MATLAB Implementation of Bayesian Information Criterion (BIC)

Bayesian Information Criterion (BIC) is an evaluation metric for statistical model selection that balances model goodness-of-fit against model complexity, helping researchers choose the optimal model among multiple candidates.

Implementing BIC in MATLAB typically involves the following computational logic: first calculate the maximum likelihood estimate of the model, then apply penalty adjustments based on sample size and number of model parameters. The core concept of BIC imposes stronger penalties on complex models to prevent overfitting. Specifically, the BIC calculation formula is generally expressed as:

[ \text{BIC} = -2 \cdot \log(\text{likelihood function value}) + k \cdot \log(n) ]

where ( k ) represents the number of model parameters and ( n ) is the sample size. A smaller BIC value indicates better balance between model fit and complexity.

In practical applications, BIC is commonly used in regression analysis, time series modeling, and machine learning model comparisons—for instance, in ARIMA model selection or determining optimal cluster numbers in cluster analysis. While MATLAB's Statistics and Machine Learning Toolbox provides built-in functions for BIC calculation, manual implementation allows better customization for specific models. The implementation typically involves computing the log-likelihood function using MATLAB's probability distribution functions and applying the penalty term calculation through basic arithmetic operations.

The main advantage of BIC lies in its simplicity and broad applicability, making it particularly suitable for robust model screening when dealing with large sample sizes.