Mallows model with intractable constant

The paper Probabilistic Preference Learning with the Mallows Rank Model by Vitelli et al. was published last year in JMLR which may be why I missed it. It brings yet another approach to the perpetual issue of intractable  normalising constants. Here, the data is made of rankings of n objects by N experts, with an assumption of a latent ordering ρ acting as “mean” in the Mallows model. Along with a scale α, both to be estimated, and indeed involving an intractable normalising constant in the likelihood that only depends on the scale α because the distance is right-invariant. For instance the Hamming distance used in coding. There exists a simplification of the expression of the normalising constant due to the distance only taking a finite number of values, multiplied by the number of cases achieving a given value. Still this remains a formidable combinatoric problem. Running a Gibbs sampler is not an issue for the parameter ρ as the resulting Metropolis-Hastings-within-Gibbs step does not involve the missing constant. But it poses a challenge for the scale α, because the Mallows model cannot be exactly simulated for most distances. Making the use of pseudo-marginal and exchange algorithms presumably impossible. The authors use instead an importance sampling approximation to the normalising constant relying on a pseudo-likelihood version of Mallows model and a massive number (10⁶ to 10⁸) of simulations (in the humongous set of N-sampled permutations of 1,…,n). The interesting point in using this approximation is that the convergence result associated with pseudo-marginals no long applies and that the resulting MCMC algorithm converges to another limiting distribution. With the drawback that this limiting distribution is conditional to the importance sample. Various extensions are found in the paper, including a mixture of Mallows models. And an round of applications, including one on sushi preferences across Japan (fatty tuna coming almost always on top!). As the authors note, a very large number of items like n>10⁴ remains a challenge (or requires an alternative model).

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