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Understanding predictive information criteria for Bayesian models
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“Although information criteria are computationally convenient, they are based on asymptotic results and assumptions about the data that may not always hold [89].”
From Comparing Bayesian and non-Bayesian accounts of human confidence reports (2018) · cites this paper as [89]
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“WAIC is a more Bayesian approach to information criteria that adds a correction for the effective number of parameters [89].”
From Comparing Bayesian and non-Bayesian accounts of human confidence reports (2018) · cites this paper as [89]
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“There is also something called the Bayesian information criterion (a misleading name, we believe) that adjusts for the number of fitted parameters with a penalty that increases with the sample size, n (Schwarz 1978).”
BackgroundModel selection and multimodel inference: a practical information-theoretic approach.
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“In this asymptotic limit, the posterior is dominated by the likelihood—the prior contributes only one factor, while the likelihood contributes n factors, one for each data point—and so the likelihood function also approaches the same normal distribution.”
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“When several candidate models are available, they can be compared and averaged using Bayes factors (which is equivalent to embedding them in a larger discrete model) or some more practical approximate procedure (Hoeting et al., 1999) or continuous model expansion (Draper, 1999).”
MethodInformation Theory and an Extension of the Maximum Likelihood Principle
Akaike, H. (1992) · Springer Series in Statistics · doi:10.1007/978-1-4612-0919-5_38 · Cited by 3,028 · cited 5×
“For practical reasons alone, there remains a place for simple bias corrections such as AIC (Akaike, 1973), DIC (Spiegelhalter et al., 2002, van der Linde, 2005), and, more recently, WAIC (Watanabe, 2010), and all these can be viewed as approximations to different versions of cross-validation (Stone,…”
BackgroundMethodInfluentialWatanabe, S. (2010) · Journal of machine learning research · doi:10.5555/1756006.1953045 · Cited by 2,921 · cited 5×
“Bayesian cross-validation works also with singular models, and Bayesian LOO-CV has been proven to asymptotically equal to WAIC (Watanabe, 2010).”
BackgroundMethodInfluentialBayesian Data Analysis, 2nd edn
Brix, A. (2005) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1111/j.1467-985x.2004.00347_4.x · Cited by 2,588 · cited 3×
“As is well known in hierarchical modeling (see, e.g., Spiegelhalter et al. 2002; Gelman et al. 2003), the line separating prior distribution from likelihood is somewhat arbitrary and is related to the question of what aspects of the data will be changed in hypothetical replications.”
Background
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“Standard selection criteria include the Bayesian information criterion (Weakliem, 1999) and the Akaike information criterion (Gelman et al., 2014).”
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“109 The WAIC is defined as 5) Here, S is the number of simulations draws and n is the number of available data-points.”
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“Information criterion can be used to choose the optimal number of clusters by evaluating the number of clusters that best balances accuracy and complexity, such as the Cross-Validation Information Criterion used in (Young et al., 2018).”
MethodA dynamic computational model of gaze and choice in multi-attribute decisions.
Yang, X.-Z., & Krajbich, I. (2022) · Psychology Review · doi:10.1037/rev0000350 · Cited by 59
“We evaluated the models based on Watanabe-Akaike information criteria (WAIC; Vehtari et al., 2017; Watanabe, 2010), which is based on the posterior distribution rather than a point estimate (Gelman et al., 2014).”
BackgroundMethodBecher, M., Stegmueller, D., Brouard, S., & Kerrouche, É. (2021) · Social Science Quarterly · doi:10.1111/ssqu.13035 · Cited by 79
The effects of long-term high-altitude exposure on cognition: A meta-analysis
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dockerHDDM: A User-Friendly Environment for Bayesian Hierarchical Drift-Diffusion Modeling
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“In contrast, WAIC does not rely on this strict assumption and uses the full posterior distribution, offering greater flexibility and accuracy but at a higher computational complexity (Gelman et al., 2014).”
