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Density estimation & likelihood ratios

Section status

Contributor: Unclaimed — this section has no named author yet. See Contribute.

Last reviewed: —

Status: Stub — a starting point only. The curated list is not yet written.

Overview

This is the single most reused idea in machine learning for particle physics: train a classifier to separate two samples, and its output — suitably calibrated — approximates the ratio of the two densities that produced them. Because a likelihood ratio is the optimal test statistic, and because reweighting one sample into another is exactly a ratio problem, that one result covers a remarkable amount of the field.

Three application areas that look unrelated are the same technique in different clothing. In simulation-based inference the ratio is between simulations at different parameter points, and it gives you a likelihood. In unfolding, the ratio reweights simulation to data, iteratively, and it gives you a measurement. In anomaly detection the ratio is between a signal region and sidebands, and it gives you a search.

The complementary tool is explicit density estimation — normalizing flows and diffusion models that provide a tractable likelihood rather than a ratio. The distinction that matters in practice is that ratios are often easier to estimate accurately than the densities themselves, since the hard normalization cancels, but densities can be sampled from, and ratios cannot. Which you want depends on whether you are measuring or generating.

The open problem across all of it is calibration. A classifier that ranks correctly but is not calibrated gives a monotone function of the likelihood ratio rather than the ratio itself, which is fine for a cut but fatal for an interval.

  • The frontier of simulation-based inference, Cranmer et al. (2019) (arXiv:1911.01429) — the review that frames ratio estimation as the core of simulation-based inference
  • Learning Likelihood Ratios with Neural Network Classifiers, Rizvi et al. (2023) (arXiv:2305.10500) — specifically on learning likelihood ratios with neural classifiers, and the practical pitfalls
  • Modern Machine Learning for LHC Physicists, Plehn et al. (2022) (arXiv:2211.01421) — lecture notes covering classifiers, reweighting and density estimation in one place

Curated paper list

This list is a seed, not a curated selection

A few landmark papers are listed to give the section a starting shape. A proper curated list — thematically grouped, with an annotation on every entry — is what this section still needs. See Contribute.

  • Approximating Likelihood Ratios with Calibrated Discriminative Classifiers, Cranmer et al. (2015) (arXiv:1506.02169) — the founding paper: a calibrated discriminative classifier approximates the likelihood ratio
  • OmniFold: A Method to Simultaneously Unfold All Observables, Andreassen et al. (2020) (arXiv:1911.09107) — OmniFold, showing the same idea used iteratively as a reweighting scheme
  • Anomaly Detection with Density Estimation, Nachman et al. (2020) (arXiv:2001.04990) — ANODE, using estimated densities in signal region and sidebands to construct a ratio for searches
  • Classifying Anomalies THrough Outer Density Estimation (CATHODE), Hallin et al. (2021) (arXiv:2109.00546) — CATHODE, which replaces the direct ratio with sampling from an interpolated density and outperforms it

Benchmarks, datasets & software

Not yet compiled for this section.

Open questions

What is settled here, what is contested, and what remains unsolved? This is the part a bibliography structurally cannot provide, and often the most useful paragraph on the page.

Further reading

Relevant work that is not an entry point — too specialized, too recent, or simply not where a newcomer should start. Suggestions that do not fit the curated list above belong here rather than being turned away.

Nothing listed yet.

Cross-references