Understanding predictive information criteria for Bayesian models

Gelman, Andrew; Hwang, Jessica; Vehtari, Aki

doi:10.1007/s11222-013-9416-2

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

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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Information criteria are based on asymptotic results and assumptions about the data that may not always hold.
“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]
Section 3.4 (p. 10) says other information criteria rely on Fisher’s asymptotic theory and assume a regular model; Section 2.2 (p. 4) explains why those assumptions fail for singular models. The claim overgeneralizes to information criteria as a whole, since the paper says WAIC also works with singular models. (LLM, conf=0.86)
Experimental — accuracy not yet measured
WAIC is a more Bayesian approach to information criteria that adds a correction for the effective number of parameters.
“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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Estimating the Dimension of a Model
Schwarz, G. (1978) · The Annals of Statistics · doi:10.1214/aos/1176344136 · Cited by 47,033 · cited 1× · see citing works
“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× · see citing works
“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., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013) · doi:10.1201/b16018 · Cited by 27,276 · see citing works
Self-citation
Pattern Recognition and Neural Networks
Ripley, B. D. (1996) · Cambridge University Press eBooks · doi:10.1017/cbo9780511812651 · Cited by 6,446 · see citing works
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× · see citing works
“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× · see citing works
“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× · see citing works
“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× · see citing works
“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
Bayesian Theory
Bernardo, J. M., & Smith, A. F. M. (1994) · Wiley Series in Probability and Statistics · doi:10.1002/9780470316870 · Cited by 2,178 · see citing works
BAYESIAN THEORY
Restrepo-Montoya, D. (2008) · doi:10.1007/3-540-28820-1_2 · Cited by 1,824 · cited 1×
“Bernardo and Smith (1994) argue that suitable scoring rules for prediction are proper and local: propriety of the scoring rule motivates the decision maker to report his or her beliefs honestly, and for local scoring rules predictions are judged only on the plausibility they assign to the event…”
Background
Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician
Rubin, D. (1984) · The Annals of Statistics · doi:10.1214/aos/1176346785 · Cited by 1,503 · cited 2× · see citing works
“The ﬁt of model to data can be assessed using posterior predictive checks (Rubin, 1984), prior predictive checks (when evaluating potential replications involving new parameter values), or, more generally, mixed checks for hierarchical models (Gelman, Meng, and Stern, 2006).”
BackgroundMethod
Bayesian Model Choice: Asymptotics and Exact Calculations
Gelfand, A., & Dey, D. (1994) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1994.tb01996.x · Cited by 1,399 · cited 1× · see citing works
“In some ways, our paper is similar to the review article by Gelfand and Dey (1994), except that they were focused on model choice whereas our goal is more immediately to estimate predictive accuracy for the goal of model comparison.”
Method
Bayesian Data Analysis.
Fearn, T., Gelman, A., Carlin, J., Stern, H., & Rubin, D. B. (1996) · Biometrics · doi:10.2307/2533081 · Cited by 1,322 · see citing works
Self-citation
An asymptotic equivalence of choice of model by cross validation and Akaike's criterion
Stone, M. (1977) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1977.tb01603.x · Cited by 1,286 · cited 2× · see citing works
“…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, 1977).”
Method
Making and Evaluating Point Forecasts
Gneiting, T. (2011) · Journal of the American Statistical Association · doi:10.1198/jasa.2011.r10138 · Cited by 1,174 · cited 3× · see citing works
“We use the squared error as an example scoring function for point prediction, because the squared error and its derivatives seem to be the most common scoring functions in predictive literature (Gneiting, 2011).”
BackgroundMethodInfluential
A Predictive Approach to Model Selection
Geisser, S., & Eddy, W. (1979) · Journal of the American Statistical Association · doi:10.1080/01621459.1979.10481632 · Cited by 1,079 · see citing works
Deviance information criteria for missing data models
Celeux, G., Forbes, F., Robert, C., & Titterington, D. (2006) · Bayesian Analysis · doi:10.1214/06-ba122 · Cited by 965 · cited 1× · see citing works
“Various diﬃculties have been noted with DIC (see Celeux et al., 2006, Plummer, 2008, and much of the discussion of Spiegelhalter et al., 2002) but there has been no consensus on an alternative.”
