12d\) (or \(d < N/12\)) for Gaussian PID. The prior Venkatesh correction left residual errors as large as 30%–40% of the true joint information value.
- In real spiking datasets, uncorrected joint information was biased upward by ≈30%–50%, redundancy was almost unbiased, and synergy bias differences were approximately 60%, 200%, and 800% of target values across three datasets. In one dataset, uncorrected estimates would have wrongly concluded synergy > redundancy, whereas corrected estimates found redundancy higher.
- For human fMRI with \(d = 20\) ROIs and 256 time points, synergy bias was ≈35% of the target value and redundancy bias ≈10%, both well corrected. In MEG, corrections removed spurious pre-stimulus information; post-stimulus discrete bias subtraction was 146% for joint information and 1,669% for synergy, after which synergy and redundancy were small and non-significant. Three-source PID showed three-way synergy bias far larger than three-way redundancy, with a larger gap than for two-source PID.
Methods
- Simulations generated discrete spike-count and Gaussian processes with known ground-truth PID components, varying number of samples \(N\), discrete bins \(R,S\), Gaussian dimension \(d\), and information level. Bias was computed as the average empirical estimate minus ground truth, typically over 100 simulations, with mean ± 3 SEM.
- Bias corrections included quadratic extrapolation (QE), shuffle subtraction, resampling, and a merged weighted correction. For discrete PID, QE fits a \(1/N\) polynomial using half and quarter data; shuffle subtraction permutes target values; merged combines QE and shuffle. For Gaussian PID, resampling replaces QE, and merged combines resampling and shuffle.
- Empirical validation used five previously published datasets: mouse A1 two-photon calcium imaging (\(n = 6,209\) pairs, mean 140 trials), mouse CA1 calcium imaging (\(n = 36,158\) pairs, 870 trials), monkey A1 electrophysiology (\(n = 83\) pairs, 108 trials), human HCP resting-state fMRI (\(n = 100\) participants, 256 time points, \(d = 20\) ROIs), and human MEG (\(n = 30\) participants, 519–864 trials). Analyses used discrete BROJA PID, Gaussian gPID, or three-source MMI PID.
Limitations
- Stated: the authors tested BROJA, Imin, and IMMI PIDs but say other PIDs, including continuous non-Gaussian distributions, should be tested. They also state that the bias-correction procedures are heuristic and lack theoretical guarantees for the sign of residual errors.
- Stated: for cellular spiking activity, they tested pairs and triplets of neural units but did not systematically test large populations; scaling up would require coupling with dimensionality-reduction techniques. They also note that combining bias-correction and compensatory procedures remains to be explored.
- Evident from the text: three-source analyses used MMI because discrete BROJA and Gaussian gPID are not defined for more than two sources, so the three-source bias results do not cover those PID definitions.
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80 Own result · 37 Own interpretation · 32 No basis given · 24 Check pending · 9 Own method · 6 No full text of the sources · 1 Sources not found
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In the paper
“This bias is much larger for synergy than for redundancy.”
- Also stated in: § Introduction · boosted
“We find that synergy is far more biased than redundancy, especially in the discrete case.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Abstract
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- Also stated in: § Introduction · boosted
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In the paper
“Here, we demonstrate that empirical measures of synergy and redundancy based on either Gaussian or discrete probability estimators suffer from a substantial limited-sampling estimation bias.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Abstract
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“Fourth, and unlike what was assumed in a previous study,30 the bias is highly uneven across PID components.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
What the sources say
- Check pendingcites [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
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“The gap between them increases with the number of parameters specifying the probability distributions.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Abstract
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“Third, the bias of the synergy is larger both proportionally and in absolute terms for lower information levels (Figure 2A, right).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“Instead, the synergy and redundancy bias very slightly increases with the information level.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“We develop procedures that effectively correct for the bias and provide rules of thumb for the sample sizes required to obtain unbiased estimates.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Abstract
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“Synergy and redundancy are also biased upward.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“We develop procedures that effectively correct for the bias and provide rules of thumb for the sample sizes required to obtain unbiased estimates.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Abstract
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“As a result, the values of information estimated from a finite amount of samples are not centered around the underlying ground-truth information value (Figures 1B and 1D).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“We show that, when used on empirical brain datasets, they successfully remove large synergy biases across species, recording modalities, and experimental designs.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Abstract
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“Importantly, the bias of the Gaussian PID is also highly uneven between synergy and redundancy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“First, as for the Shannon joint information, synergy and redundancy have an upward bias decreasing with N (Figure 2A, left).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“Second, for the tens of samples usually available in neuroscience experiments, the bias could be as large as, or even larger than, the target quantities (Figures 2A and S5).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“As predicted by an exact analytical bias expression valid for N > 3d,80,81 the joint information bias depends only on N and d but is independent of the information level (Figure 2B, right).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
What the sources say
- Check pendingcites [80] Cai, 2015, Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions, J. Multivar. Anal. source
- Check pendingcites [81] Goodman, 1963, The distribution of the determinant of a complex wishart distributed matrix, Ann. Math. Stat. source
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“The bias increases much faster with d for the joint information (scaling as a supralinear function of 3d, as it contains bias of the entropy of the joint distribution of two sources and one target) than for the single-source information (scaling as a supralinear function of 2d, as it contains bias of the entropy of the joint distribution of one source and one target).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“When real-data probabilities are estimated from the finite number of available samples N, both the probabilities to be estimated for the discrete PID and the covariance matrices needed for the Gaussian PID have finite sampling fluctuations around their true values, which make the distributions of sources typically more dependent on the target than they really are (Figures 1A and 1C).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“Our work also reported an MEG-based multivariate analysis of sensory information coding across multiple brain regions in different anatomical groups, which reveals the presence of higher-order redundancy in stimulus representations within and across cortical groups.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“we found that our bias corrections provide accurate estimates of population redundancy and synergy even for small trial numbers (Note S7 and Figure S18).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“Because of the opposite trend of their residual errors, we also averaged the resample and shuffle corrections to obtain a merged correction, which further reduces the residual bias.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“From post-stimulus activity (Figure 8C, bottom right), the bias corrections removed from the plugin estimates biases that were similar in size to those identified in pre-stimulus activity”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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“the synergy bias (and to a lesser extent the redundancy bias) grows rapidly with the dimensionality of the problem under investigation (related to either the number of sources used or the number of dimensions used to describe each source)”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“For example, considering simulations with ground-truth values of redundancy larger than synergy, not correcting for the bias would lead to incorrectly estimating synergy larger than redundancy for low trial numbers (Figure S5A, third row).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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“As we show in Figure S17, these low-dimensional Gaussian estimates of redundant and synergistic interactions across brain regions have negligible bias.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
