Thank you so much for this post, I’ve found it really useful. I’m about to implement some method to calculate the marginal likelihood of an econometric model and now I see that the harmonic mean estimator is not an attractive solution. What do you think about Geweke’s (1999) modified harmonic mean estimator (p.46 here: http://www.tandfonline.com/doi/pdf/10.1080/07474939908800428)?

Thank you for your help in advance!

Best,

Greg

The likelihood isn’t a distribution, but I assume you mean to sample from a distribution whose density is proportional to the likelihood. It’s possible that there is no such distribution (the likelihood can integrate to infinity), but let’s suppose there is.

I think this won’t work in general, since you generally don’t know the normalizing constant needed to turn the likelihood into a density function. The goal is to find the integral of the prior times the likelihood over the parameter space. If you estimate this by sampling proportionally to the likelihood, you need to average the prior times the normalizing constant for the likeihood. So you haven’t really reduced to an easier problem, in most cases.

]]>Marginal likelihoods are *supposed* to be sensitive to the prior distribution used. I think the problem is that you don’t actually want to compare models by marginal likelihood / Bayes factors.

Setting g to a normal distribution may not be appropriate for my models. However, the bigger problem, as least it seems to me, is the influence of the magnitude of the prior variance on the marginal likelihood. Non-informative priors were used for my models, and one model contains one or two additional parameters relative to the other model to be compared with. Setting different prior precisions for these parameters would change the marginal likelihood, resulting in different estimates of the Bayes factor. That is really annoying!

]]>That sounds about right, on a brief glance, except that I don’t think setting g to a normal distribution with mean and covariance matching the posterior is going to work, in general, since it will likely give substantial probability to points that have negligible probability in the posterior (unless the posterior is actually normal). I think the estimator you’re using will then give the wrong answer, for approximately same reason that the harmonic mean estimator doesn’t work well.

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