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Two well-known (and related) measures of model complexity from statistics are the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

When might AIC = BIC?

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    $\begingroup$ You should try writing down the formulas and setting them equal to each other :) You will get the answer immediately. $\endgroup$ – guy Mar 14 at 13:18
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As a reminder:

$$AIC = - 2 \log \mathcal{L}(\hat{\theta}|X)+2k $$

$$BIC = - 2 \log \mathcal{L}(\hat{\theta}|X)+k \ln(n)$$

So for what values of $n$ is $2 = \ln(n)$?

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    $\begingroup$ (+1) I noticed that for $BIC$ you write $\log$ and $\ln$ in the same expression. Is this distinction necessary? $\endgroup$ – Sycorax Mar 14 at 19:23
  • $\begingroup$ Both logarithms have $e$ as their basis. It is just that log-likelihood (instead of ln-likelihood) is the term we use to describe the natural logarithm of the likelihood. $\endgroup$ – Stats Mar 14 at 19:27
  • $\begingroup$ @Sycorax: I guess it's to signify that for the $\log$ it really doesn't matter (as long as you're consistent) whereas for the $\ln$, well it has to be $e$. $\endgroup$ – Mehrdad Mar 15 at 0:37
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    $\begingroup$ But $n$ should be an integer, right? $\endgroup$ – innisfree Mar 15 at 5:44
  • $\begingroup$ @innisfree Not always. E.g., for generalized additive models the estimated degrees of freedom are usually not integers. $\endgroup$ – Roland Mar 15 at 7:12

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