Srikrishnan, V., Lafferty, D. C., Wong, T. E., Lamontagne, J., Quinn, J. D., Sharma, S., Molla, N., Herman, J. D., Sriver, R. L., Morris, J., et al. (2022) · Earth s Future · doi:10.1029/2021ef002644 · Cited by 72
“Model selection and averaging techniques based on optimizing out-of-sample pre-553 dictive performance (Gelman et al., 2014; Vehtari et al., 2017; Yao et al., 2018) may help,554 but still require the model structures under consideration to be capable of capturing ap-555 propriate changes to…”
MethodCan humans perceive the metabolic benefit provided by augmentative exoskeletons?
Medrano, R. L., Thomas, G., & Rouse, E. J. (2022) · Journal of NeuroEngineering and Rehabilitation · doi:10.1186/s12984-022-01002-w · Cited by 50
“This metric evaluates the predictive power of each psychometric model [62] and corrects for the number of parameters to favor parsimony.”
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“As previously re-ported, the optimal number of subtypes was determined using information criteria calculated through cross-validation 48 to balance model complexity with internal model accuracy.”
MethodPrinciples and Operation of Virtual Brain Twins
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“The posterior provides all the necessary information for evaluating accuracy and measuring out-of-sample predictive performance [209], [210].”
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Improved information criteria for Bayesian model averaging in lattice field theory
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“Other information criteria which include the same first term include the predictive information criterion (PIC) [38, 40] and the WatanabeAkaike information criteria (WAIC) [46].”
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“We log-transformed the number of years as log(year + 1) as it improved the quality of the model fitting to the data, decreasing the Watanabe–Akaike information criterion 66,67 by −733 on average (−11”
MethodTrends in methamphetamine use, markets and harms in Australia, 2003–2019
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“Models were compared using the deviance information criterion (DIC) as a predictive measure of choice in Bayesian applications [29,30].”
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Societies of strangers do not speak less complex languages
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“We compare all competing models in our analyses based on obtained widely applicable information criterion values [WAIC; ( 46 , 47 )].”
Wildlife impacts and changing climate pose compounding threats to human food security
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“As for the AICM, it was derived based on an analogy with the AIC, by relying on a definition of the effective number of parameters of a model based on the Monte Carlo variance of the log likelihood.”
BackgroundMethodInfluentialTraditional kriging versus modern Gaussian processes for large‐scale mining data
Christianson, R., Pollyea, R., & Gramacy, R. (2022) · Statistical analysis and data mining · doi:10.1002/sam.11635 · Cited by 34
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“The Deviance Information Criterion (DIC) and the Watanabe–Akaike Information Criterion (WAIC) were applied to select the best-fitting model from a variety of models [40,41].”
MethodSensitivity-Aware Amortized Bayesian Inference
Elsemüller, L., Olischläger, H., Schmitt, M., Bürkner, P.-C., Köthe, U., & Radev, S. T. (2023) · Trans. Mach. Learn. Res. · doi:10.48550/arxiv.2310.11122 · Cited by 31
Contrasting effects of human settlement on the interaction among sympatric apex carnivores
Penjor, U., Astaras, C., Cushman, S. A., Kaszta, Ż., & Macdonald, D. W. (2022) · Proceedings of the Royal Society B: Biological Sciences · doi:10.1098/rspb.2021.2681 · Cited by 42
“Model convergence was diagnosed using the Gelman–Rubin statistic (R̂) for all parameters and visual inspection of trace plots [53].”
MethodForcada, J., Hoffman, J., Gimenez, O., Staniland, I., Bucktrout, P., & Wood, A. G. (2023) · Global Change Biology · doi:10.1111/gcb.16947 · Cited by 30
“Selection of best models was subsequently based on WAIC (Gelman et al., 2014; Watanabe, 2010 ), and the IPMs were fitted with the best mark– recapture modelling options.”
