# Bayesian Information Criterion (BIC) for large samples

The Bayesian information criterion is defined as $BIC = -2 \text{ln}(L) + k\text{ln}(n)$, where $L$ is the maximized likelihood of the data, and where $n$ is the sample size.

In case of a huge sample size, BIC tend to $\infty$.

Is there any transformation that needs to be done in order to compute the BIC for large samples?

Thank you,

• @Mariam, but just for clarification: what is your confuse confuse about $\mathrm{BIC} \to \infty$? – agronskiy Oct 4 '13 at 15:42
Normalize BIC by dividing By 2n. Let the likelihood function $$L = \prod_{i=1}^n p( x_i | \theta)$$. Then when the sample size $$n$$ is large then the influence of the $$K \log n$$ term becomes negligible and the normalized BIC converges To the cross entropy of the fitted Model with the density which generated the data. To see this note $$\frac{BIC}{2n} = -(1/n) \log L+ (k/n) \log (n) = -(1/n) \sum_{i=1}^n \log p( x_i |\theta) + (k/n) \log (n)$$ Under usual regularity conditions First normalized log likelihood term on right hand side converges to a number and second term on right hand side converges to zero.