Harmonic means, again again

Another arXiv posting I had had no time to comment is Nial Friel’s and Jason Wyse’s “Estimating the model evidence: a review“. This is a review in the spirit of two of our papers, “Importance sampling methods for Bayesian discrimination between embedded models” with Jean-Michel Marin (published in Jim Berger Feitschrift, Frontiers of Statistical Decision Making and Bayesian Analysis: In Honor of James O. Berger, but not mentioned in the review) and “Computational methods for Bayesian model choice” with Darren Wraith (referred to by the review). Indeed, it considers a series of competing computational methods for approximating evidence, aka marginal likelihood:

• Laplace approximation
• harmonic mean estimator
• Chib’s method
• annealed importance sampling (à la Neal,  2001)
• nested sampling
• power posteriors (which is actually a form of path sampling, à la Friel and Pettitt)

The paper correctly points out the difficulty with the naïve harmonic mean estimator. (But it does not cover the extension to the finite variance solutions found in”Importance sampling methods for Bayesian discrimination between embedded models” and in “Computational methods for Bayesian model choice“.)  It also misses the whole collection of bridge and umbrella sampling techniques covered in, e.g., Chen, Shao and Ibrahim, 2000 . In their numerical evaluations of the methods, the authors use the Pima Indian diabetes dataset we also used in “Importance sampling methods for Bayesian discrimination between embedded models“. The outcome is that the Laplace approximation does extremely well in this case (due to the fact that the posterior is very close to normal), Chib’s method being a very near second. The harmonic mean estimator does extremely poorly (not a suprise!) and the nested sampling approximation is not as accurate as the other (non-harmonic) methods. If we compare with our 2009 study, importance sampling based on the normal approximation (almost the truth!) did best, followed by our harmonic mean solution based on the same normal approximation. (Chib’s solution was then third, with a standard deviation ten times larger.)