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“the amount of bias subtracted by the corrections was a large fraction of the value of the corresponding quantities after bias subtraction.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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“our systematic analysis of the limited-sampling bias of PID analyses reveals that synergy is far more biased than redundancy”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“For discrete PID, the sufficient-sampling regime requires that the number of samples N satisfy N > 4 × S × R2, where R and S are the number of possible discrete values for the sources and target, respectively.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“Finally, when applying the merged correction and the other corrections in the sufficient-sampling regime, accurate estimates with small residual biases are obtained regardless of the information level (Figure S8A), highlighting the robustness of the results.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“The Gaussian PID in the one-dimensional sources had very small bias, as shown by the fact that the plugin (uncorrected) and bias-subtracted estimators have very similar values.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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“We find that both corrections reduce considerably the bias of the joint information and of the PID components (especially of the synergy, which is more biased).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“We chose it to replace the QE (used in the discrete PID) for the Gaussian PID, as we verified that QE does not work well for the Gaussian information because its bias is non-polynomial.81”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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- Check pendingcites [81] Goodman, 1963, The distribution of the determinant of a complex wishart distributed matrix, Ann. Math. Stat. source
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“As predicted by our simulations, redundancy is almost unbiased (shown by the fact that it does not change value after bias corrections; Figures 6A–6C).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
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“In Figure 5, the correction left residual errors of the order of 0.1 bits or smaller against original uncorrected bias values that were of the order of several bits.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“These findings hold also for the Unique information (Figure S2).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“In our simulations, the Venkatesh bias correction left a residual error that could be as big as 30%–40% of the true joint information value (Figures 4B, 5B, S9, and S10).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“The Venkatesh correction fails to remove a large part of the synergy and redundancy bias, as predicted by the theoretical and simulation results (Figure S17).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
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“For the Gaussian PID, the sufficient-sampling regime requires that the number of samples N satisfy N > 12d, where d is the dimension of the sources and the target.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“Based on the fact that (Figure 2A) the bias of each PID term decreases smoothly with the number of samples N”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“Across simulations, the merged, resampling, and shuffle corrections consistently leave a negligible residual error for N > 4 × 3d or d < N/12 (Figures 4B, 5B, S3, S8B, S9, S10, and S11), a condition that we term the “sufficient-sampling regime for Gaussian PID.””
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“For the Gaussian PID, the bias of the synergy also grows faster with the source dimension d than that of the redundancy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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In the paper
“Conclusions based on plugin estimates would have led to the wrong conclusion that in this system both synergy and redundancy are present, with synergy larger than redundancy”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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In the paper
“Application of our bias corrections successfully removed the spurious pre-stimulus information values, and all corrections gave consistent results.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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In the paper
“All corrections gave consistent results, and the amount of bias subtracted by the corrections was a large fraction of the value of the resulting quantities after bias subtraction (146% and 1,669% for joint information and synergy, respectively).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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In the paper
“All our Gaussian bias corrections improve considerably the existing Venkatesh correction.30”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
What the sources say
- Check pendingcites [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
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In the paper
“Because the shuffle subtraction and the QE (for discrete) or resampling (for Gaussian) tend to have residual errors of opposite sign (see Figures 4, 5, and S3–S10), we introduced a merged bias correction that estimates the bias as the weighted average of the two above-described procedures:”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
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In the paper
“As shown by the smaller values obtained after correcting for the bias, the plugin (uncorrected) joint information is biased upward by ≈30%–50% (Figures 6A–6C).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
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In the paper
“that in practical applications the bias may be as large as the target information values”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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In the paper
“in the pre-stimulus activity (compatible with the above theoretical predictions that the bias of synergy and joint information is larger for lower information values in the discrete case).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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In the paper
“In most analyzed conditions, the majority (or at least a large part) of the synergy computed without bias corrections (plugin estimator) reflected just the limited-sampling bias and not real neural synergy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“The bias-correction procedures appear to be highly effective also for the three-source PID.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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“The merged correction is highly effective because it evens out the opposite tendencies of the QE and the shuffle subtraction (Figures 4A, 5A, and S8A).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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“We use discrete PID because it is natural for spiking activity and discrete stimuli and because Gaussian PID gave extremely low information values for two of these datasets (Figure S13 and Note S5).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
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In the paper
“However, we found that the union information is much less biased upward than the joint information (Figure 3).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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In the paper
“Bias-corrected estimates were very similar between the discrete and Gaussian PID when using bias corrections”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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In the paper
“The good performance of these bias corrections within this regime was found across all simulation parameters and across all considered PID types (Figures 4A, 5A, and S3–S7).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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In the paper
“For the three-source PID analyses (Figures 7 and 8), we used an equal weighting between the shuffle and the QE correction because in the range of simulations we explored, we did not have major shifts in information levels.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
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In the paper
“For a fixed number of samples, using any of the bias corrections, but especially the merged correction, extends considerably the range of discrete number of bins for which information can be estimated accurately.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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In the paper
“The most effective bias-correction method is what we termed the merged correction, which combines linearly two different bias-correction methods with opposite tendencies for residual errors.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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In the paper
“and that it is often large enough to be comparable to or larger than the target values of the quantities.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
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“In this setting, the value of each individual source, or the combination of any pair of sources, does not provide any information about the target variable, while the combination of the three sources fully determines Y.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of the simulations used to test the three-source PID
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In the paper
“This implies that the bias must be corrected for in real-data analyses.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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In the paper
“While the problem of null models for information-theoretic quantities is gaining recognition,29 to our knowledge only two studies have investigated PID sampling bias, specifically for a Gaussian PID30 and for a PID with a mix of discrete and Gaussian variables.31”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Introduction
What the sources say
- Check pendingcites [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
- Check pendingcites [31] Barà, 2025, Partial information decomposition for discrete target and continuous source random variables, Phys. Rev. E source
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In the paper
“Earlier applications of information theory to neuroscience have recognized that Shannon information measures from neural activity suffer from a prominent limited-sampling bias.24,25,26”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Introduction
What the sources say
- Not checked — no full text of the sourcecites [24] Panzeri, 1996, Analytical estimates of limited sampling biases in different information measures, Network source
- Not checked — no full text of the sourcecites [25] Paninski, 2003, Estimation of entropy and mutual information, Neural Comput. source
- Check pendingcites [26] Nemenman, 2004, Entropy and information in neural spike trains: Progress on the sampling problem, Phys. Rev. E source
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In the paper
“However, to date the problem of whether and how synergy or redundancy components of PID are biased has not been studied systematically.”