MethodPick, J. L., Kasper, C., Allegue, H., Dingemanse, N. J., Dochtermann, N. A., Laskowski, K. L., Lima, M. R., Schielzeth, H., Westneat, D. F., WRIGHT, J. M., et al. (2023) · Methods in Ecology and Evolution · doi:10.1111/2041-210x.14200 · Cited by 43
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Ru, Z., Liu, J.-P., Kadziński, M., & Liao, X.-W. (2023) · European Journal of Operational Research · doi:10.1016/j.ejor.2023.05.007 · Cited by 30
Unified landslide hazard assessment using hurdle models: a case study in the Island of Dominica
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“This procedure was based on numerical techniques such as the Deviance Information Criterion and the Watanabe-Akaike information criterion (DIC and WAIC, respectively; Meyer 2014; Gelman et al. 2014), and graphical methods such as the probability integral transform (PIT; Gneiting et al.”
MethodControlled backfill in oncology dose-finding trials.
Dehbi, H., O'Quigley, J., & Iasonos, A. (2021) · Contemporary Clinical Trials · doi:10.1016/j.cct.2021.106605 · Cited by 30
Bayesian ordinal regression for multiple criteria choice and ranking
Ru, Z., Liu, J.-P., Kadziński, M., & Liao, X.-W. (2021) · European Journal of Operational Research · doi:10.1016/j.ejor.2021.09.028 · Cited by 30
Yadav, R., Huser, R., Opitz, T., & Lombardo, L. (2022) · doi:10.48550/arxiv.2205.09908 · Cited by 29
“…models (M1, M 0 1, M2, M 0 2) and different mark distributions using the continuous ranked probability score (CRPS; Gneiting and Raftery, 2007) and the deviance information criterion (DIC; Gelman et al., 2014), leading to very similar interpretations; see Table 4 in the Supplementary Material.”
Bodenmann, L., Baker, J., & Stojadinović, B. (2023) · Natural Hazards and Earth System Sciences · doi:10.5194/nhess-23-2387-2023 · Cited by 28
“Sci., 23, 2387–2402, 2023 https://doi.org/10.5194/nhess-23-2387-2023 where ψM,r is the rth sample from the posterior p(ψM|Dtrain) obtained via MCMC (Gelman et al., 2014).”
MethodInfluentialDeveloping a Hidden Markov model for occupancy prediction in high-density higher education buildings
Alfalah, B., Shahrestani, M., & Shao, L. (2023) · Journal of Building Engineering · doi:10.1016/j.jobe.2023.106795 · Cited by 28
Chalmandrier, L., Stouffer, D., Purcell, A. S. T., Lee, W. G., Tanentzap, A., & Laughlin, D. (2021) · bioRxiv · doi:10.1101/2021.07.12.448750 · Cited by 28
“We compared the fits of the calibrated models using two metrics: the Deviance information criterion (DIC, Gelman et al., 2014) and Nagelkerke's pseudo R 2 metric (Nagelkerke, 1991), which lends itself well to models that use a beta distribution and gives an indication of the variance they explain…”
MethodObesity rise plateaus in developed nations and accelerates in developing nations
Zhou, B., Phelps, N. H., Galeazzi, A., O'Driscoll, O. N., Bennett, J. E., Jain, L., Brisis, Y. D. D., Barradas‐Pires, A., Deo, F., Stevens, G. A., et al. (2026) · Nature · doi:10.1038/s41586-026-10383-0 · Cited by 42
“We had a target of eight converged MCMC chains for generating our estimates, which is twice the recommended minimum number to assess convergence using the Rhat diagnostic 91,95 .”