Background
A widely applicable Bayesian information criterion
Watanabe, S. (2012) · Journal of machine learning research · doi:10.5555/2567709.2502609 · Cited by 963 · cited 1×
“Watanabe (2013) has also proposed a widely applicable Bayesian information criterion (WBIC) which works also in singular and unrealizable cases.”
Background
Philosophy and the practice of Bayesian statistics
Gelman, A., & Shalizi, C. R. (2013) · British Journal of Mathematical and Statistical Psychology · doi:10.1111/j.2044-8317.2011.02037.x · Cited by 825 · see citing works
Self-citation
Expected Information as Expected Utility
Bernardo, J. (1979) · The Annals of Statistics · doi:10.1214/aos/1176344689 · Cited by 705 · cited 2× · see citing works
“The logarithmic score is the unique (up to an aﬃne transformation) local and proper scoring rule (Bernardo, 1979), and appears to be the most commonly used scoring rule in model selection.”
Background
A comparative study of ordinary cross-validation, <i>v</i>-fold cross-validation and the repeated learning-testing methods
BURMAN, P. (1989) · Biometrika · doi:10.1093/biomet/76.3.503 · Cited by 634 · cited 2× · see citing works
“For large n the difference is negligible, but for small n (or when using k-fold cross-validation) we can use a first order bias correction b by estimating how much better predictions would be obtained if conditioning on n data points (Burman, 1989):”
Method
Penalized loss functions for Bayesian model comparison
Plummer, M. (2008) · Biostatistics · doi:10.1093/biostatistics/kxm049 · Cited by 395 · cited 1× · see citing works
“Various diﬃculties have been noted with DIC (see Celeux et al., 2006, Plummer, 2008, and much of the discussion of Spiegelhalter et al., 2002) but there has been no consensus on an alternative.”
Background
A survey of Bayesian predictive methods for model assessment, selection and comparison
Vehtari, A., & Ojanen, J. (2012) · Statistics Surveys · doi:10.1214/12-ss102 · Cited by 339 · cited 5× · see citing works
“There are extensions of AIC which have an adjustment related to the eﬀective number of parameters (see Vehtari and Ojanen, 2012, section 5.5, and references therein) but these are seldom used due to stability problems and computational diﬃculties, issues that have motivated the construction of the…”
BackgroundMethodInfluentialSelf-citation
Estimation in Parallel Randomized Experiments
Rubin, D. B. (1981) · Journal of Educational Statistics · doi:10.3102/10769986006004377 · Cited by 269 · cited 3× · see citing works
“From Rubin (1981).”
MethodInfluential
A cross-validatory method for dependent data
Burman, P., Chow, E., & Nolan, D. (1994) · Biometrika · doi:10.1093/biomet/81.2.351 · Cited by 245 · cited 4× · see citing works
“Regular cross-validation and information criteria can be used for time series in case of stationary Markov process and squared error or a scoring function or rule which is well approximated by a quadratic form (Akaike, 1973, Burman et al., 1994).”
MethodInfluential
The direct use of likelihood for significance testing
Dempster, A. P. (1997) · Statistics and Computing · doi:10.1023/a:1018598421607 · Cited by 220 · cited 1× · see citing works
“Figure 2 is reminiscent of the direct likelihood methods of Dempster (1974) and Aitkin (2010).”
Method
Bayesian Model Assessment and Comparison Using Cross-Validation Predictive Densities
Vehtari, A., & Lampinen, J. (2002) · Neural Computation · doi:10.1162/08997660260293292 · Cited by 198 · cited 3× · see citing works
“A practical estimate of related sampling uncertainty can be obtained by analyzing the variation in the expected log predictive densities ̂ elppdi using parametric or nonparametric approaches (Vehtari and Lampinen, 2002).”
BackgroundMethodInfluentialSelf-citation
Local Model Influence
McCulloch, R. (1989) · doi:10.1080/01621459.1989.10478793 · Cited by 192 · cited 2×
“One way to understand such diﬀerences is to calibrate based on simpler models (McCulloch, 1989).”
Method
Statistical aspects of model selection
Shibata, R. (1989) · Springer eBooks · doi:10.1007/978-3-642-75007-6_5 · Cited by 155 · cited 2× · see citing works
“DIC is a variation of the regularized information criteria which have been shown to be asymptotically equal to LOO-CV using plug-in predictive densities (Shibata 1989).”
