Approximate Profile Maximum Likelihood

Abstract

We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML has appealing theoretical properties, but is difficult to compute exactly. Inspired by observations gleaned from exactly solvable cases, we look for an approximate PML solution, which, intuitively, clumps comparably frequent symbols into one symbol. This amounts to lower-bounding a certain matrix permanent by summing over a subgroup of the symmetric group rather than the whole group during the computation. We extensively experiment with the approximate solution, and find the empirical performance of our approach is competitive and sometimes significantly better than state-of-the-art performance for various estimation problems.

References in corpus (2)

• Chebyshev polynomials, moment matching, and optimal estimation of the unseen
• Estimation of KL Divergence: Optimal Minimax Rate

Cited by in corpus (6)

• Learning to be a Statistician: Learned Estimator for Number of Distinct Values
• The Bethe and Sinkhorn Permanents of Low Rank Matrices and Implications for Profile Maximum Likelihood
• Non-Bayesian Parametric Missing-Mass Estimation
• The Broad Optimality of Profile Maximum Likelihood
• PLM4NDV: Minimizing Data Access for Number of Distinct Values Estimation with Pre-trained Language Models
• Instance Based Approximations to Profile Maximum Likelihood