# Maximum Likelihood Estimation for Learning Populations of Parameters

arXiv: https://arxiv.org/abs/1902.04553 · HTML: https://mlsanity.com/paper/1902.04553/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2019.

## Built on by (most cited first, 11 of 11)

- [Near-Optimal Mean Estimation with Unknown, Heteroskedastic Variances](https://mlsanity.com/paper/2312.02417/) (2023)
- [Sharper Bounds for Chebyshev Moment Matching, with Applications](https://mlsanity.com/paper/2408.12385/) (2024)
- [Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints](https://mlsanity.com/paper/2202.05112/) (2022)
- [A Generative Framework for Personalized Learning and Estimation: Theory, Algorithms, and Privacy](https://mlsanity.com/paper/2207.01771/) (2022)
- [Generalized Identifiability Bounds for Mixture Models with Grouped Samples](https://mlsanity.com/paper/2207.11164/) (2022)
- [Fisher-Pitman permutation tests based on nonparametric Poisson mixtures with application to single cell genomics](https://mlsanity.com/paper/2106.03022/) (2021)
- [NOTMAD: Estimating Bayesian Networks with Sample-Specific Structures and Parameters](https://mlsanity.com/paper/2111.01104/) (2021)
- [Testing with Non-identically Distributed Samples](https://mlsanity.com/paper/2311.11194/) (2023)
- [When Can Voting Help, Hurt, or Change Course? Exact Structure of Binary Test-Time Aggregation](https://mlsanity.com/paper/2605.05592/) (2026)
- [Nonparametric mixture MLEs under Gaussian-smoothed optimal transport distance](https://mlsanity.com/paper/2112.02421/) (2021)
- [Causal Inference with Unobserved Confounding: A Mixture Learning Perspective](https://mlsanity.com/paper/2609.23219/) (2026)

## Abstract (arXiv)

Consider a setting with N independent individuals, each with an unknown parameter, pi ∈ [0, 1] drawn from some unknown distribution P^star. After observing the outcomes of t independent Bernoulli trials, i.e., Xi ~ Binomial(t, pi) per individual, our objective is to accurately estimate P^star. This problem arises in numerous domains, including the social sciences, psychology, health-care, and biology, where the size of the population under study is usually large while the number of observations per individual is often limited. Our main result shows that, in the regime where t ll N, the maximum likelihood estimator (MLE) is both statistically minimax optimal and efficiently computable. Precisely, for sufficiently large N, the MLE achieves the information theoretic optimal error bound of O(frac1t) for t < clogN, with regards to the earth mover's distance (between the estimated and true distributions). More generally, in an exponentially large interval of t beyond c logN, the MLE achieves the minimax error bound of O(frac1√tlog N). In contrast, regardless of how large N is, the naive "plug-in" estimator for this problem only achieves the sub-optimal error of Theta(frac1√t).