Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ Bayesian information criterion

Theorem: Consider a linear regression model $m$

\[\label{eq:mlr} m: \; y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V) \; .\]

Then, the Bayesian information criterion for this model is

\[\label{eq:mlr-bic} \mathrm{BIC}(m) = n \log\left( \frac{\mathrm{wRSS}}{n} \right) + n \left[ 1 + \log(2\pi) \right] + \log|V| + \log(n) \, (p + 1)\]

where $\mathrm{wRSS}$ is the weighted residual sum of squares, $p$ is the number of regressors in the design matrix $X$ and $n$ is the number of observations in the data vector $y$.

Proof: The Bayesian information criterion is defined as

\[\label{eq:bic} \mathrm{BIC}(m) = -2 \, \mathrm{MLL}(m) + k \log(n)\]

where $\mathrm{MLL}(m)$ is the maximum log-likelihood, $k$ is the number of free parameters in $m$ and $n$ is the number of observations.

The maximum log-likelihood for multiple linear regression is given by

\[\label{eq:mlr-mll} \mathrm{MLL}(m) = - \frac{n}{2} \log\left( \frac{\mathrm{wRSS}}{n} \right) - \frac{n}{2} \left[ 1 + \log(2\pi) \right] - \frac{1}{2} \log|V|\]

and the number of free paramters in multiple linear regression is $k = p + 1$, i.e. one for each regressor in the design matrix $X$, plus one for the noise variance $\sigma^2$.

Thus, the BIC of $m$ follows from \eqref{eq:bic} and \eqref{eq:mlr-mll} as

\[\label{eq:mlr-bic-qed} \mathrm{BIC}(m) = n \log\left( \frac{\mathrm{wRSS}}{n} \right) + n \left[ 1 + \log(2\pi) \right] + \log|V| + \log(n) \, (p + 1) \; .\]

Metadata: ID: P308 | shortcut: mlr-bic | author: JoramSoch | date: 2022-02-11, 06:43.