vertical likelihood Monte Carlo integration

A few months ago, Nick Polson and James Scott arXived a paper on one of my favourite problems, namely the approximation of normalising constants (and it went way under my radar, as I only became aware of it quite recently!, then it remained in my travel bag for an extra few weeks…). The method for approximating the constant Z draws from an analogy with the energy level sampling methods found in physics, like the Wang-Landau algorithm. The authors rely on a one-dimensional slice sampling representation of the posterior distribution and [main innovation in the paper] add a weight function on the auxiliary uniform. The choice of the weight function links the approach with the dreaded harmonic estimator (!), but also with power-posterior and bridge sampling. The paper recommends a specific weighting function, based on a “score-function heuristic” I do not get. Further, the optimal weight depends on intractable cumulative functions as in nested sampling. It would be fantastic if one could draw directly from the prior distribution of the likelihood function—rather than draw an x [from the prior or from something better, as suggested in our 2009 Biometrika paper] and transform it into L(x)—but as in all existing alternatives this alas is not the case. (Which is why I find the recommendations in the paper for practical implementation rather impractical, since, were the prior cdf of L(X) available, direct simulation of L(X) would be feasible. Maybe not the optimal choice though.)

“What is the distribution of the likelihood ordinates calculated via nested sampling? The answer is surprising: it is essentially the same as the distribution of likelihood ordinates by recommended weight function from Section 4.”

The approach is thus very much related to nested sampling, at least in spirit. As the authors later demonstrate, nested sampling is another case of weighting, Both versions require simulations under truncated likelihood values. Albeit with a possibility of going down [in likelihood values] with the current version. Actually, more weighting could prove [more] efficient as both the original nested and vertical sampling simulate from the prior under the likelihood constraint. Getting away from the prior should help. (I am quite curious to see how the method is received and applied.)

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