Method
Bayesian Measures of Model Complexity and Fit
Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van Der Linde, A. (2002) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/1467-9868.00353 · Cited by 151 · cited 5× · see citing works
“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
DIC in variable selection
Linde, A. (2005) · Statistica Neerlandica · doi:10.1111/j.1467-9574.2005.00278.x · Cited by 151 · cited 1× · see citing works
“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,…”
Method
Statistical Inference: An Integrated Bayesian/Likelihood Approach
Aitkin, M. (2010) · doi:10.1201/ebk1420093438 · Cited by 69 · cited 2×
“So, in this case, DIC and AIC are the same.”
BackgroundMethod
Intrinsic losses
Robert, C. P. (1996) · Theory and Decision · doi:10.1007/bf00133173 · Cited by 59 · cited 4× · see citing works
“Regular cross-validation and information criteria can be used for time series in case of stationary Markov process and squared error or a scoring function or rule which is well approximated by a quadratic form (Akaike, 1973, Burman et al., 1994).”
MethodInfluential
Predictive likelihood for Bayesian model selection and averaging
Ando, T., & Tsay, R. (2010) · International Journal of Forecasting · doi:10.1016/j.ijforecast.2009.08.001 · Cited by 57 · cited 1× · see citing works
“Ando and Tsay (2010) have proposed an information criterion for the joint prediction, but its bias correction has the same computational diﬃculties as many other extensions of AIC and it can not be compared to cross-validation, since it is not possible to leave n data points out in the…”
Background
Implications of the ‘bread and peace’ model for the 2008 US presidential election
Hibbs, D. (2008) · doi:10.1007/s11127-008-9333-7 · Cited by 47 · cited 1×
“It is based on the ‘bread and peace’ model created by political scientist Douglas Hibbs (see Hibbs, 2008, for a recent review) to forecast elections based solely on economic growth (with corrections for wartime, notably Adlai Stevenson’s exceptionally poor performance in 1952 and Hubert Humphrey’s…”
Improved Probabilistic Prediction of Healthcare Performance Indicators using Bidirectional Smoothing Models
Jones, H. E., & Spiegelhalter, D. J. (2012) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1111/j.1467-985x.2011.01019.x · Cited by 16 · cited 2× · see citing works
“Hence the plan is to like to start with (5) and then apply some sort of bias correction to get a reasonable estimate of (2).”
Background
Rejoinder to discussion of 'Philosophy and the practice of Bayesian statistics'.
Gelman, A., & Shalizi, C. (2013) · British Journal of Mathematical & Statistical Psychology · doi:10.1111/j.2044-8317.2012.02066.x · Cited by 12
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“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).”
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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).”
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“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…”
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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.”
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“The posterior provides all the necessary information for evaluating accuracy and measuring out-of-sample predictive performance [209], [210].”
Societal determinants of flood-induced displacement
Vestby, J., Schutte, S., Tollefsen, A., & Buhaug, H. (2024) · Proceedings of the National Academy of Sciences of the United States of America · doi:10.1073/pnas.2206188120 · Cited by 46 · see citing works
Improved information criteria for Bayesian model averaging in lattice field theory
Neil, E. T., & Sitison, J. W. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.109.014510 · Cited by 47 · see citing works
“Other information criteria which include the same first term include the predictive information criterion (PIC) [38, 40] and the WatanabeAkaike information criteria (WAIC) [46].”
