Cited by 2,104Statistical Methods and InferenceStatistical Methods and Bayesian InferenceStatistical Methods and Applications

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Understanding predictive information criteria for Bayesian models

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Gelman, Andrew, Hwang, Jessica, Vehtari, Aki·2014·Statistics and Computing

DOI 10.1007/s11222-013-9416-2

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Estimating the Dimension of a Model
Schwarz, G. (1978) · The Annals of Statistics · doi:10.1214/aos/1176344136 · Cited by 47,033 · cited 1×
“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).”
Background
Model selection and multimodel inference: a practical information-theoretic approach.
Guthery, F. S., Burnham, K. P., & Anderson, D. R. (2003) · The Journal of Wildlife Management · doi:10.2307/3802723 · Cited by 41,495 · cited 1×
“The log predictive density has an important role in statistical model comparison because of its connection to the Kullback-Leibler information measure (see Burnham and Anderson, 2002, and Robert, 1996).”
Method
Bayesian Data Analysis
Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2003) · doi:10.1201/9780429258480 · Cited by 27,276
Self-citation
Pattern Recognition and Neural Networks
Ripley, B. D. (1996) · Cambridge University Press eBooks · doi:10.1017/cbo9780511812651 · Cited by 6,500
Strictly Proper Scoring Rules, Prediction, and Estimation
Gneiting, T., & Raftery, A. E. (2007) · Journal of the American Statistical Association · doi:10.1198/016214506000001437 · Cited by 5,793 · cited 1×
“Examples include the quadratic, logarithmic, and zero-one scores, whose properties are reviewed by Gneiting and Raftery (2007).”
Background
Optimal Statistical Decisions
Groot, M. D. (1970) · doi:10.2307/2987329 · Cited by 4,527 · cited 2×
“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.”
Background
Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors
Hoeting, J. A., Madigan, D., Raftery, A. E., & Volinsky, C. T. (1999) · Statistical Science · doi:10.1214/ss/1009212519 · Cited by 4,118 · cited 1×
“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).”
Method
Information 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,…”
BackgroundMethodInfluential
Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory
Watanabe, 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).”
BackgroundMethodInfluential
Bayesian 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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Draper, D.: Model uncertainty yes, discrete model averaging maybe. Stat. Sci. 14, 405–409 (1999)
Efron, B., Tibshirani, R.: An Introduction to the Bootstrap. Chapman & Hall, New York (1993) · DOI · Paper page
Gelman, A., Meng, X.L., Stern, H.S.: Posterior predictive assessment of model fitness via realized discrepancies (with discussion). Stat. Sin. 6, 733–807 (1996)
Watanabe, S.: Algebraic Geometry and Statistical Learning Theory. Cambridge University Press, Cambridge (2009) · DOI

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“Using information criteria, all models are then ranked by predictive accuracy [41].”
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“ELPD indicates how much credence the model ascribes to out- of-sample data and hence is a relative index of predictive performance (Gelman et al., 2014 ).”
Background
Community stability is related to animal diversity change
Jarzyna, M. A., Norman, K., LaMontagne, J. M., Helmus, M. R., Li, D., Parker, S. M., Rocha, M. P., Record, S., Sokol, E. R., Zarnetske, P. L., et al. (2022) · Ecosphere · doi:10.1002/ecs2.3970 · Cited by 19
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“We used the widely applicable information criterion (WAIC) to compare models with and without interactions (Gelman, Hwang, & Vehtari, 2013).”
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“We first compared the prediction accuracy of the Rotational Dual Gaussian model with the original Dual Gaussian model using the Widely Applicable Information Criterion (WAIC) [14].”
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“We also calculated the Widely Applicable Information Criterion (WAIC) of each model to compare their expected predictive accuracy on out- of-sample data (Gelman et al., 2014 ).”
Method
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“We used the widely applicable information criterion to compare models with and without interactions (Gelman et al., 2014 ).”
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Evaluation of predictive capability of Bayesian spatio-temporal models for Covid-19 spread
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“The WAIC is a standard relative goodness-of-fit criterion and is now replacing the commonly used deviance Information criterion (DIC) as it can be stably estimated within McMC applications [12].”
Method
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“This process is repeated for each data point to obtain an estimate of the model’s predictive accuracy [64].”