Towards Efficient MCMC Sampling in Bayesian Neural Networks by Exploiting Symmetry
Wiese, J. G., Wimmer, L., Papamarkou, T., Bischl, B., Günnemann, S., & Rügamer, D. (2023) · ECML/PKDD · doi:10.48550/arxiv.2304.02902 · Cited by 27
“…performance of our MCMC-based PPD approximation, we measure the goodness-of-fit on the test data using the log point-wise predictive density (LPPD) [Gelman et al. , 2014] , where θ (1) , . . . , θ ( G ) are G samples obtained across all chains via MCMC sampling from the parameter posterior density…”
MethodFully Bayesian estimation of virtual brain parameters with self-tuning Hamiltonian Monte Carlo
Jha, J., Hashemi, M., Vattikonda, A. N., Wang, H., & Jirsa, V. (2022) · Machine Learning Science and Technology · doi:10.1088/2632-2153/ac9037 · Cited by 29
“…sampling algorithms such as Metropolis-Hastings, Gibbs sampling and slice-sampling [27] generally fail to explore the parameter space efficiently when applied to large-scale inverse problems [36–38], as often encountered in the application of whole-brain imaging for clinical diagnoses.”
MethodGomez, S., Berger, E., Nicholl, M., Blanchard, P., & Hosseinzadeh, G. (2022) · Astrophysical Journal · doi:10.3847/1538-4357/ac9842 · Cited by 27
Bakhshi, A. K., & Ahmed, M. M. (2022) · Transportation Research Part C: Emerging Technologies · doi:10.1016/j.trc.2021.103539 · Cited by 27
SEAM: An Integrated Activation-Coupled Model of Sentence Processing and Eye Movements in Reading
Rabe, M., Paape, D., Mertzen, D., Vasishth, S., & Engbert, R. (2023) · Journal of Memory and Language · doi:10.1016/j.jml.2023.104496 · Cited by 26
“This would involve creating k (usually, in machine learning, k = 10) subsets of the data to train on, and then use the k held-out data sets for evaluation; this would allow us to compute a quantitative measure of average fit, such as expected log pointwise density (e.g., Gelman et al., 2014).”
MethodGeorgiou, G. (2022) · Behavioral Science · doi:10.3390/bs12120469 · Cited by 26
Robust Leave-One-Out Cross-Validation for High-Dimensional Bayesian Models
Silva, L., & Zanella, G. (2022) · Journal of the American Statistical Association · doi:10.1080/01621459.2023.2257893 · Cited by 26
“See for example Gelman et al. (2014) for arguments in favor of using the ELPD metric in (3) and its LOO-CV estimator in (4).”
Li, W., Gu, X.-B., Fang, H., Zhao, T., Yin, R., Cheng, Z.-K., Tan, C., Zhou, Z., & Du, Y. (2024) · European Journal of Agronomy · doi:10.1016/j.eja.2023.127026 · Cited by 25
A mixed methods study of health care professionals’ attitudes towards vaccination in 15 countries
Alasmari, A., Larson, H., & Karafillakis, E. (2022) · Vaccine: X · doi:10.1016/j.jvacx.2022.100219 · Cited by 25
Zhou, Y., Jiang, X., Fu, C.-Y., Liu, H.-Y., & Zhang, G.-P. (2022) · Accident Analysis and Prevention · doi:10.1016/j.aap.2022.106756 · Cited by 25
Strebel, N., Kéry, M., Guélat, J., & Sattler, T. L. (2022) · Journal of Biogeography · doi:10.1111/jbi.14335 · Cited by 29
Big Problems in Spatio-Temporal Disease Mapping: Methods and Software
Orozco‐Acosta, E., Adín, A., & Ugarte, M. D. (2022) · Computer Methods and Programs in Biomedicine · doi:10.1016/j.cmpb.2023.107403 · Cited by 23
Scanpath Regularity as an Index of Reading Comprehension
Meziere, D., Yu, L., McArthur, G., Reichle, E. D., & von der Malsburg, T. (2023) · Scientific Studies of Reading · doi:10.1080/10888438.2023.2232063 · Cited by 24
“The models were then evaluated and compared within each set (one per test and one with averaged data) using leave-one-out cross-validation ( LOO ; Gelman et al., 2014; Vehtari et al., 2017).”
