  We propose the Pseudo-Mallows distribution over the set of all permutations
of $n$ items, to approximate the posterior distribution with a Mallows
likelihood. The Mallows model has been proven to be useful for recommender
systems where it can be used to learn personal preferences from highly
incomplete data provided by the users. Inference based on MCMC is however slow,
preventing its use in real time applications. The Pseudo-Mallows distribution
is a product of univariate discrete Mallows-like distributions, constrained to
remain in the space of permutations. The quality of the approximation depends
on the order of the $n$ items used to determine the factorization sequence. In
a variational setting, we optimise the variational order parameter by
minimising a marginalized KL-divergence. We propose an approximate algorithm
for this discrete optimization, and conjecture a certain form of the optimal
variational order that depends on the data. Empirical evidence and some theory
support our conjecture. Sampling from the Pseudo-Mallows distribution allows
fast preference learning, compared to alternative MCMC based options, when the
data exists in form of partial rankings of the items or of clicking on some
items. Through simulations and a real life data case study, we demonstrate that
the Pseudo-Mallows model learns personal preferences very well and makes
recommendations much more efficiently, while maintaining similar accuracy
compared to the exact Bayesian Mallows model.