Background
Salvage Logging Strongly Affects Woodpecker Abundance and Reproduction: a Meta-analysis
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“We used weakly informative priors [55,56] with a Student-t’s distribution with 3 degrees of freedom, a mean of 0 and a standard deviation of 2.”
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“Posterior predictive checks were done to determine whether the posterior distributions resulted in a good resemblance between simulated and observed data (Gelman et al., 2014).”
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“…model including its ability to deal with small samples of participants (see Escudero et al., 2020; Georgiou, 2023b; van Weakly informative priors were used, namely, student’s t -distribution with 3 degrees of freedom, a mean of 0, and a standard deviation of 2.5 (see Gelman et al., 2014, 2015).”
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Personalised regional modelling predicts tau progression in the human brain
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“The ELPD is used for estimating the out-of-sample predictive accuracy and is adapted from [69].”
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“However, there are potentially much better weighting strategies available for Bayesian model ﬁ ts (Gelman et al., 2014).”
Background
Increased birth rank of homosexual males: disentangling the older brother effect and sexual antagonism hypothesis
Raymond, M., Turek, D. B., Durand, V., Nila, S., Suryobroto, B., Vadez, J., Barthès, J., Apostoulou, M., & Crochet, P. (2023) · bioRxiv · doi:10.1101/2022.02.22.481477 · Cited by 16
“These functions describing the OBE effect were compared by evaluating model fit to the sibling data using WAIC (Gelman et al., 2014; Watanabe, 2013): the mean of ten independent chains was used, each with a length of 20,000 samples and a”
MethodInfluential
Decision‐making biases in suicide attempters with major depressive disorder: A computational modeling study using the balloon analog risk task (BART)
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“The LOOIC is an index of out‐of‐sample prediction accuracy based on a fitted Bayesian model derived from the log‐likelihood estimated from a posterior distribution (Gelman et al., 2014; Vehtari et al., 2017).”
Method
A Review and Roadmap of Deep Learning Causal Discovery in Different Variable ParadigmsPreprint
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“Based on the directed acyclic graph (DAG) [22], [23] structure and Bayesian models [24], [25], it aims to learn the statistical relationship between two observed variables under the inﬂuence of another variable.”
Background
Bayesian beta regression for bounded responses with unknown supports
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Spatial voting models in circular spaces: A case study of the U.S. House of Representatives
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“To conclude this Section, we present in Table 4 the value of the Watanabe-Akaike Information Criteria (WAIC, see Watanabe, 2010, 2013; Gelman et al., 2014) associated with both the circular and one-dimensional Euclidean models fitted to the data from 100th to 116th U.”
MethodInfluential
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“Bayesian inference is carried out by MCMC, where we sample from the posterior distribution by the HamiltonianMonte Carlo algorithm (Gelman et al. 2013).”
Method
Human-wildlife conflict is amplified during periods of drought
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“We ran all models for 70,000 iterations, with a 10,000-iteration burn-in across three chains and used NIMBLE custom samplers to increase the efficiency of MCMC mixing ( 65 ).”
Converting carbonyl index values in microplastics studies.
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Saturated hydraulic conductivity (Ksat) and topographic controls on baseflow contribution in high-altitude aquifers with complex geology
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“In contrast, the application of traditional cutoff values for BCFI and BTLI is not endorsed by Garnier-Villarreal and Jorgensen (2020) [54] because the criteria were not applicable to all samples.”
Method
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“Models with a lower WAIC indicate a more parsimonious model [43].”
Method
A Bayesian change point analysis re-examines the 4.2 ka BP event in southeast Europe and southwest Asia
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Hydrology is a major influence on amphibian abundance in a large European floodplain
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“The nine models were compared using the Watanabe– Akaike information criterion (WAIC; Watanabe, 2010 ), computed by the meth-ods of Gelman et al. ( 2014 ).”
Method
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Transferability of Alzheimer's disease progression subtypes to an independent population cohort
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Random effects in dynamic network actor models
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“The information criteria mentioned above aim to estimate the expected out-of-sample error using a bias-corrected adjustment of within-sample error (Gelman et al., 2014).”
Method
Roundtrip, free-floating and peer-to-peer carsharing: A Bayesian behavioral analysis
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Analysis of spatial and temporal impact differences of birth rate in mainland China
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Effect of transit-oriented design on pedestrian and cyclist safety using bivariate spatial models.