- Also stated in: § Bias of synergy and redundancy for discrete and Gaussian estimators · hedged
“Although the bias of Shannon information has been studied extensively for both discrete24,25,27 and Gaussian estimators,80,81 the bias of individual PID components has remained largely unexplored.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Introduction
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- Also stated in: § Bias of synergy and redundancy for discrete and Gaussian estimators · hedged
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In the paper
“The union information depends only on marginal distributions of the target and one source also for the BROJA PID.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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In the paper
“The fact that the bias of the synergy is consistently larger than that of the redundancy and that their difference increases with R for the discrete PID and with d for the Gaussian PID can be understood from analytical considerations, developed in Note S6 and briefly summarized here.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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In the paper
“This bias not only affects the accuracy of the estimate but can also skew comparisons between the information carried by simpler (less biased) and more complex (more biased) neural representations defined by a larger number of parameters.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Introduction
What the sources say
- Not checked — no full text of the sourcecites [27] Panzeri, 2007, Correcting for the sampling bias problem in spike train information measures, J. Neurophysiol. source
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In the paper
“The resulting second-order equations (plotted in Figure S15) allow users to easily obtain an estimation of the magnitude of the residual bias of both synergy and redundancy for the discrete and Gaussian PID as a function of N, R, and d.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
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In the paper
“In the Gaussian case, analytical considerations (see Note S6) indicate that the union information bias depends only on the dimensionality of the covariance matrices d and on the number of trials N and not on the magnitude of the union information itself (as implicitly assumed in Equation 17 of Venkatesh et al.30).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
What the sources say
- Not matchedcites Note S6
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Check pendingcites Venkatesh et al. [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
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In the paper
“it empowers experimenters with bias-correction methods that greatly improve PID estimates with respect to uncorrected measures or other available methods.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
What the sources say
- Check pendingcites [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
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In the paper
“The uneven distribution of bias between synergy and redundancy is practically important because it implies that conclusions taken from limited empirical data without correcting for the bias may offer a distorted view of information coding, artificially skewed toward synergy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
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In the paper
“Thus, the resulting PID terms can be taken as an estimation of their bias, which we subtracted from their plugin estimates.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
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In the paper
“For the Gaussian MMI, closed-form solutions for PID exist, since the mutual information terms appearing in Equation 3 can be expressed in terms of covariance matrices as in Equation 20.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Gaussian PID measures
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In the paper
“This suggests that computing both QE and shuffle subtraction from real data provides reasonable empirical bounds to the information estimate for synergy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
hedgedstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“Our work highlights and explains the presence of a major and previously neglected difference in the sampling bias between synergy and redundancy, which can inflate synergy disproportionately.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus, the bias problem cannot be neglected in real-data neuroscience applications.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Applying this procedure to the discrete PID would have been not possible because it assumes an analytical knowledge of the bias, which is only available for the Gaussian case.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
What the sources say
- Check pendingcites Venkatesh et al. [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“Because the Venkatesh procedure relies on these incorrect proportional-bias assumptions, it misassigns the bias among PID components, especially when all PID terms are non-zero and the true union information does not match any of the Shannon information quantities (so that the rectification steps in Equations 106 and 107 of Venkatesh et al.30 do not correct the error).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
What the sources say
- Check pendingcites Venkatesh et al. [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“our analyses of real fMRI recordings with Gaussian PID confirm that previous reports of synergy in cross-area brain interactions are genuine and not due to biased estimates”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
What the sources say
- Check pendingcites [23] Luppi, 2024, Information decomposition and the informational architecture of the brain, Trends Cogn. Sci. source
- Check pendingcites [49] Luppi, 2022, A synergistic core for human brain evolution and cognition, Nat. Neurosci. source
- Check pendingcites [95] Varley, 2023, Partial entropy decomposition reveals higher-order information structures in human brain activity, Proc. Natl. Acad. Sci. USA source
- Check pendingcites [99] Varley, 2023, Multivariate information theory uncovers synergistic subsystems of the human cerebral cortex, Commun. Biol. source
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In the paper
“The rule of thumb is that the information estimation is accurate (with a small tolerance value that can be easily estimated from the data) in the so-called sufficient-sampling regime.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Our results show that, by assuming that the bias of the union information is proportional to the bias of the joint information, this approach can severely misattribute bias across PID components.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus, the bias of three-way synergy Syn3 and three-way redundancy Red3 with the MMI PID can be estimated from the bias of Shannon information quantities.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Synergy and redundancy bias in three-source PID
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“These bias corrections also greatly extend the dimensionality of the problems that can be accurately investigated empirically, allowing, for example, highly multivariate analyses of synergy and redundancy across many brain regions.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“Our simulations and analytical considerations were accompanied by a thorough validation of the performance of our bias-correction algorithms on five different neuroscientific datasets.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“and that their use allows formulating and addressing neuroscientific questions about multisource encoding that would be out of reach without bias corrections.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: unclear — It is unclear whether it could be checked against evidence.
