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Wikipedia states that under certain circumstances, $\mathrm{BIC} $ can be calculated as: $$\mathrm{BIC} = n \ln(\mathrm{RSS}/n) + k \ln(n),$$

where $\mathrm{RSS}$ is the residual sum of squares. However, I was not able to find a reliable source for it. This question gives another source, however that source is just a copy of the Wikipedia article.

Yaffee and McGee state that

$$\mathrm{BIC} = n \ln(\mathrm{MSE}) + k \ln(n),$$

which IMHO would be identical to the Wikipedia definition, if they did not specify

$$\mathrm{MSE} = \frac{1}{n-k}(RSS).$$

That's a definition of $\mathrm{MSE}$ that I have never seen before (due to $-k$) and does not make a lot of sense to me.

Any references to reliable sources for BIC with residual sum of squares or an explanation of the strange definition of $\mathrm{MSE}$ is greatly appreciated.

Thank you.

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  • $\begingroup$ The $-k$ is the standard adjustment for creating an unbiased estimator of residual variance in a regression, and serves the same purpose (well is exactly the same thing) as the $-1$ in the usual calculation of variance, where you divide by $n-1$. I'm a bit surprised you haven't run into it before :) $\endgroup$
    – jbowman
    Commented Mar 4, 2020 at 19:08
  • $\begingroup$ @jbowman Of course! Ok, I withdraw my hasty statement that I have never seen it before. Of course I have, I just did not make the connection because I'm used to only $1$ being subtracted. $\endgroup$
    – Nos
    Commented Mar 4, 2020 at 19:29

1 Answer 1

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The formulas given by Yaffee and McGee and Wikipedia are indeed identical, except for Bessel's correction ($-k$). Whether Bessel's correction has to be applied depends on the use case.

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