generalised ratio of uniforms

A recent arXiv posting of the paper “On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling” by Martino, Luengo, and Míguez from Madrid rekindled my interest in this rather peculiar simulation method. The ratio of uniforms samples uniformly on the subgraph

\mathcal{A}=\{(v,u);\,0\le u\le\sqrt{p(v/u)}\}

to produce simulations from p as the ratio v/u. The proof is straightforward first year calculus but I do not find the method intuitive as, say, accept/reject…. The paper gives a very detailed background on those methods, as well as on the “inverse of density method”, which is like looking at the uniform simulation over the subgraph, but with both axes inverted (slice sampling is the same on both). (A minor point of contention or at least misunderstanding: when using the inverse of density method, the authors claim that using the unormalised and the normalised versions of the target leads to the same outcome. While it is true for the direct method, I have trouble seeing the equivalent in the inverse case…) The paper also stresses that the optimal case for accept-reject is when the target is bounded because the uniform can then be used as a proposal. I agree this is a simpler solution but fail to see any optimality in the matter. The authors then study ways of transforming unbounded subgraphs into bounded domains (i.e. bounded pdfs and supports). This imposes conditions on the transform f, which must have finite limits for p(x)/f'(x) or p-1(x)/f'(x) at the boundaries. (An optimal choice is when f is the cdf of p, since then the transform is uniform.)

The remainder (and more innovative) part of the paper is less clear in that I do not get a generic feeling on what it is about! The generalisation of the above is to consider uniform sampling from

\mathcal{A}_g=\big\{(v,u);\,0\le u\le g^{-1}\{c p[v/g'(u)]\}\big\}

for a generic increasing function g such that g(0)=0. And c a positive constant. (Any positive constant?!) But this is from a 1991 paper by Jon Wakefield, Alan Gelfand, and Adrian Smith. The extension is thus in finding g such that the above region is bounded and can be explored by uniform sampling over a box.. And in noticing that “the generalized Ratio-of-Uniform method is a combination of the transformed rejection method applied to the inverse density with the extended inverse-of-density method” (p.27).

I wonder at the applicability of the approach for costly target functions p. And at the extension to larger dimensions. And wish I had more time (or more graduate students) to look at possible adaptive constructions of the transform g. An interesting and fruitful read, nonetheless!

One Response to “generalised ratio of uniforms”

  1. Dan Simpson Says:

    It always feels like there should be some sort of potential theory-ish thing going on behind this method. It’s certainly fascinating! I also wonder about the connection between the g() function here and transformations used for high dimensional integration (where you want the integration points, which are defined over a box, to usually fall within the region of interest). I also wonder if such a link (if it exists) can be used in reverse to get some sort of information about hpd regions (which are defined through such an integral)…

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