# What is the difference between 'Laplace approximation' and 'Modified harmonic mean'?

this question is about Bayesian and computational statistics. I am learning them right now, I have two very common output from my software, one is Laplace approximation and the other is Modified harmonic mean. Both of them are used for approximation of log marginal likelihood, usually they have very close value. What are their differences? And could anyone provide me some background knowledge no these methods? Thanks!

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$$f(y)\approx (2\pi)^{d}|\widetilde{\Sigma}|^{1/2}f(y|\widetilde{\theta})f(\widetilde{\theta})$$
Where $\widetilde{\theta}$ is a posterior mode (can be estimated from MCMC output) and $\widetilde{\Sigma}$ is inverse of Hessian matrix based on log-likelihood and evaluated at posterior mode (this can be evaluated from MCMC output as well), $f(y|\theta)$ your likelihood and $f(\theta)$ is prior distribution. This approximation works well if your posterior distribution is similar to normal distribution. The second estimator, modified (generalized) harmonic mean estimator is derived by the following identity: $$\frac{1}{f(y)}=\int \frac{1}{f(y)}g(\theta)d \theta=\int \frac{1}{f(y|\theta)f(\theta)}f(\theta|y)g(\theta)d \theta=\int \frac{g(\theta)}{f(y|\theta)f(\theta)}f(\theta|y)d \theta=E_{f(\theta|y)}\left [ \frac{g(\theta)}{f(y|\theta)f(\theta)} \right ]$$ Here $g(\theta)$ is any "importance" density and has to be close to the posterior. So these two, Laplace and generalized harmonic mean, estimators are for marginal likelihood value estimation.
An anonymous editor contends that there is an error here:$$f(y)\approx n^{-d/2}(2\pi)^{d/2}|\widetilde{\Sigma}|^{1/2}f(y|\widetilde{\theta})f( \widetilde{ \theta})$$, citing this pdf –  gung Aug 23 at 21:18