# Is it possible that AIC = BIC?

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?

• You should try writing down the formulas and setting them equal to each other :) You will get the answer immediately.
– guy
Commented Mar 14, 2019 at 13:18

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 $$2 = \ln(n)$$?

• (+1) I noticed that for $BIC$ you write $\log$ and $\ln$ in the same expression. Is this distinction necessary?
– Sycorax
Commented Mar 14, 2019 at 19:23
• 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. Commented Mar 14, 2019 at 19:27
• @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 $e$. Commented Mar 15, 2019 at 0:37
• But $n$ should be an integer, right? Commented Mar 15, 2019 at 5:44
• @innisfree Not always. E.g., for generalized additive models the estimated degrees of freedom are usually not integers. Commented Mar 15, 2019 at 7:12