# Approximate Profile Maximum Likelihood

arXiv: https://arxiv.org/abs/1712.07177 · HTML: https://mlsanity.com/paper/1712.07177/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2017.

## Built on by (most cited first, 4 of 4)

- [Finite-Sample Symmetric Mean Estimation with Fisher Information Rate](https://mlsanity.com/paper/2306.16573/) (2023)
- [Learning to be a Statistician: Learned Estimator for Number of Distinct Values](https://mlsanity.com/paper/2202.02800/) (2022)
- [On the Efficient Implementation of High Accuracy Optimality of Profile Maximum Likelihood](https://mlsanity.com/paper/2210.06728/) (2022)
- [PLM4NDV: Minimizing Data Access for Number of Distinct Values Estimation with Pre-trained Language Models](https://mlsanity.com/paper/2504.00608/) (2025)

## Abstract (arXiv)

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.