Bayes factor

A Bayes factor is the ratio of the marginal likelihoods of observed data under two competing models. It updates prior odds into posterior odds and measures relative evidential support.

Marginal likelihood: The probability of observed data under a model, averaged over its parameter values using the prior distribution. Each model’s marginal likelihood supplies one side of the Bayes factor.
Bayesian probability: An interpretation of probability as a degree of belief updated with evidence. Bayes factors quantify evidential change within this probabilistic framework.
Null hypothesis significance testing: A statistical framework that evaluates data against a null hypothesis using a test statistic and significance threshold. Its p-values answer a different question from a Bayes factor’s relative model support.
P-value: The probability, under a specified null hypothesis, of obtaining a test statistic at least as extreme as observed. A p-value is not the probability of a model or the ratio of two models’ evidence.
Jeffreys–Lindley paradox: A conflict in which a significance test and Bayesian model comparison can favor different conclusions as sample size grows. It exposes how prior assumptions and sample size can make Bayes factors diverge from p-value intuitions.
Prior odds: The ratio of prior probabilities assigned to two hypotheses or models. Multiplying prior odds by the Bayes factor yields posterior odds.
Statistical model: A mathematical description of how data could have been generated. A Bayes factor requires explicit competing models that assign probabilities to data.
Bayesian hypothesis testing: The evaluation of hypotheses using prior probabilities and the likelihood of observed data. Bayes factors are a standard way to compare discrete hypotheses in this framework.
Likelihood-ratio test: A frequentist test that compares maximized likelihoods under nested statistical models. It uses best-fitting parameter values, whereas a Bayes factor averages over parameter uncertainty.
Prior sensitivity analysis: An assessment of how conclusions change when prior distributions or assumptions are varied. Diffuse or concentrated parameter priors can substantially change marginal likelihoods and Bayes factors.

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