MethodLin, W.-C., Lu, Y., Li, G.-Y., Jiang, X.-D., & Lu, D. (2022) · GIScience & Remote Sensing · doi:10.1080/15481603.2022.2044139 · Cited by 24
“Thus HBA is suitable for complex multilevel models involving nonlinear relationships between variables and abnormal distribution at multiple levels (Gelman, Hwang, and Vehtari 2013).”
BackgroundLee, S. (2022) · Mathematics · doi:10.20944/preprints202201.0407.v1 · Cited by 24
“For a detailed explanation of the criteria, refer to [17, 115].”
BackgroundDecisions Under Uncertainty as Bayesian Inference on Choice Options
Vieider, F. M. (2024) · Management Science · doi:10.1287/mnsc.2023.00265 · Cited by 31
“I test model performance based on their out-of-sample predictive ability (Gelman et al. 2014a, Vehtari et al. 2017).”
MethodThe role of teacher support in students' academic performance in low- and high-stakes assessments
Ansong, D., Okumu, M., Amoako, E. O., Appiah‐Kubi, J., Ampomah, A. O., Koomson, I., & Hamilton, E. R. (2023) · Learning and Individual Differences · doi:10.1016/j.lindif.2023.102396 · Cited by 32
Eltanamly, H., Leijten, P. H. O., van Roekel, E., Mouton, B., Pluess, M., & Overbeek, G. J. (2022) · Child Development · doi:10.1111/cdev.13848 · Cited by 34
“Because the first half of each chain is discarded as burn- in, and because we used a thinning rate of every 10th iteration (meaning that only 1 in 10 iterations is saved), our results here are based on 4500 iterations (see Gelman et al., 2014).”
MacTavish, R., Bixby, H., Cavanaugh, A., Agyei-Mensah, S., Bawah, A., Owusu, G., Ezzati, M., Arku, R., Robinson, B., Schmidt, A. M., et al. (2023) · World Development · doi:10.1016/j.worlddev.2023.106253 · Cited by 23
“The model fit was also evaluated using the Watanabe-Akaike Information Criterion (WAIC) which is the generalized version of the Akaike Information Criterion that estimates prediction error while considering model simplicity to prevent overfitting (Evans, 2019; Gelman et al., 2014).”
MethodBayesian neural networks with physics‐aware regularization for probabilistic travel time modeling
Olivier, A., Mohammadi, S., Smyth, A., & Adams, M. (2023) · Comput. Aided Civ. Infrastructure Eng. · doi:10.1111/mice.13047 · Cited by 22
“The LPPD on the other hand is computed as (Gelman et al., 2014): Both B3 and B4 seem to perform adequately with respect to these measures of in-distribution accuracy with about 60 s RMSE for test data L, which is similar to results from the MAP NNs (see Table 3, where B_prior simply refers to an…”
ResultHopkins, W. A., Case, B. F., Groffen, J., Brooks, G. C., Jachowski, C. M. B., Button, S. T., Hallagan, J. J., O’Brien, R., & Kindsvater, H. K. (2023) · The American Naturalist · doi:10.1086/724819 · Cited by 24
“Similar to other information criteria, LOOIC includes a penalty that accounts for the effective number of parameters (Gelman et al. 2014).”
MethodThe role of vision and proprioception in self-motion encoding: An immersive virtual reality study
Bayramova, R., Valori, I., McKenna-Plumley, P. E., Callegher, C. Z., & Farroni, T. (2021) · Attention Perception & Psychophysics · doi:10.3758/s13414-021-02344-8 · Cited by 30
“The Watanabe-Akaike Information Criterion (WAIC, Gelman et al. 2014; Vehtari et al. 2017) was used as information criteria to evaluate the models.”