BackgroundInfluential
The Upper Edge of the Neptune Desert Is Stable Against Photoevaporation
Vissapragada, S., Knutson, H., Greklek-McKeon, M., Oklopčić, A., Dai, F., dos Santos, L. D., Jovanovic, N., Mawet, D., Millar-Blanchaer, M., Paragas, K., et al. (2022) · Astronomical Journal · doi:10.3847/1538-3881/ac92f2 · Cited by 46 · see citing works
Endemic persistence of a highly contagious pathogen: Foot-and-mouth disease in its wildlife host
Jolles, A., Gorsich, E., Gubbins, S., Beechler, B. R., Buss, P., Juleff, N., de Klerk-Lorist, L., Maree, F., Pérez-Martín, E., van Schalkwyk, O. L., et al. (2021) · Science · doi:10.1126/science.abd2475 · Cited by 46
Evaluating Forecasts with scoringutils in R (preprint)
Bosse, N., Gruson, H., Cori, A., van Leeuwen, E., Funk, S., & Abbott, S. (2022) · doi:10.48550/arxiv.2205.07090 · Cited by 45
Does water availability affect the critical N dilution curves in crops? A case study for maize, wheat, and tall fescue crops
Ciampitti, I., Makowski, D., Fernández, J. A., Lacasa, J., & Lemaire, G. (2021) · Field crops research (Print) · doi:10.1016/j.fcr.2021.108301 · Cited by 44
Past and recent anthropogenic pressures drive rapid changes in riverine fish communities
Danet, A., Giam, X., Olden, J. D., & Comte, L. (2024) · Nature Ecology & Evolution · doi:10.1038/s41559-023-02271-x · Cited by 55 · see citing works
“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”
Method
Trends in methamphetamine use, markets and harms in Australia, 2003–2019
Man, N. W. Y., Sisson, S. A., McKetin, R., Chrzanowska, A., Bruno, R., Dietze, P., Price, O. H., Degenhardt, L. J., Gibbs, D., Salom, C., et al. (2022) · Drug and Alcohol Review · doi:10.1111/dar.13468 · Cited by 44 · see citing works
“Models were compared using the deviance information criterion (DIC) as a predictive measure of choice in Bayesian applications [29,30].”
Method
A household-based community health worker programme for non-communicable disease, malnutrition, tuberculosis, HIV and maternal health: a stepped-wedge cluster randomised controlled trial in Neno District, Malawi
Wroe, E. B., Nhlema, B., Dunbar, E. L., Kulinkina, A. V., Kachimanga, C., Aron, M., Dullie, L., Makungwa, H., Chabwera, B., Phiri, B., et al. (2021) · BMJ Global Health · doi:10.1136/bmjgh-2021-006535 · Cited by 36 · see citing works
On the Application of Bayesian Leave-one-out Cross-validation to Exoplanet Atmospheric Analysis
Welbanks, L., McGill, P., Line, M., & Madhusudhan, N. (2022) · Astronomical Journal · doi:10.3847/1538-3881/acab67 · Cited by 40
Identifying spatio-temporal seizure propagation patterns in epilepsy using Bayesian inference
Vattikonda, A. N., Hashemi, M., Sip, V., Woodman, M., Bartolomei, F., & Jirsa, V. (2021) · Communications Biology · doi:10.1038/s42003-021-02751-5 · Cited by 40 · see citing works
I Will Survive: Predicting Business Failures from Customer Ratings
Naumzik, C., Feuerriegel, S., & Weinmann, M. (2021) · Marketing Science · doi:10.1287/mksc.2021.1317 · Cited by 46 · see citing works
Revised EU and WHO air quality thresholds: Where does Europe stand?
Beloconi, A., & Vounatsou, P. (2023) · Atmospheric Environment · doi:10.1016/j.atmosenv.2023.120110 · Cited by 38
Societies of strangers do not speak less complex languages
Shcherbakova, O., Michaelis, S., Haynie, H., Passmore, S., Gast, V., Gray, R., Greenhill, S. J., Blasi, D. E., & Skirgård, H. (2023) · Science Advances · doi:10.1126/sciadv.adf7704 · Cited by 37
“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
Salerno, J., Stevens, F. R., Gaughan, A. E., Hilton, T., Bailey, K. M., Bowles, T. M., Cassidy, L., Mupeta‐Muyamwa, P., Biggs, D., Pricope, N. G., et al. (2021) · Current Biology · doi:10.1016/j.cub.2021.08.074 · Cited by 45 · see citing works
A cross-comparison of different extreme value modeling techniques for traffic conflict-based crash risk estimation
Niu, D., Sayed, T., Fu, C., & Mannering, F. (2024) · Analytic Methods in Accident Research · doi:10.1016/j.amar.2024.100352 · Cited by 35
Identifying the Best Approximating Model in Bayesian Phylogenetics: Bayes Factors, Cross-Validation or wAIC?
Lartillot, N. (2023) · Systematic Biology · doi:10.1093/sysbio/syad004 · Cited by 45 · see citing works
“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.”
BackgroundMethodInfluential
Traditional 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
Investigating the association between neighbourhood characteristics and e-scooter safety
Heydari, S., Forrest, M., & Preston, J. (2022) · Sustainable cities and society · doi:10.1016/j.scs.2022.103982 · Cited by 34
Mindfulness facets predict quality of life and sleep disturbance via physical and emotional distresses in Chinese cancer patients: A moderated mediation analysis.
