news from PCI

Max Ent at Max Plank

a come-back of the harmonic mean estimator

Are we in for a return of the harmonic mean estimator?! Allen Caldwell and co-authors arXived a new document that Allen also sent me, following a technique that offers similarities with our earlier approach with Darren Wraith, the difference being in the more careful and practical construct of the partition set and use of multiple hypercubes, which is the smart thing. I visited Allen’s group at the Max Planck Institut für Physik (Heisenberg) in München (Garching) in 2015 and we confronted our perspectives on harmonic means at that time. The approach followed in the paper starts from what I would call the canonical Gelfand and Dey (1995) representation with a uniform prior, namely that the integral of an arbitrary non-negative function [or unnormalised density] ƒ can be connected with the integral of the said function ƒ over a smaller set Δ with a finite measure measure [or volume]. And therefore to simulations from the density ƒ restricted to this set Δ. Which can be recycled by the harmonic mean identity towards producing an estimate of the integral of ƒ over the set Δ. When considering a partition, these integrals sum up to the integral of interest but this is not necessarily the only exploitation one can make of the fundamental identity. The most novel part stands in constructing an adaptive partition based on the sample, made of hypercubes obtained after whitening of the sample. Only keeping points with large enough density and sufficient separation to avoid overlap. (I am unsure a genuine partition is needed.) In order to avoid selection biases the original sample is separated into two groups, used independently. Integrals that stand too much away from the others are removed as well. This construction may sound a bit daunting in the number of steps it involves and in the poor adequation of a Normal to an hypercube or conversely, but it seems to shy away from the number one issue with the basic harmonic mean estimator, the almost certain infinite variance. Although it would be nice to be completely certain this doom is avoided. I still wonder at the degenerateness of the approximation of the integral with the dimension, as well as at other ways of exploiting this always fascinating [if fraught with dangers] representation. And comparing variances.

sanpshot from München [#2]

trip to München

While my train ride to the fabulous De Gaulle airport was so much delayed that I had less than ten minutes from jumping from the carriage to sitting in my plane seat, I handled the run through security and the endless corridors of the airport in the allotted time, and reached Munich in time for my afternoon seminar and several discussions that prolonged into a pleasant dinner of Wiener Schnitzel and Eisbier.  This was very exciting as I met physicists and astrophysicists involved in population Monte Carlo and parallel MCMC and manageable harmonic mean estimates and intractable ABC settings (because simulating the data takes eons!). I wish the afternoon could have been longer. And while this is the third time I come to Munich, I still have not managed to see the centre of town! Or even the nearby mountains. Maybe an unsuspected consequence of the Heisenberg principle…