Asymmetric Likelihood-Free Hypothesis Testing
Abstract
In likelihood-free hypothesis testing (LFHT), given samples from two unknown distributions P,Q on [n] with dTV(P,Q)≥ε, and from a third distribution D∈{P,Q}, the goal is to identify whether D=P or D=Q. We study asymmetric likelihood-free hypothesis testing, where the sample budgets sP and sQ for P and Q may differ, and characterize the feasible triples (sP,sQ,sD) up to logarithmic factors. We establish that unequal reference budgets can give a polynomial saving. For instance, for constant separation, n samples from P and n3/4 from Q allow testing with Õ(n1/4) samples from D, whereas both existing baselines---symmetric LFHT and unequal-sample closeness testing---require Ω(√n) samples from D. We further apply our results to non-adaptive distribution 2-clustering, where the sample allocation among k distributions must be fixed before observing any data. We obtain matching upper and lower bounds for this problem up to logarithmic factors in terms of the domain size, number of distributions, smaller cluster size, and separation, both with one cluster distribution known and with both unknown. Comparing with known bounds for adaptive sample allocation gives a worst-case adaptivity gap of Θ̃(k1/4).
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BibTeX
@misc{kumar2026asymmetric,
title = {{Asymmetric Likelihood-Free Hypothesis Testing}},
author = {Gunjan Kumar and Yash Pote and Jonathan Scarlett},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29798},
eprint = {2610.29798},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29798},
}
RIS
TY - UNPB
TI - Asymmetric Likelihood-Free Hypothesis Testing
AU - Gunjan Kumar
AU - Yash Pote
AU - Jonathan Scarlett
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29798
UR - https://arxiv.org/abs/2610.29798
AB - In likelihood-free hypothesis testing (LFHT), given samples from two unknown distributions P,Q on [n] with d_TV(P,Q)≥ε, and from a third distribution D∈{P,Q}, the goal is to identify whether D=P or D=Q. We study asymmetric likelihood-free hypothesis testing, where the sample budgets s_P and s_Q for P and Q may differ, and characterize the feasible triples (s_P,s_Q,s_D) up to logarithmic factors. We establish that unequal reference budgets can give a polynomial saving. For instance, for constant separation, n samples from P and n^3/4 from Q allow testing with Õ(n^1/4) samples from D, whereas both existing baselines---symmetric LFHT and unequal-sample closeness testing---require Ω(√n) samples from D. We further apply our results to non-adaptive distribution 2-clustering, where the sample allocation among k distributions must be fixed before observing any data. We obtain matching upper and lower bounds for this problem up to logarithmic factors in terms of the domain size, number of distributions, smaller cluster size, and separation, both with one cluster distribution known and with both unknown. Comparing with known bounds for adaptive sample allocation gives a worst-case adaptivity gap of Θ̃(k^1/4).
ER -