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| Bayesian Information Criterion | |
|---|---|
| Name | Bayesian Information Criterion |
| Abbreviation | BIC |
| Introduced | 1978 |
| Developer | Gideon E. P. Schwarz |
| Field | Statistics |
| Related | Akaike information criterion, Likelihood ratio test, Bayesian model selection |
Bayesian Information Criterion The Bayesian Information Criterion is a model selection criterion used to compare statistical models while penalizing complexity. It balances goodness of fit against model parsimony to favor models that generalize, and it is widely employed across empirical fields including econometrics, biostatistics, and machine learning. Its development and use intersect with influential figures and institutions in statistical theory and applied science.
The Bayesian Information Criterion quantifies model quality by combining the maximized log-likelihood of a model with a penalty term proportional to the number of estimated parameters and the sample size. In common form, it evaluates candidate models using the likelihood evaluated at maximum likelihood estimates and an explicit complexity penalty; practitioners compute and compare BIC values to choose among nested or non-nested alternatives. The criterion is often presented alongside alternative selection rules developed by statisticians and researchers at institutions such as Columbia University, University of Chicago, Princeton University, Harvard University, and University of California, Berkeley.
Schwarz derived the criterion within a Bayesian asymptotic framework, approximating the marginal likelihood of a model under regularity conditions and large-sample limits. This derivation relates to Laplace's method and asymptotic expansions used by mathematicians and statisticians connected to Cambridge University, Stanford University, Oxford University, École Normale Supérieure, and University of Tokyo. The theoretical basis leverages connections to the Bayes factor, integrating prior distributions and likelihoods in a manner reminiscent of work by Thomas Bayes, Pierre-Simon Laplace, Ronald Fisher, Jerzy Neyman, and modern contributors associated with Institute for Advanced Study and Max Planck Society research groups.
BIC is consistent under certain regularity assumptions: as sample size increases, it tends to select the true model with probability approaching one when the true model lies among candidates. This property contrasts with finite-sample behavior explored by scholars at London School of Economics, University of Cambridge, Yale University, Massachusetts Institute of Technology, and University of Oxford. The penalty term grows with sample size, imparting stronger parsimony pressure than some alternatives, a feature noted in comparative studies involving researchers from National Bureau of Economic Research, International Monetary Fund, World Bank, and research teams at Bell Labs and IBM Research.
Practitioners use BIC to rank models estimated with maximum likelihood or comparable estimation frameworks applied in software developed by groups at RStudio', Microsoft Research, Google Research, Amazon Web Services, and academic departments such as Department of Statistics, University of Washington and Department of Biostatistics, Johns Hopkins University. It is applied to regression, time series, mixture modeling, and hierarchical models in fields influenced by institutions like National Institutes of Health, Centers for Disease Control and Prevention, European Commission, and industry labs at Tesla, Inc. and Siemens. Users routinely compare BIC values across models, preferring models with lower BIC, and often validate selections using cross-validation techniques promoted at Carnegie Mellon University, University of Pennsylvania, and New York University.
BIC is often contrasted with the Akaike Information Criterion developed by Hirotugu Akaike and with likelihood-ratio tests formalized by scientists associated with Royal Statistical Society and American Statistical Association. Compared to AIC, BIC imposes a larger penalty for complexity as sample size grows, leading to different selections in practice; this divergence has been examined by researchers from Princeton University, Columbia University, Stanford University, University of Michigan, and Cornell University. Other alternatives include cross-validation methods propagated at Imperial College London and minimum description length principles discussed by scholars at University of California, Santa Barbara and Santa Fe Institute.
BIC has been applied to model choice in macroeconomic models developed at Federal Reserve Board, European Central Bank, and academic programs at London School of Economics; in phylogenetic model selection studies by groups at Smithsonian Institution and Natural History Museum, London; and in genomics and proteomics projects sponsored by Wellcome Trust, Howard Hughes Medical Institute, and university centers at Broad Institute. Example applications include selecting autoregressive orders in time series at National Oceanic and Atmospheric Administration, choosing mixture components in clustering work at Facebook AI Research, and determining variable sets in epidemiological studies conducted at World Health Organization and Bill & Melinda Gates Foundation-funded consortia.
Critics note that BIC relies on large-sample approximations and assumed model regularity; departures from these assumptions—such as model misspecification, high-dimensional settings, or complex hierarchical structures—can undermine its reliability, a concern investigated by teams at University of California, Los Angeles, Duke University, University of Toronto, and McGill University. In high-dimensional contexts common in genetics and machine learning, alternatives and modifications developed by researchers at Broad Institute, DeepMind, Allen Institute for AI, and university departments offer more robust performance. Debates about prior sensitivity, asymptotic justification, and empirical calibration have been advanced in literature from International Biometric Society, Royal Statistical Society, Institute of Mathematical Statistics, and leading journals associated with American Association for the Advancement of Science.
Category:Statistical model selection