Fong, T. C. T., & Ho, R. T. H. (2020) · Psycho-Oncology · doi:10.1002/pon.5363 · Cited by 30 · see citing works
Characteristics of the Third COVID-19 Pandemic Wave with Special Focus on Socioeconomic Inequalities in Morbidity, Mortality and the Uptake of COVID-19 Vaccination in Hungary
Oroszi, B., Juhász, A., Nagy, C., Horváth, J. K., Komlos, K., Túri, G., McKee, M., & Ádány, R. (2022) · Journal of Personalized Medicine · doi:10.3390/jpm12030388 · Cited by 33 · see citing works
“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].”
Method
Sensitivity-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, Z. M., & Macdonald, D. W. (2022) · Proceedings of the Royal Society B Biological Sciences · doi:10.1098/rspb.2021.2681 · Cited by 38 · see citing works
“Model convergence was diagnosed using the Gelman–Rubin statistic (R̂) for all parameters and visual inspection of trace plots [53].”
Method
Ninety years of change, from commercial extinction to recovery, range expansion and decline for Antarctic fur seals at South Georgia
Forcada, 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.”
Method
Describing posterior distributions of variance components: Problems and the use of null distributions to aid interpretation
Pick, 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 39 · see citing works
Reconstructing the impact of COVID-19 on the immunity gap and transmission of respiratory syncytial virus in Lombardy, Italy
Ang, H. J., Menegale, F., Preziosi, G., Pariani, E., Migliari, M., Pellegrinelli, L., Sechi, G., Buoro, S., Merler, S., Cereda, D., et al. (2023) · EBioMedicine · doi:10.1016/j.ebiom.2023.104745 · Cited by 30
Probabilistic ordinal regression methods for multiple criteria sorting admitting certain and uncertain preferences
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
Bryce, E., Lombardo, L., van Westen, C., Tanyaş, H., & Castro‐Camilo, D. (2022) · Stochastic Environmental Research and Risk Assessment · doi:10.1007/s00477-022-02239-6 · Cited by 34 · see citing works
“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.”
Method
Controlled 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
Joint modeling of landslide counts and sizes using spatial marked point processes with sub-asymptotic mark distributions
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.”
Accounting for path and site effects in spatial ground-motion correlation models using Bayesian inference
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).”
MethodInfluential
Developing 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
Predictions of biodiversity are improved by integrating trait-based competition with abiotic filtering
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…”
Method
Obesity 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 36 · see citing works
“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…”
Method
Fully 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 33 · see citing works
“…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.”
Method
Luminous Supernovae: Unveiling a Population between Superluminous and Normal Core-collapse Supernovae
Gomez, S., Berger, E., Nicholl, M., Blanchard, P., & Hosseinzadeh, G. (2022) · Astrophysical Journal · doi:10.3847/1538-4357/ac9842 · Cited by 27 · see citing works
Bayesian extreme value analysis of kinematic-based surrogate measure of safety to detect crash-prone conditions in connected vehicles environment: A driving simulator experiment
Bakhshi, A. K., & Ahmed, M. M. (2022) · Transportation Research Part C: Emerging Technologies · doi:10.1016/j.trc.2021.103539 · Cited by 27 · see citing works
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 · see citing works
“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).”
Method
The Acquisition of /ɪ/–/iː/ Is Challenging: Perceptual and Production Evidence from Cypriot Greek Speakers of English
Georgiou, 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 citing works
“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).”
Optimizing nitrogen application rate by establishing a unified critical nitrogen dilution curve for maize under different mulching planting patterns
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 · see citing works
Bayesian spatial correlation, heterogeneity and spillover effect modeling for speed mean and variance on urban road networks.
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
Spatiotemporal modelling of abundance from multiple data sources in an integrated spatial distribution model
Strebel, N., Kéry, M., Guélat, J., & Sattler, T. (2022) · Journal of Biogeography · doi:10.1111/jbi.14335 · Cited by 25
Big problems in spatio-temporal disease mapping: Methods and software
Orozco-Acosta, E., Adin, A., & Ugarte, M. (2022) · Comput. Methods Programs Biomed. · doi:10.1016/j.cmpb.2023.107403 · Cited by 25
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).”
