Particle physics operates at the boundary of the detectable. Experiments at colliders like the Large Hadron Collider (LHC) at CERN search for rare signals buried in enormous backgrounds, requiring statistical methods that can rigorously quantify evidence for or against new physics. While the field has historically been dominated by frequentist methods — largely due to concerns about prior dependence — Bayesian approaches have steadily gained ground, and the interplay between the two frameworks has driven some of the most sophisticated statistical methodology in any science.
The Higgs Discovery: A Statistical Landmark
The July 2012 announcement of the Higgs boson discovery was framed in frequentist terms: a local significance exceeding 5σ (p-value below 3 × 10⁻⁷). But behind the scenes, Bayesian methods were used extensively. The profile likelihood ratio, while frequentist in spirit, has a Bayesian interpretation. And the systematic uncertainties — detector calibration, background modeling, theoretical cross-sections — were treated as nuisance parameters with prior distributions, a fundamentally Bayesian technique sometimes called the "hybrid" approach.
where μ is the signal strength parameter, θ are nuisance parameters,
and the significance is Z = √q₀ (in units of standard deviations σ).
Exclusion Limits and the CLs Method
When searches fail to find new particles, physicists set upper limits on production cross-sections. The CLs method, while not fully Bayesian, was designed to avoid the counter-intuitive exclusion of signals to which an experiment has no sensitivity — a problem that afflicts pure frequentist limits. Fully Bayesian limits integrate the posterior distribution of the signal strength parameter, yielding 95% credible intervals. The choice of prior on the signal strength — flat, reference, or physically motivated — remains debated.
When searching for a new particle across a mass range, a local excess may be significant but globally unremarkable — the "look-elsewhere effect." Bayesian model comparison naturally handles this through the prior predictive probability: a model that predicts the signal could appear at any mass is penalized relative to one that predicts a specific mass. The Bayes factor between signal and background hypotheses provides a global measure of evidence that automatically accounts for the trials factor.
Systematic Uncertainties as Nuisance Parameters
Modern particle physics analyses involve hundreds of systematic uncertainty sources: jet energy scales, luminosity measurements, parton distribution functions, Monte Carlo generator settings. The standard LHC approach treats each as a nuisance parameter with a prior distribution (typically Gaussian or log-normal), then marginalizes — integrating over — these parameters. This is Bayesian marginalization, even when the final result is presented in frequentist terms. The RooFit and HistFactory software frameworks implement this hybrid Bayesian-frequentist approach.
Bayesian Unfolding
Detector effects smear and distort the true particle distributions. Bayesian unfolding, introduced to particle physics by D'Agostini, treats the true distribution as a parameter with a prior and uses Bayes' theorem to infer it from the observed data. Iterative Bayesian unfolding has become a standard technique for measuring differential cross-sections and comparing with theoretical predictions.
"In particle physics, we use Bayesian methods every day while calling ourselves frequentists. The field would benefit from greater honesty about this, and greater comfort with the subjective elements that are already present." — Louis Lyons, Oxford physicist and organizer of the PhyStat conference series
Current Frontiers
Machine learning classifiers used for signal-background separation are increasingly evaluated with Bayesian calibration. Simulation-based inference methods enable Bayesian analysis when the likelihood is intractable (as in many BSM searches). And Bayesian optimal experimental design is being explored for guiding the next generation of collider experiments and detector upgrades.