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In the paper
“Altogether, this application suggests that our bias corrections are useful to extend the Gaussian estimation of synergistic and redundant interactions to distributed multidimensional brain networks, even with a limited number of samples at different time points, as is common in neuroimaging (especially for clinical applications).91,92”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
What the sources say
- Check pendingcites [91] Van Dijk, 2010, Intrinsic functional connectivity as a tool for human connectomics: theory, properties, and optimization, J. Neurophysiol. source
- Not checked yetcites [92] Birn, 2013, The effect of scan length on the reliability of resting-state fMRI connectivity estimates, Neuroimage source
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“and that this problem can artificially skew comparisons of synergy and redundancy in the brain.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“they show that bias corrections can improve the accuracy of interaction analyses with data sizes typical of clinical applications.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This extends the applicability of PID to, for example, clinical rather than laboratory settings.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“these bias-correction methods are generally applicable to a wide range of experimental situations and types of brain recordings, crucially encompassing both discrete and continuous cases.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Therefore, by construction, only the three-way synergy Syn3 PID component is non-zero.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of the simulations used to test the three-source PID
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“The target variable (visual sample orientation) was drawn from a Gaussian distribution by experimental design.77”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Three-source PID analysis of real MEG data
What the sources say
- Check pendingcites [77] Park, 2025, Confirmation bias through selective readout of information encoded in human parietal cortex, Nat. Commun. source
plainstates a cause- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This system has a ground-truth non-zero redundancy component only.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of the simulations used to test the three-source PID
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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- It may state two separate claims as one. 81
- Probably not a substantial claim. 20
- The quote only partly states this claim. 1
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When varying, at fixed N, the number of discrete bins R, S of each source and target variable (keeping R = S for simplicity), the bias of the synergy increases rapidly and supralinearly with R (Figure 2A, middle), whereas the bias of the redundancy grows much more slowly and is much smaller.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“While variance can be reduced by averaging (e.g., across experimental subjects or groups of neural units), the limited-sampling bias (shortened to bias hereafter) cannot.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The Gaussian MMI and the gPID synergy bias are thus dominated by the bias of entropies of 3d-dimensional distributions that scale as a supralinear function of 3d, whereas the redundancy bias contains only biases of entropies of 2d-dimensional distributions that scale as a supralinear function of 2d.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The bias of the sum of the Unique information terms is plotted in Figure S2B. It increases with 1/N and d and is in most cases intermediate between that of synergy and redundancy in terms of size and rate of growth with 1/N and d.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These properties are explained by analytical results (obtained in the large-N limit; Note S6) showing that the bias decreases with N as powers of 1/N and that it is proportional to the number of discrete bins R, S.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This leads to both a systematic error (limited-sampling bias) and a statistical error (limited-sampling variance) in information estimates.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In sum, for both MMI and BROJA discrete PID the synergy bias scales as (S × R2)/N, whereas redundancy bias grows at most as (S × R)/N as R increases.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The Gaussian joint information has an upward bias that decreases with the number of samples N (Figure 2B, left) and increases with the dimensionality d of the sources and target (Figure 2B, middle).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The bias of the joint information scales as (S × R2)/N (as it requires the sampling of the joint distribution of s, r1, r2), while the bias of single-source information scales as (S × R)/N (it requires the sampling of the joint distribution of s and of one source).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“In the Gaussian case, an exact expression is available81 for the mutual information bias, which grows steeply and supralinearly when increasing the dimension d of sources and targets and decreases when increasing N (Figure S14).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
What the sources say
- Check pendingcites [81] Goodman, 1963, The distribution of the determinant of a complex wishart distributed matrix, Ann. Math. Stat. source
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“As previously reported,27,60 the plugin estimate of the joint Shannon information has an upward bias, which decreases with the number of samples N (Figures 1B and 2A [left]), increases with the number of discrete bins R, S of each source and target variable (Figure 2A, middle, where we keep R = S for simplicity), and decreases with the information level (Figure 2A, right).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
What the sources say
plainstates a causen = 100- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The MMI union bias equals the bias of the maximum single-source information (Equation S12) and thus scales as (S × R)/N and is smaller than the joint information bias (Figure 3A).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When varying d at fixed N (Figure 2B, middle), the bias of both synergy and redundancy increases with d, but it increases faster for synergy than for redundancy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The bias of the sum of the Unique information terms (which is the difference between the bias of the joint information and the biases of synergy and redundancy) is plotted in Figure S2A. It increases with 1/N and R and is in most cases intermediate between that of synergy and redundancy in terms of size and rate of growth with 1/N and R.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“our bias-correction methods reduce by orders of magnitude the number of samples needed to reach a certain estimation precision or the magnitude of the residual errors in PID estimation from small numbers of samples.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We verified (Figure S12) that this optimized weighted merged procedure improved slightly a simpler version with equal weighting of the shuffle and QE.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since shuffle subtraction performs considerably better than QE at low information levels, and the opposite holds at higher information levels, for each simulated data point from all four discrete simulation scenarios shown in Figure S5, we computed”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates an association- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It leads to very small residual biases throughout the sufficient-sampling regime for both synergy and redundancy, and it still works well even in conditions of severe undersampling—well outside the sufficient-sampling regime.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Results using the maximum available number of time points (1,189), showing a still significant but much lower bias, are reported in Figure S17.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
plainstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In the sufficient-sampling regime, the small residual error for synergy and joint information is mostly upward for QE (intuitively because bias terms of higher order in 1/N, not included in the QE, are all positive83) and downward for the shuffle correction (because lower information levels have larger upward bias).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Our systematic analysis of the information carried by tens of thousands of pairs of neurons across different cortical areas revealed widespread presence of both synergy and redundancy across brain areas and species in neuron-to-neuron interactions, with regional variations in the prevalence of synergy vs. redundancy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates an association- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Application of our bias corrections successfully removed the spurious pre-stimulus information values, and all corrections gave consistent results.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The resampling correction works well for the Gaussian but less so for the discrete PID, because the overall bias of the joint Gaussian information does not depend much on the information level (Figures S9 and S10).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Using this bias correction leads to much more precise estimates than using no bias corrections and extends the range of applicability of the method to larger numbers of bins and smaller number of samples.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Simulations corroborate these analytical predictions (Figures 4, 5, S3, and S8–S11): the Venkatesh correction typically underestimates the bias of synergy and redundancy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We found that synergy, and to a lesser extent redundancy, suffer from a considerable limited-sampling bias.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“whereas after bias corrections it becomes clear that both synergy and redundancy are small and neither reaches significance.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
boostedSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“To recapitulate, for the discrete PID, the bias of the synergy scales with the number of bins as (S × R2)/N and is much larger than the bias of redundancy, which scales as (S × R)/N.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“All corrections gave consistent results, and the amount of bias subtracted by the corrections was a large fraction of the value of the resulting quantities after bias subtraction (146% and 1,669% for joint information and synergy, respectively).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“both joint information and synergy and redundancy values were relatively large (approximately 0.1 and 0.03 bits, respectively)”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Despite these datasets having a good number of trials compared to typical experiments, the synergy bias is major, as the difference between bias-corrected and plugin synergy estimates is approximately 60%, 200%, and 800% of the target values across the three datasets (Figures 6A–6C).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A first scenario (“bit-of-all”) had all PID components with non-zero values (as it happened in all the four neuroscientific datasets we analyzed with two-source PID).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of simulations used to test the bias properties
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Comparing Figure 2A (left) with Figure 4A shows that not correcting for the bias would have led to large errors even in the sufficient-sampling regime and that to obtain, without using bias corrections, information measures as accurate as the ones corrected with the merged correction would have needed orders-of-magnitude more samples.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Also in the Gaussian case, the merged correction is the correction that works better and provides accurate estimates (small residual biases) even for numbers of samples or dimension values well outside the sufficient-sampling regime.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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In the paper
“Redundancy is much less biased (10% of the target value) and also well corrected for (Figure 6D).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
boosted- Asserted: passes — The authors put it forward as true.
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In the paper
“The merged correction extends the applicability of the PID to much smaller sample sizes and a much higher number of dimensions compared to uncorrected estimates and to the previous available Venkatesh correction (Figure 5B).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“We applied QE only to the discrete PID because the bias of the Gaussian PID was clearly not polynomial (see Note S6).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
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In the paper
“We consistently find across simulations that both QE and shuffle subtraction worked well for both synergy and redundancy as long as N is larger than 4 × S × R × R (dashed vertical line, Figure 4A) or equivalently that R<√(N/4S) (Figure 5A).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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In the paper
“Given that, in the discrete scenario, the resampling bias correction exhibits the same sign of residual bias as QE but consistently poorer performance (Figures S5–S7), we did not include this correction in”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“As expected intuitively, the shuffle correction works better than resampling for lower but not for higher information levels (Figure S10).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
plainstates an association- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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In the paper
“An important property of this dataset is that both the sources (MEG activity) and the target (presented orientation of the visual stimulus) follow approximately a Gaussian distribution (see methods), and thus both the discrete and the Gaussian PID can be meaningfully computed.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates a causen = 30- Asserted: passes — The authors put it forward as true.
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In the paper
“Further, when using bias corrections (but not when using uncorrected estimates), results are almost identical when running the analysis on half of the datasets (Figure S16), further confirming the robustness of the bias corrections.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Because of the above-described dependence of the bias of the PID components on the information level (Figure 2), the small residual error for synergy and redundancy information was typically downward for the resampling correction and typically upward for the shuffle correction.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
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In the paper
“With the plugin (uncorrected) estimator (Figure 8A, left), both joint information and synergy values were relatively large, indicating a significant bias.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We find that on simulated data, the bias of the three-way synergy is far larger than the bias of the three-way redundancy, with a larger gap than that found for the synergy and redundancy obtained with the two-source PID (Figure 7) for both discrete and Gaussian PID.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“As a result, in the Gaussian case, the synergy is more biased than the redundancy, and the gap in bias between the two grows supralinearly with d.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“Both corrections reduce by orders of magnitude the bias of the joint information and of synergy and redundancy, leaving a residual bias that was just a small fraction of the uncorrected bias (Figures 4B, 5B, S3, S8B, S9, and S10).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“After bias corrections, the PID reveals significant redundancy between the activity of the anatomical groups but no or little synergy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“We also developed a robust method for correcting for the bias, which significantly reduces the number of samples needed for accurate estimates.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In this setting, the value of each individual source, or the combination of any pair of sources, does not provide any information about the target variable, while the combination of the three sources fully determines Y.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of the simulations used to test the three-source PID
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Because the Gaussian joint information bias does not depend on the information level, the bias of synergy and redundancy does not increase for lower information levels as in the discrete case (Figure 2B, right).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“It satisfies many desirable properties including additivity of all PID components for independent systems of sources and targets (proven for discrete33,46 and empirically validated for Gaussian PID30) and has been widely applied to neural data.13,36,37,47,48”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Background: PID and its neuroscience use
What the sources say
- Check pendingcites [33] Bertschinger, 2014, Quantifying unique information, Entropy source
- Check pendingcites [46] Rauh, 2023, Continuity and additivity properties of information decompositions, Int. J. Approx. Reason. source
- Check pendingcites [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Our bias corrections extend the range of neuroscience questions and experimental designs addressable with PID and allow accurate comparisons between synergy and redundancy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Abstract
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“The information-theoretic neuroscience literature has long recognized the importance of correcting for the limited-sampling bias and has provided effective tools for correcting the Shannon information sampling bias.24,25,26,27,28”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Introduction
What the sources say
- Not checked — no full text of the sourcecites [24] Panzeri, 1996, Analytical estimates of limited sampling biases in different information measures, Network source
