Bayesian Information Criterion (BIC)

Abstract

Synonyms Schwarz criterion ; Schwarz information criterion (SIC) Definition In statistics, the Bayesian information criterion (BIC) (Schwarz 1978 ) is a model selection criterion. It is a selection criterion for choosing between different models with different numbers of parameters. The BIC is an asymptotic result derived under the assumption that the data distribution belongs to the exponential family. Suppose that: \\\\( x \\\\) = the observed data. \\\\( n \\\\) = the number of data points in \\\\( x \\\\) , or equivalently, the sample size. \\\\( k \\\\) = the number of free parameters to be estimated. \\\\( p(\\\\left. x ight|k) \\\\) = the probability of the observed data given the number of parameters. \\\\( L \\\\) = the maximized value of the likelihood function for the estimated model. The formula for the BIC is: $$ - 2\\\\ln p(\\\\left. x ight|k) \\\\approx BIC = - 2\\\\ln L + k\\\\ln (n) $$ Given any two estimated models, the model with the lower BIC value is the one to be preferred.