I've recently seen a few papers in physics using the Bayesian information criterion (BIC) to evaluate models. I'm much more familiar with Bayesian evidence, $p(x|M)$.

I've read in a few places, e.g. Wikipedia, that $$ \text{BIC} \approx -2\cdot\ln p(x|M) $$ provided a few conditions are met. Wikipedia, however, is a bit confusing concerning the conditions, as it appears to offer a tautology:

The BIC is an asymptotic result derived under the assumptions that the data distribution is in the exponential family. That is, the integral of the likelihood function $p(x|\theta,M)$ times the prior probability distribution $p(\theta|M)$ over the parameters $\theta$ of the model $M$ for fixed observed data $x$ is approximated as $$ {-2 \cdot \ln{p(x|M)}} \approx \mathrm{BIC} = {-2 \cdot \ln{\hat L} + k \cdot (\ln(n) - \ln(2 \pi))}. \ $$

I read this as saying the approximation holds if the approximation holds, as the approximation is that the evidence may be approximately written as a member of the exponential family (which is basically the approximation in question).

My instinct is the BIC could only approximate the evidence in very special cases - after all, BIC cannot approximate arbitrary evidence integrals with arbitrary priors - and that BIC is probably misused in physics.

What exactly are the conditions under which the BIC might approximate the evidence? It can't just be e.g. flat priors, because I could reparameterise any model such that priors were flat.

  • $\begingroup$ I guess it could be: likelihood is approximately gaussian function in parameters in which priors are flat? $\endgroup$ – innisfree Feb 8 '16 at 12:43
  • $\begingroup$ It is not really Bayesian and it does not aproximate "anything" quoting Gelman: andrewgelman.com/2008/10/23/i_hate_bic_blah :) $\endgroup$ – Tim Feb 8 '16 at 12:51

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