Method
A comparative analysis of modeling approaches and canopy height-based data sources for mapping forest growing stock volume in a northern subtropical ecosystem of China
Lin, 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).”
Background
Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and ApplicationsPreprint
Lee, S. (2022) · Mathematics · doi:10.20944/preprints202201.0407.v1 · Cited by 24 · see citing works
“For a detailed explanation of the criteria, refer to [17, 115].”
Background
Decisions Under Uncertainty as Bayesian Inference on Choice Options
Vieider, F. M. (2024) · Management Sciences · doi:10.1287/mnsc.2023.00265 · Cited by 23 · see citing works
“I test model performance based on their out-of-sample predictive ability (Gelman et al. 2014a, Vehtari et al. 2017).”
Method
The 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 · see citing works
Strengthening parental self‐efficacy and resilience: A within‐subject experimental study with refugee parents of adolescents
Eltanamly, H., Leijten, P., van Roekel, E., Mouton, B., Pluess, M., & Overbeek, G. (2022) · Child Development · doi:10.1111/cdev.13848 · Cited by 23 · see citing works
“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).”
Identifying deprived “slum” neighbourhoods in the Greater Accra Metropolitan Area of Ghana using census and remote sensing data
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 · see citing works
“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).”
Method
Bayesian 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…”
Result
Filial Cannibalism Leads to Chronic Nest Failure of Eastern Hellbender Salamanders (Cryptobranchus alleganiensis)
Hopkins, 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 · see citing works
“Similar to other information criteria, LOOIC includes a penalty that accounts for the effective number of parameters (Gelman et al. 2014).”
Method
The 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 28 · see citing works
“The Watanabe-Akaike Information Criterion (WAIC, Gelman et al. 2014; Vehtari et al. 2017) was used as information criteria to evaluate the models.”
Method
An application of Bayesian inference to examine student retention and attrition in the STEM classroom
Bertolini, R., Finch, S. J., & Nehm, R. H. (2023) · Frontiers in Education · doi:10.3389/feduc.2023.1073829 · Cited by 25 · see citing works
A Bayesian Approach with Urban-Scale Energy Model to Calibrate Building Energy Consumption for Space Heating: A Case Study of Application in Beijing
Na, W., & Wang, M.-M. (2022) · Social Science Research Network · doi:10.2139/ssrn.3879557 · Cited by 21
Computing Joint Action Costs: Co-Actors Minimize the Aggregate Individual Costs in an Action Sequence
Török, G., Stanciu, O., Sebanz, N., & Csibra, G. (2021) · Open Mind · doi:10.1162/opmi_a_00045 · Cited by 21
“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 · see citing works
A Poisson Lognormal-Lindley Model for Simultaneous Estimation of Multiple Crash-types: Application of Multivariate and Pooled Univariate Models
Tahir, H. B., Yasmin, S., & Haque, M. M. (2023) · Analytic Methods in Accident Research · doi:10.1016/j.amar.2023.100315 · Cited by 20
Integrated gut microbiome and metabolome analyses identified fecal biomarkers for bowel movement regulation by Bifidobacterium longum BB536 supplementation: A RCT
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 20 · see citing works
“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].”
Method
Estimating growth from length frequency distribution: comparison of ELEFAN and Bayesian approaches for red endeavour prawns (Metapenaeus ensis)
Zhou, 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
Estimating the duration of antibody positivity and likely time of Leptospira infection using data from a cross-sectional serological study in Fiji
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 21 · see citing works
“The convergence criterion for potential scale reduction (PSR) assesses MCMC convergence (Gelman et al., 2014).”
Background
Neurocomputational 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 · see citing works
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
Integrated modeling of waterfowl distribution in western Canada using aerial survey and citizen science (eBird) data
Adde, A., Amat, C. C., Mazerolle, M., Darveau, M., Cumming, S., & O'Hara, R. (2021) · Ecosphere · doi:10.1002/ecs2.3790 · Cited by 20
Re-examining the chatBot Usability Scale (BUS-11) to assess user experience with customer relationship management chatbots
Borsci, S., & Schmettow, M. (2024) · Personal and Ubiquitous Computing · doi:10.1007/s00779-024-01834-4 · Cited by 24 · see citing works
“Using information criteria, all models are then ranked by predictive accuracy [41].”

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 · 2012: 1
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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