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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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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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
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“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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“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
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“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
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“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
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“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
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“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
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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
“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
plaincorrelational- 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
“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.
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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
“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.
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- 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.
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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
“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.
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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
“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
plaindirect- 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
“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
plaindirect- 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
“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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- 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. The quote only partly states this claim.
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 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
plaindirect- 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 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
boosteddirect- 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 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 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
plaindirect- 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
“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
plaincorrelational- 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
“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
plaincorrelational- 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
“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
plaindirectn = 30- 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
“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.
- 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
“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
plaincorrelational- 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
“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
hedgeddirect- 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
“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
plaincorrelational- 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
“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
plaindirect- 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
“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
plaindirect- 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
“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
plaincorrelational- 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
“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
plaindirect- 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
“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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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
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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.
- 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
“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
plaindirect- 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 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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- 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
“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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- 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
“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
plaindirect- 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
“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.
- 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
“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
plaindirect- 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.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
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
boosteddirect- Asserted: passes — The authors put it forward as true.
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- 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 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.
- 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 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
plaindirect- 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 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
plaindirect- 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 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
hedgeddirect- 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
“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.
- 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
“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
hedgeddirect- 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
“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
plaindirect- 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 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.
- 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
“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
plaindirect- 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 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
boosteddirect- 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
“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.
- 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 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
plaincorrelational- 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 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
boosteddirect- 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 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
plaincorrelational- 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
“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
plaindirect- 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
hedgeddirect- 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
plaindirect- 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
hedgeddirect- 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
plaindirect- 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
hedgeddirectSingle 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
plaindirect- 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
plaindirect- 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
boosteddirect- 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
plaindirect- 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.
plaincorrelational- 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
plaincorrelational- 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
plaindirect- 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
plaindirect- 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
plaincorrelational- 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
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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
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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
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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
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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
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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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How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Sampling bias corrections for discrete and Gaussian partial information decompositions. https://app.cochat.ai/papers/10.1016/j.patter.2026.101619
- The paper
- 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. (2026). Sampling bias corrections for discrete and Gaussian partial information decompositions. Patterns. https://doi.org/10.1016/j.patter.2026.101619