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Non-Separable Spatio-Temporal Models via Transformed Multivariate Gaussian Markov Random Fields
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“A broader discussion of these criteria can be found in Gelman et al. (2014).”
Background
Elevation, disturbance, and forest type drive the occurrence of a specialist arboreal folivore
Lindenmayer, D., McBurney, L., Blanchard, W., Marsh, K. J., Bowd, E., Watchorn, D., Taylor, C., & Youngentob, K. N. (2022) · PLoS ONE · doi:10.1371/journal.pone.0265963 · Cited by 15
“We chose the most parsimonious model, that is, the simplest model within two LOOIC units of the best fitting model.”
MethodInfluential
Two faces of perceptual awareness during the attentional blink: Gradual and discrete.
Karabay, A., Wilhelm, S. A., de Jong, J., Wang, J., Martens, S., & Akyürek, E. G. (2021) · Journal of Experimental Psychology General · doi:10.1037/xge0001156 · Cited by 20
“WAIC penalizes models on the number of free parameters in a modest manner and lower WAIC scores indicate better ﬁ t (Gelman et al., 2014).”
Method
Spatial variation in bioclimatic relationships for a snow‐adapted species along a discontinuous southern range boundary
Sultaire, S. M., Humphreys, J. M., Zuckerberg, B., Pauli, J., & Roloff, G. (2021) · Journal of Biogeography · doi:10.1111/jbi.14279 · Cited by 15
Dimension-Grouped Mixed Membership Models for Multivariate Categorical Data
Gu, Y.-Q., Erosheva, E., Xu, G.-J., & Dunson, D. (2021) · Journal of machine learning research · doi:10.48550/arxiv.2109.11705 · Cited by 15
“Gelman et al. (2014) contains a comprehensive review of various predictive information criteria for evaluating Bayesian models.”
Human Mobility to Parks Under the COVID‐19 Pandemic and Wildfire Seasons in the Western and Central United States
Yang, A.-N., Yang, J., Yang, D., Xu, R.-T., He, Y., Aragon, A., & Qiu, H. (2021) · GeoHealth · doi:10.1029/2021gh000494 · Cited by 15
“Particularly, several studies have suggested that the usage of public parks and open spaces were impacted by the COVID-19 pandemics and its NPI policies (Gelman et al., 2014; Shoari et al., 2020; Volenec et al., 2021; Xie et al., 2020).”
Background
Bridging Traditional-Statistics and Machine-Learning Approaches in Psychology: Navigating Small Samples, Measurement Error, Nonindependent Observations, and Missing Data
Lavelle-Hill, R. E., Smith, G., & Murayama, K. (2025) · Advances in Methods and Practices in Psychological Science · doi:10.1177/25152459251345696 · Cited by 11
“Other methods include Bayesian estimation (Gelman et al., 2014) and inverse probability weighting (L. Li et al., 2013; Seaman & White, 2013).”
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“Multiple sequences produced independently for assessing convergence were estimated (Gelman et al., 2014).”
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“Such procedures are able to produce an almost unbiased estimate of a model’s predictive ability (Watanabe 2010), and have been reviewed extensively by Arlot and Celisse (2010), and by Gelman et al. (2014) in a Bayesian context.”
Self-citation
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Wang, X., Zhang, S., Gasperi, L., Robertson, S. J., & Gómez, M. (2023) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2023.113698 · Cited by 13
MSM with HIV: Improving prevalence and risk estimates by a Bayesian small area estimation modelling approach for public health service areas in the Netherlands
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Pesticide use in vineyards is affected by semi-natural habitats and organic farming share in the landscape
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“It was acceptable for all of the parameters (R 1.1, Gelman et al., 2014).”
BackgroundInfluential
Association of Poor Mental-Health Days With COVID-19 Infection Rates in the U.S.
Ransome, Y., Luan, H., Song, I., Fiellin, D., & Galea, S. (2021) · American Journal of Preventive Medicine · doi:10.1016/j.amepre.2021.08.032 · Cited by 14
“The Watanabe−Akaike Information Criterion was used to compare model fit; a lower value indicates better model fit.(28) This study used county-level data and therefore is not considered Human Subjects Research, and was thus is exempt based on the guidelines from Yale University Institutional Review Board (IRB).”
