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Given the basic marginal likelihood identity:

$$ ln \; m(y)=ln \; p(y|\theta^*)+ln\; \pi(\theta^*)-ln \pi(\theta^*|y) $$

is there a way to derive from this the Bayesian Information Criterion?

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As proposed by Chib (1995), the marginal likelihood can be computed from the marginal likelihood identity: $$m(y)=\frac{\phi(y|\theta^*)\pi(\theta^*)}{\pi(\theta^*|y)}$$ where $\theta^*$ can be any admissible value. The natural logarithm of this equation presents a computationally convenient expression: $$log \; m(y) = log\; \pi(\theta^*)+log \; \phi(y|\theta^*)-log \; \pi(\theta^*|y)$$ This expression can also be seen as something corrected by $-log\; \pi(\theta^*|y)$. The assumption on the posterior density: $\int_A \pi_\alpha(\theta|\alpha)\; d\theta \xrightarrow{n\to \infty} \int_A N(\theta|\mu, \Sigma) \; d\theta$ which can also be scale by $\sqrt{n}(\theta-\hat{\theta})$ will converge to a standard normal. Therefore we have: $\theta|y\sim N(\hat{\theta}, \frac{1}{n}\nu_\infty)$ where $\nu_\infty=\left(ln \; \frac{1}{n}\frac{d^2 log \; p(y|\theta}{d\theta d\theta} \right)^{-1}$ is a bounded term $Op(1)$. We can use this $\theta|y$ put it in the identity: $$m(y)\approx\frac{\pi(\theta)\phi(y|\theta)}{N(\theta|\hat{\theta},\frac{1}{n} \nu_\infty)}$$ and evaluate at MLE: $$m(y)\approx\frac{\pi(\hat{\theta_n})\phi(y|\hat{\theta}_n)}{(2\pi)^{-\frac{1}{2}}|\frac{1}{n}\nu_\infty|^{-\frac{1}{2}} e^0}=\frac{\pi(\hat{\theta_n})\phi(y|\hat{\theta}_n)}{(2\pi)^{-\frac{1}{2}}\left(\frac{1}{n}\right)^{-\frac{d}{2}}|\nu_\infty|^{-\frac{1}{2}} }$$ given that $|kA|=k^d|A|$ Given that $\pi(\hat{\theta_n})$, $(2\pi)^{-\frac{1}{2}}$ and $|\nu_\infty|^{-\frac{1}{2}}$ are $Op(1)$, then we have: $$log \; m(y)= log \; \phi(y|\hat{\theta_n})-\frac{d}{2} log \; n + Op(1)$$ which is the BIC ($m(y)=BIC[1+Op(1)]$) which includes a penalty so that high dimensional model not necessarily are better.

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