MethodBertolini, R., Finch, S. J., & Nehm, R. H. (2023) · Frontiers in Education · doi:10.3389/feduc.2023.1073829 · Cited by 29
Na, W., & Wang, M.-M. (2022) · Social Science Research Network · doi:10.2139/ssrn.3879557 · Cited by 21
Török, G., Stanciu, O., Sebanz, N., & Csibra, G. (2021) · Open Mind · doi:10.1162/opmi_a_00045 · Cited by 32
“To compare models, we calculated leave-one-out cross-validation measures (LOO-CV; Gelman et al., 2014; Vehtari et al., 2017) using the loo package in R (Vehtari et al., 2020).”
Quantifying drivers of decline: A case study of long-term changes in arboreal marsupial detections
Lindenmayer, D., Bowd, E., Youngentob, K. N., & Evans, M. (2024) · Biological Conservation · doi:10.1016/j.biocon.2024.110589 · Cited by 20
Tahir, H. B., Yasmin, S., & Haque, M. M. (2023) · Analytic Methods in Accident Research · doi:10.1016/j.amar.2023.100315 · Cited by 20
Nakamura, Y., Suzuki, S., Murakami, S., Nishimoto, Y., Higashi, K., Watarai, N., Umetsu, J., Ishii, C., Ito, Y., Mori, Y., et al. (2022) · Computational and Structural Biotechnology Journal · doi:10.1016/j.csbj.2022.10.026 · Cited by 24
“WAIC approximates the prediction error (generalization error) for unknown data in the model; WAIC has also been used for a model in which the posterior distribution of parameters cannot be approximated by a normal distribution [56].”
MethodZhou, S., Hutton, T., Lei, Y., Miller, M., van der Velde, T. D., & Deng, R. (2022) · ICES Journal of Marine Science · doi:10.1093/icesjms/fsac131 · Cited by 20
Rees, E., Lau, C. L., Kama, M., Reid, S., Lowe, R., & Kucharski, A. (2022) · PLoS Neglected Tropical Diseases · doi:10.1371/journal.pntd.0010506 · Cited by 20
“Model selection was based on the lowest value of the widely applicable information criterion (WAIC), which balances the goodness of fit of the model with model complexity, and therefore aims to balance the risks of overfitting and underfitting [40,41].”
The role of organisational compassion in knowledge hiding and thriving at work
Koon, V. (2022) · Knowledge Management Research & Practice · doi:10.1080/14778238.2022.2062470 · Cited by 22
“The convergence criterion for potential scale reduction (PSR) assesses MCMC convergence (Gelman et al., 2014).”
BackgroundNeurocomputational mechanisms underlying the subjective value of information
Goh, A. X.-A., Bennett, D., Bode, S., & Chong, T. T. ‐. (2021) · Communications Biology · doi:10.1038/s42003-021-02850-3 · Cited by 25
Spatio-temporal models of bovine tuberculosis in the Irish cattle population, 2012-2019.
Madden, J., McGrath, G., Sweeney, J., Murray, G., Tratalos, J., & More, S. (2021) · Spatial and Spatio-temporal Epidemiology · doi:10.1016/j.sste.2021.100441 · Cited by 20
Metadata
- Authors
- Gelman, A., Hwang, J., & Vehtari, A.
- Affiliations
- Columbia University (US)
- Harvard University (US)
- Aalto University (FI)
- Corresponding author
- Gelman, Andrew
- Venue
- Statistics and Computing
- Published
- 2013-08-20
- Volume
- 24
- Issue
- 6
- Pages
- 997-1016
- ISSN
- 0960-3174 (ISSN-L)
- 1573-1375
- OpenAlex
- W2130902307
- Cited by
- 2,104
- References
- 42
- Citations by year
- 2026: 153 · 2025: 202 · 2024: 198 · 2023: 203 · 2022: 227 · 2021: 266 · 2020: 213 · 2019: 195 · 2018: 154 · 2017: 136 · 2016: 90 · 2015: 44 · 2014: 16 · 2013: 3
- FWCI
- 45.30
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Statistics and Computing (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- Institute of Education Sciences
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Keywords
- information criteria, Akaike information criterion, deviance information criterion, Bayesian model selection, out-of-sample prediction error, bias correction, model comparison
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