- Not checked — no full text of the sourcecites [25] Paninski, 2003, Estimation of entropy and mutual information, Neural Comput. source
- Check pendingcites [26] Nemenman, 2004, Entropy and information in neural spike trains: Progress on the sampling problem, Phys. Rev. E source
- Not checked — no full text of the sourcecites [27] Panzeri, 2007, Correcting for the sampling bias problem in spike train information measures, J. Neurophysiol. source
- Check pendingcites [28] Kraskov, 2004, Estimating mutual information, Phys. Rev. E source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
In the paper
“For the discrete case, exact expressions for the bias are not known. However, approximate expressions are available in the large-N limit.24”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
What the sources say
- Not checked — no full text of the sourcecites [24] Panzeri, 1996, Analytical estimates of limited sampling biases in different information measures, Network source
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“The decrease of the bias with the information level can also be explained from these analytical calculations,24 because the bias increases with the number of discrete responses that are possible (i.e., with non-zero probability). Low information levels correspond to less concentrated stimulus-specific distributions and, thus, have a larger number of possible responses and are more biased.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
What the sources say
- Not checked — no full text of the sourcecites [24] Panzeri, 1996, Analytical estimates of limited sampling biases in different information measures, Network source
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“To our knowledge, no general, closed-form expression exists for Gaussian PIDs with a multi-dimensional target, and such systems can yield results that deviate from IMMI.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Gaussian PID measures
hedged- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A parameter α regulates the strength of stimulus tuning of the spike counts of the individual neurons or, in other words, the separation in response strength between least and most effective stimuli (higher values of α leading to higher values of individual neuron information and, thus, higher values of joint information)”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of simulations used to test the bias properties
plainstates a cause- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“For all datasets, analysis parameters and trial numbers meet the sufficient-sampling regime.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This analysis confirms the effectiveness of the bias-correction procedures and demonstrates that they are essential for empirical studies.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Importantly and reassuringly, estimates of both redundancy and synergy were highly consistent across methods within each dataset (Figures 6A–6C), suggesting that these estimates on real data are precise and unbiased.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
hedged- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This question cannot be addressed by the discrete PID (as it does not scale to very large multivariate analyses)”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates a cause- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If uncorrected, the bias problem can artificially lead to inaccurate estimates and skew comparisons in favor of synergy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Using the uncorrected estimator would have thus led to a considerable overestimation of the synergy and to errors in conclusions.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“when considering spiking activity at the cellular level (which requires discrete PID), we tested synergy and redundancy between pairs and triplets of neural units, but we have not systematically tested large populations.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Bias-uncorrected estimates would lead to highly imprecise and possibly misleading conclusions about neural synergy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“The Venkatesh correction assumes that the union information has the same proportional bias as the joint information (and thus of the synergy, see methods), and it then rescales all PID accordingly (Equations 3, 4, and 5).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
What the sources say
- Check pendingcites [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“indicating that the bias corrections provide reliable and uninflated values for both the discrete and Gaussian PID.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“The direct calculation of information is very precise for small populations, but it does not scale up with population size.27”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
What the sources say
- Not checked — no full text of the sourcecites [27] Panzeri, 2007, Correcting for the sampling bias problem in spike train information measures, J. Neurophysiol. source
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“indicating a significant bias and proving that this question could not be addressed without bias-correction techniques even when making Gaussian approximations.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Since in this dataset MEG activity was recorded both before and after stimulus presentation, we used pre-stimulus MEG activity (for which the ground-truth value of joint information about the stimulus, and its synergy and redundancy, must all be zero) as a first important test of the bias-correction performance.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“whereas the three-way redundancy has terms with bias scaling as the bias of the single-source information (which scales as (S × R)/N for the discrete PID and as a sum of digamma functions with up to 2d terms for the Gaussian PID).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates an association- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Together, these results confirm that bias corrections are both necessary and highly effective for three-source PID”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The magnitude of the bias removed from post-stimulus estimates was comparable to, but smaller than, that observed in the pre-stimulus activity (compatible with the above theoretical predictions that the bias of synergy and joint information is larger for lower”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates an association- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“increasing γ increases synergy because it makes the two neurons more strongly correlated—because the individual process changes its rate with different stimuli, having a shared process γ > 0 also affects the amount of synergistic information about the stimulus that can be gained by only measuring the joint responses of the two neurons”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of simulations used to test the bias properties
What the sources say
- Not checked — no full text of the sourcecites [20] Pola, 2003, An exact method to quantify the information transmitted by different mechanisms of correlational coding, Network source
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“while IMMI and Imin have been defined for an arbitrary number of source variables,22,97 the BROJA PID is currently defined only for the case of two source variables.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > BROJA discrete PID
What the sources say
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In datasets 2 and 3 (Figures 6B and 6C), using uncorrected estimates would have led to a very major overestimation of the real importance of synergy, and in dataset 3 it would have led to the wrong conclusions that synergy was higher than redundancy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In this way, all information about the target is lost and all PID terms should be zero but for the finite sampling effect.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“However, we did not provide theoretical guarantees of the sign of residual errors of different bias-correction procedures.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Second, the bias-correction procedures are heuristic, although we support them with analytical considerations.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, this procedure is expected to lead often to a major underestimation of union information, which in turn leads to a severe overestimation of synergy and redundancy (see methods), which persists also for large N.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Corrections for the limited-sampling bias of synergy and redundancy
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“our work could be readily applied more widely, as PID using the discrete or Gaussian estimators studied here are often employed in real-data analyses in non-neuroscience fields”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Discussion
What the sources say
- Not checked yetcites [100] Chan, 2017, Gene regulatory network inference from single-cell data using multivariate information measures, Cell Syst. source
- Not checked yetcites [101] Wollstadt, 2023, A rigorous information-theoretic definition of redundancy and relevancy in feature selection based on (partial) information decomposition, J. Mach. Learn. Res.
