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In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; models with lower BIC are generally preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC). When fitting models, it is possible to increase the maximum likelihood by adding parameters, but doing so may result in overfitting. Both BIC and AIC attempt to resolve this problem by introducing a penalty term for the number of parameters in the model; the penalty term is larger in BIC than in AIC for sample sizes greater than 7. The BIC was developed by Gideon E. Schwarz and published in a 1978 paper, where he gave a Bayesian argument for adopting it. Definition The BIC is formally defined as B I C = k ln ⁡ ( n ) − 2 ln ⁡ ( L ^ ) .   {\displaystyle \mathrm {BIC} =k\ln(n)-2\ln({\widehat {L}}).\ } where L ^ {\displaystyle {\hat {L}}} = the maximized value of the likelihood function of the model M {\displaystyle M} , i.e. L ^ = p ( x ∣ θ ^ , M ) {\displaystyle {\hat {L}}=p(x\mid {\widehat {\theta }},M)} , where θ ^ {\displaystyle {\widehat {\theta }}} are the parameter values that maximize the likelihood function and x {\displaystyle x} is the observed data; n {\displaystyle n} = the number of data points in x {\displaystyle x} , the number of observations, or equivalently, the sample size; k {\displaystyle k} = the number of parameters estimated by the model. For example, in multiple linear regression, the estimated parameters are the intercept, the q {\displaystyle q} slope parameters, and the constant variance of the errors; thus, k = q + 2 {\displaystyle k=q+2} . Derivation The BIC can be derived by integrating out the parameters of the model using Laplace's method, starting with the following model evidence:: 217  p ( x ∣ M ) = ∫ p ( x ∣ θ , M ) π ( θ ∣ M ) d θ {\displaystyle p(x\mid M)=\int p(x\mid \theta ,M)\pi (\theta \mid M)\,d\theta } where π ( θ ∣ M ) {\displaystyle \pi (\theta \mid M)} is the prior for θ {\displaystyle \theta } under model M {\displaystyle M} . The log-likelihood, ln ⁡ ( p ( x | θ , M ) ) {\displaystyle \ln(p(x|\theta ,M))} , is then expanded to a second order Taylor series about the MLE, θ ^ {\displaystyle {\widehat {\theta }}} , assuming it is twice differentiable as follows: ln ⁡ ( p ( x ∣ θ , M ) ) = ln ⁡ ( L ^ ) − n 2 ( θ − θ ^ ) T I ( θ ) ( θ − θ ^ ) + R ( x , θ ) , {\displaystyle \ln(p(x\mid \theta ,M))=\ln({\widehat {L}})-{\frac {n}{2}}(\theta -{\widehat {\theta }})^{\operatorname {T} }{\mathcal {I}}(\theta )(\theta -{\widehat {\theta }})+R(x,\theta ),} where I ( θ ) {\displaystyle {\mathcal {I}}(\theta )} is the average observed information per observation, and R ( x , θ ) {\displaystyle R(x,\theta )} denotes the residual term. To the extent that R ( x , θ ) {\displaystyle R(x,\theta )} is negligible and π ( θ ∣ M ) {\displaystyle \pi (\theta \mid M)} is relatively linear near θ ^ {\displaystyle {\widehat {\theta }}} , we can integrate out θ {\displaystyle \theta } to get the following: p ( x ∣ M ) ≈ L ^ ( 2 π n ) k 2 | I ( θ .... Discover the Sadanori Konishi Genshiro Kitagawa popular books. Find the top 100 most popular Sadanori Konishi Genshiro Kitagawa books.