A numeric ratio comparing the marginal likelihoods of two competing statistical models given the same data; it quantifies relative evidence provided by the data in favor of one model versus another under specified priors.

Principle

Compute each model's integrated likelihood (marginalizing parameters under their priors) and form their ratio to measure how much more the data support one model over the other.

Demonstration

For coin data with k heads in n tosses, compare M0: p = 1/2 (no parameter) to M1: p ~ Beta(α,β). The Bayes factor is P(data|M0) / ∫ P(data|p) π(p) dp, evaluating the single-point likelihood versus the marginal under the Beta prior.

Misapplication

Treating the Bayes factor as posterior model probability without combining it with prior model odds, or using improper (non-normalizable) priors that make marginal likelihoods undefined.

Consequence

When applied correctly it yields an interpretable scale of evidence for model selection and can update prior model odds to posterior odds; but results depend sensitively on prior choices for model parameters.

Reversal

Interpreting the inverse ratio compares evidence in favor of the alternate model; expressing results as posterior odds requires multiplying the Bayes factor by prior odds.

Boundary

Applies only when both models have well-defined marginal likelihoods (proper priors or integrable likelihoods); not suitable when priors are improper or marginalization is undefined; does not directly assess predictive calibration.

Semantic Tension

Often confused with p-values or likelihood-ratio tests: Bayes factors compare marginalized model evidence under priors, while p-values measure extremeness under a null and likelihood-ratio tests compare pointwise likelihoods without integrating priors.

Synthesis

A Bayes factor is the prior-weighted likelihood ratio of two models that quantifies how data update relative belief in those models through marginalization over model parameters.