Method
Comparative performance of catch-based and surplus production models on evaluating largehead hairtail (Trichiurus lepturus) fishery in the East China Sea
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“DIC is the information criterion used for most Bayesian analyses (Gelman et al. 2014).”
Method
Reliability Modeling and Parameter Estimation for High-Speed Train Wheels Subject to Multi-Dimensional Degradation Processes Considering Mutual Dependency
Men, T., Liu, B., Li, Y.-F., Lin, Y.-H., & Zhang, Y. (2024) · IEEE Transactions on Reliability · doi:10.1109/tr.2023.3315294 · Cited by 13
“Bayesian inference techniques, such as Markov chain Monte Carlo, can estimate missing values and model parameters, given the observed data [48].”
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Affect-congruent attention modulates generalized reward expectations
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Comparing DIC and WAIC for multilevel models with missing data
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Models of underlying autotrophic biomass dynamics fit to daily river ecosystem productivity estimates improve understanding of ecosystem disturbance and resilience
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“Two WAIC criteria were computed based on the log pointwise predictive density (LPPD) with WAIC 1 = − 2 ( LLPD − p 1 ) and WAIC 2 = − 2 ( LLPD − p 2 ) where penalty terms p 1 and p 2 are deﬁned in Gelman et al. (2014) just prior to, and in, their Eq.”
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“…Bayesian bootstrap (Rubin, 1981), whose samples correspond to discrete distributions supported at {U (s)Dj} nD j=1 with weights, assuming no ties in the realisations of the diseased placement values, distributed according to a Dirichlet(nD; 1, . . . , 1) distribution (Gelman et al., 2013, p. 548).”
Method
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“We estimated the Watanabe-Akaike information criterion (Gelman et al. 2014; Watanabe 2010), which penalizes model complexity, and the deviance information criterion (Spiegelhalter et al. 2002) to compare all the models ﬁ tted.”
Method
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“The predictive performance of multiple models can be compared with leave-one-out cross validation (LOO-CV), either using WAIC (Widely Applicable Information Criterion) or ELPD (Expected Log Predictive Density), which can be calculated with the loo package (Gelman et al., 2014; Vehtari et al., 2017).”
Method
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“Moreover, the LOO-LPD is more robust against overfitting and model mis-specification than the log evidence (Jankowiak and Pleiss, 2021; Gelman et al., 2014).”
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How do ecologists estimate occupancy in practice?
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“We then compared a suite of models we deemed plausible using the widely applicable information criteria (WAIC), which approximates out of sample prediction error (70).”
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“Model comparisons can be performed in many ways, but a common paradigm is the leave-one-out cross-validation (LOO; Gelman et al., 2014).”
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BackgroundMethodInfluential
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“…measures such the Deviance Information Criteria (Spiegelhalter et al. 2002, DIC), the Watanabe Akaike information criterion (Watanabe and Opper 2010; Gelman et al. 2014, WAIC), or the cross-validation measure conditional predictive ordinate (CPO) for evaluating the predictive capacity (Pettit…”
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“For illustration purposes, here we show how model ﬁt indices can be produced, such as Watanabe–Akaike or Widely Applicable Information Criterion (WAIC) [48] and Log Pointwise Predictive Density (LPPD) [49] (Listing 12).”
BackgroundMethod
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Gilbert, B., Fiksel, J., Wilson, E., Kalter, H., Kante, A., Akum, A., Blau, D. M., Bassat, Q., Macicame, I., Gudo, E. S., et al. (2023) · American Journal of Tropical Medicine and Hygiene · doi:10.4269/ajtmh.22-0319 · Cited by 12
“22 WAIC also harmonizes well with Bayesian inference, making use of the entire posterior distribution available from the Markov Chain Monte Carlo runs.”

Authors: Gelman, A., Hwang, J., & Vehtari, A.
Affiliations: Columbia University (US)
Harvard University (US)
Aalto University (FI)
Corresponding author: Gelman, Andrew
ORCID: Gelman, Andrew
Hwang, Jessica
Vehtari, Aki
Venue: Statistics and Computing
Publisher: Springer Science and Business Media LLC
Published: 2013-08-20
Volume: 24
Issue: 6
Pages: 997-1016
ISSN: 0960-3174 (ISSN-L)
1573-1375
DOI: 10.1007/s11222-013-9416-2
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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