- Check pendingcites [102] Tax, 2017, The partial information decomposition of generative neural network models, Entropy source
- Check pendingcites [103] Scagliarini, 2020, Synergistic information transfer in the global system of financial markets, Entropy source
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
In the paper
“Among the advantages of the BROJA PID, we note that this definition ensures that all PID terms satisfy the additivity property”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > BROJA discrete PID
What the sources say
- Check pendingcites [33] Bertschinger, 2014, Quantifying unique information, Entropy source
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In dataset 1 (Figure 6A), using the uncorrected plugin estimator would have led to the erroneous conclusion that synergy was larger than redundancy, whereas all three bias-corrected estimates consistently detected that redundancy was higher than synergy.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
hedgedstates a causeSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“We chose to fix the bin edges for each simulation to ensure that differences across trial numbers in the estimated values of the PID components were due only to trial numerosity and not to other reasons.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of the binning procedures used to compute the discrete PID
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
In the paper
“Although neither the discrete BROJA PID33 nor its Gaussian counterpart gPID30 have been defined for more than two sources as yet”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
What the sources say
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“a parameter β, which increases the amount of redundancy because it regulates the dissimilarity of tuning of the spike count of each individual neurons to the two stimulus features (β close to zero or one means independent tuning, i.e., each neuron encodes a different feature; higher values of β close to β = 0.5 indicate that both neurons encode the features similarly and redundantly, i.e., the two”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of simulations used to test the bias properties
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Moreover, the fact that the synergy obtained with the conservative shuffle subtraction is positive proves that there is genuine synergy in all datasets that cannot be due to finite sampling artifacts.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Evaluation of bias-correction procedures on real neural data
boostedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
In the paper
“α = 1 corresponds to choosing parameter values exactly equal to those reported in Venkatesh et al.30 α > 1 reduces the total mutual information.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of simulations used to test the bias properties
What the sources say
- Check pendingcites Venkatesh et al. [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“Furthermore, the ratio between the biases of union and joint information is not fixed: it varies across sampling regimes, with joint information exhibiting disproportionately larger bias at low trial counts N or higher dimensionality of sources and target d (see Note S6).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
What the sources say
- Not matchedcites Note S6
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plainstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This is because (see methods) the bias of the most biased terms of the three-way synergy scales like the bias of the joint information carried by three sources (which scales as (S × R3)/N for the discrete PID and as a supralinear function of 4d terms for the Gaussian PID)”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > The bias of three-source PID
plainstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“To ensure that the bias-corrected PID terms respect exactly the linear constraints of PID theory (Equations 3, 4, and 5), we computed the bias correction for the synergy, the joint, and the single-source information and then computed the bias-corrected unique and redundant information from the linear constraints (Equation 16).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Only 6.25% of distributions were significantly different from a Gaussian under a false discovery rate (FDR) of 5% (Benjamini-Hochberg FDR correction, q < 0.05), meaning deviations from Gaussianity were only slightly more frequent than expected by chance.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Three-source PID analysis of real MEG data
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: The Gaussian PID requires that the target-source distributions are approximately Gaussian.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“To ensure that the bias-corrected PID terms respect exactly the linear constraints of PID theory (Equations 3, 4, and 5), we computed the bias correction for the synergy, joint, and single-source information and then computed the bias-corrected unique and redundant information from the linear constraints (Equation 16).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
In the paper
“then making the assumptions that the union information has the same proportional bias as the joint information, thus rescaling the bias of the union information accordingly (Equation 17 of Venkatesh et al.30).”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of implementation of the bias-correction procedures
What the sources say
- Check pendingcites Venkatesh et al. [30] Venkatesh, 2023, Gaussian partial information decomposition: Bias correction and application to high-dimensional data, Adv. Neural Inf. Process. Syst. source
plainstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“We implemented a simple simulation based on Poisson responses (which are a reasonable approximation to neural firing71) that are either independent or shared across neurons.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of simulations used to test the bias properties
What the sources say
- Not checked — no full text of the sourcecites [71] Tolhurst, 1983, The statistical reliability of signals in single neurons in cat and monkey visual cortex, Vision Res. source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The authors state this as an assumption.
Shuffle subtraction assumes implicitly that the bias has similar magnitude across information levels.
“This approach, which has been employed in the mutual information literature with numerous variations of how to destroy information by data randomization,27,79,82 assumes implicitly that the bias has similar magnitude across information levels (an assumption that we will investigate below).”
The paper relies on this without support.
Other PIDs such as Imin and IMMI have been defined for, and can be readily applied to, more than two sources.
“other PIDs such as Imin and IMMI have been defined for, and can be readily applied to, more than two sources.”
The authors state this as an assumption.
The QE procedure assumes the asymptotic sampling regime N × p(s,r1,r2) ≫ 1 for all s,r1,r2, so that the bias of entropies, information, and PID terms can be accurately approximated as a second-order expansion in 1/N.
“The QE procedure60 assumes we are in the asymptotic sampling regime N × p(s,r1,r2) ≫ 1 for all s,r1,r2, so that the bias of the entropies, information, and PID terms can be accurately approximated as a second-order expansion in 1/N.”
The authors state this as an assumption.
Synergy depends on the full joint distributions and redundancy depends on the marginal but not the full joint distributions.
“Because our analytical considerations predicting a higher bias for synergy than redundancy are based on the assumption that the synergy depends on the full joint distributions and the redundancy depends on the marginal but not the full joint distributions”
The authors state this as an assumption.
Information measures from similar distributions suffer from similar biases.
“The resampling method for bias correction is grounded on the assumption that information measures from similar distributions suffer from similar biases.”
The paper relies on this without support.
The parameters chosen for individual figure panels are representative of wider trends and within the range found in real-data analyses of neuroscience experiments.
“The parameters used in individual figure panels were chosen with the criteria that these parameters were representative of wider trends, were within the range in our experience found in real-data analyses of neuroscience experiments, and illustrate well the point of interest.”
The authors state this as an assumption.
The Gaussian PID requires that the target-source distributions are approximately Gaussian.
“The Gaussian PID assumes that the target-source distributions are approximately Gaussian.”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Results > Bias of synergy and redundancy for discrete and Gaussian estimators
“R = 4 bins) independently of which of S = 2 stimuli (target) is presented.”
§ Sampling bias corrections for discrete and Gaussian partial information decompositions > Methods > Details of simulations used to test the bias properties
“In Figure 1 (zero-information case), we considered a simple scenario in which, on each simulated trial, the stimulus S and the spike count of each individual neuron R1 and R2 were drawn independently from a uniform probability distribution with four possible outcomes.”
