Random construction of interpolating sets

One of the many arXiv papers I could not discuss earlier is Huber and Schott’s “Random construction of interpolating sets for high dimensional integration” which relates to their earlier TPA paper at the València meeting. (Paper that we discussed with Nicolas Chopin.) TPA stands for tootsie pop algorithm, The paper is very pleasant to read, just like its predecessor. The principle behind TPA is that the number of steps in the algorithm is Poisson with parameter connected  to  the unknown measure of the inner set:

$N\sim\mathcal{P}(\ln[\mu(B)/\mu(B^\prime)])$

Therefore, the variance of the estimation is known as well.  This is a significant property of a mathematically elegant solution. As already argued in our earlier discussion, it however seems the paper is defending an integral approximation that sounds far from realistic, in my opinion. Indeed, the TPA method requires as a fundamental item the ability to simulate from a measure μ restricted to a level set A(β). Exact simulation seems close to impossible in any realistic problem. Just as in Skilling (2006)’s nested sampling. Furthermore, the comparison with nested sampling is evacuated rather summarily: that the variance of this alternative cannot be computed “prior to running the algorithm” does not mean it is larger than the one of the TPA method. If the proposal is to become a realistic algorithm, some degree of comparison with the existing should appear in the paper. (A further if minor comment about the introduction is that the reason for picking the relative ideal balance α=0.2031 in the embedded sets is not clear. Not that it really matters in the implementation unless Section 5 on well-balanced sets is connected with this ideal ratio…)

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