BIC

Definition

The Bayesian Information Criterion (BIC) is a model selection criterion similar to AIC but with a heavier penalty for model complexity.

$BIC=-2\log L_{\max }+k\log n$

where:

• $L_{\max }$ = maximum likelihood
• $k$ = number of parameters
• $n$ = sample size

Interpretation

• Lower BIC is better
• Penalty increases with sample size: $k\log n$
• Emphasizes consistent model selection

Model Selection

Choose the model with minimum BIC among competing models.

Comparison: AIC vs BIC

Criterion	Penalty	Best For	Asymptotic Property
AIC	$2k$	Prediction efficiency	Not consistent (may overfit)
BIC	$k\log n$	Consistent selection	Consistent (selects true model)

Key Differences

• BIC penalty grows with $n$: More data → stronger penalty for complexity
• BIC tends to select simpler models than AIC
• For $n>7$: BIC penalty exceeds AIC penalty

Practical Guidance

• Use AIC for prediction-oriented tasks
• Use BIC for explanation/interpretation

Related

• AIC Criterion
• Box-Jenkins Strategy
• Model Selection via Data Splitting