Citations
Cites
A Mathematical Theory of Communication
Shannon, C. E. (1948) · Bell System Technical Journal · doi:10.1002/j.1538-7305.1948.tb01338.x · Cited by 88,486
Elements of Information Theory
Cover, T. M., & Thomas, J. A. (2001) · Wiley eBooks · doi:10.1002/0471200611 · Cited by 37,139
The WU-Minn Human Connectome Project: An overview
Van Essen, D. C., Smith, S. M., Barch, D. M., Behrens, T. E. J., Yacoub, E., & Ugurbil, K. (2013) · NeuroImage · doi:10.1016/j.neuroimage.2013.05.041 · Cited by 5,651
The minimal preprocessing pipelines for the Human Connectome Project
Glasser, M. F., Sotiropoulos, S. N., Wilson, J. A., Coalson, T. S., Fischl, B., Andersson, J. L., Xu, J., Jbabdi, S., Webster, M., Polimeni, J. R., et al. (2013) · NeuroImage · doi:10.1016/j.neuroimage.2013.04.127 · Cited by 5,253
A multi-modal parcellation of human cerebral cortex
Glasser, M. F., Coalson, T. S., Robinson, E. C., Hacker, C. D., Harwell, J., Yacoub, E., Ugurbil, K., Andersson, J., Beckmann, C. F., Jenkinson, M., et al. (2016) · Nature · doi:10.1038/nature18933 · Cited by 5,111
Conn : A Functional Connectivity Toolbox for Correlated and Anticorrelated Brain Networks
Whitfield-Gabrieli, S., & Nieto-Castanon, A. (2012) · Brain Connectivity · doi:10.1089/brain.2012.0073 · Cited by 4,885
A component based noise correction method (CompCor) for BOLD and perfusion based fMRI
Behzadi, Y., Restom, K., Liau, J., & Liu, T. T. (2007) · NeuroImage · doi:10.1016/j.neuroimage.2007.04.042 · Cited by 4,502
Local-global parcellation of the human cerebral cortex from intrinsic functional connectivity MRI
Schaefer, A., Kong, R., Gordon, E. M., Laumann, T. O., Zuo, X.-N., Holmes, A. J., Eickhoff, S. B., & Yeo, B. T. T. (2018) · Cerebral Cortex · doi:10.1093/cercor/bhx179 · Cited by 3,657
Kraskov, A., Stögbauer, H., & Grassberger, P. (2004) · Physical Review E · doi:10.1103/physreve.69.066138 · Cited by 3,512
Van Dijk, K. R. A., Hedden, T., Venkataraman, A., Evans, K. C., Lazar, S. W., & Buckner, R. L. (2010) · Journal of Neurophysiology · doi:10.1152/jn.00783.2009 · Cited by 1,844
Research works printed but not linked: 1
- Gabrié, 2018, Entropy and mutual information in models of deep neural networks, Adv. Neural Inf. Process. Syst., 31, 1826
Cited by
Sorted by citations
Towards an informational account of interpersonal coordination
Chidichimo, E., Luppi, A. I., Mediano, P. A. M., Leong, V., Dumas, G., Canales‐Johnson, A., & Bethlehem, R. A. I. (2025) · Nature reviews. Neuroscience · doi:10.1038/s41583-025-00989-0 · Cited by 19
Self-citationConfirmation bias through selective readout of information encoded in human parietal cortex
Park, H., Arazi, A., Talluri, B. C., Celotto, M., Panzeri, S., Stocker, A. A., & Donner, T. H. (2025) · Nature Communications · doi:10.1038/s41467-025-61010-x · Cited by 10
Self-citationBoosting transfer entropy estimation accuracy with machine learning for finite-length sequences
Qiu, L., & Yang, H. (2025) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2025.117252 · Cited by 5
Towards an informational account of interpersonal coordinationPreprint
Chidichimo, E., Luppi, A., Mediano, P. A. M., Leong, V., Dumas, G., Canales‐Johnson, A., & Bethlehem, R. A. I. (2025) · PsyArXiv (OSF Preprints) · doi:10.31234/osf.io/x3nvu_v2 · Cited by 2
Self-citationJiang, F., Wang, H., Su, J., & Zhang, Z. (2025) · Eksploatacja i Niezawodnosc - Maintenance and Reliability · doi:10.17531/ein/211279 · Cited by 1
Metadata
- Authors
- Lorenz, G. M., Engel, N. M., Koçillari, L., Celotto, M., Orsenigo, D., Curreli, S., Malerba, S. B., Engel, A. K., Kayser, C., Fellin, T., Luppi, A. I., & Panzeri, S.
- Affiliations
- University Medical Center Hamburg-Eppendorf (DE)
- University of Turin (IT)
- Italian Institute of Technology (IT)
- Bielefeld University (DE)
- University of Cambridge (GB)
- Corresponding author
- Panzeri, Stefano
- ORCID
- Venue
- Patterns
- Publisher
- Elsevier BV
- Published
- 2026-09
- Volume
- 7
- Issue
- 9
- Pages
- 101619
- ISSN
- 2666-3899 (ISSN-L)
- PMC
- PMC13576656
- OpenAlex
- W7170149077
- Cited by
- 9
- References
- 119
- Citations by year
- 2026: 4
- FWCI
- 9.28
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Patterns (publishedVersion, cc-by)
- bioRxiv (Cold Spring Harbor Laboratory) (submittedVersion, cc-by-nc-nd)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- PubMed Central (submittedVersion, cc-by-nc-nd)
- Apollo (University of Cambridge) (submittedVersion, cc-by)
- Publikationen an der Universität Bielefeld (Universität Bielefeld) (submittedVersion, cc-by)
- Funders
- Wellcome Trust
- Simons Foundation Autism Research Initiative
- EU Framework Programme for Research and Innovation Marie Skłodowska-Curie Actions
- Topics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
- Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Systems and Dynamics
- Keywords
- partial information decomposition, sampling bias, bias correction, synergistic effect, redundancy, unique information, neural coding, discrete distributions, neural spiking activity, auditory cortex, posterior parietal cortex, hippocampus
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-11-11, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-11-11 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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