Other meanings of Bayes factor
Bayesian statistics
A Bayes factor is the ratio of marginal likelihoods used in Bayesian model comparison. It updates prior model odds into posterior model odds by measuring how well competing models predict the observed data after averaging over their parameter uncertainty.1
A Bayes factor compares two models through their marginal likelihoods. For models M₁ and M₂ and data D, BF₁₂ = P(D|M₁) / P(D|M₂); each marginal likelihood integrates the likelihood over the model’s prior distribution for its parameters.1 The resulting quantity is not a probability that either model is true. It is a relative measure of predictive support supplied by the data.
Bayes factors connect directly to posterior odds: posterior odds equal prior odds multiplied by the Bayes factor. Thus, a BF₁₂ of 10 multiplies the prior odds in favor of M₁ by ten, while a value below one favors M₂. Labels such as “substantial” or “strong” evidence are conventions rather than universal thresholds; the scale associated with Harold Jeffreys is widely used but should not replace substantive judgment.1
Computing a Bayes factor requires integrating over parameters, which makes prior specification central. A model receives support only when it predicts the observed data well across a meaningful region of its prior, not merely at a single best-fitting parameter value. This averaging automatically introduces a complexity penalty often described as an Occam factor: diffuse or needlessly flexible parameter priors can reduce marginal likelihood even when the model has a high maximum likelihood.1
Exact calculations are available for some conjugate models, while numerical approaches include bridge sampling, importance sampling, and nested sampling. The harmonic-mean estimator is generally unstable and is not a dependable general solution. For large samples, the Bayesian information criterion (BIC) can provide an approximation to twice the log Bayes factor under particular regularity and prior assumptions, but it is not itself an exact Bayes factor.2
A Bayes factor answers a different question from a p-value. A p-value evaluates the extremeness of data under a specified null model, whereas a Bayes factor compares the predictive performance of two models and incorporates parameter priors. Consequently, small p-values and modest Bayes factors can coexist, especially when a point-null hypothesis is compared with a broad alternative; this tension is a major topic in the study of statistical evidence.
Bayes factors also differ from posterior parameter estimates and credible intervals. They can compare a point hypothesis with a composite alternative, compare non-nested models, or assess whether a predictor should be included. In nested models, the Savage–Dickey density ratio can sometimes compute the Bayes factor for a point restriction from posterior and prior densities at the restricted value, although the result depends on compatible prior construction.3
The prior sensitivity of a Bayes factor is a feature rather than a technical footnote: data that are informative about a parameter can overwhelm reasonable prior choices, while weak data may leave conclusions strongly dependent on the prior’s scale. Reporting the prior, parameterization, computational method, and sensitivity analysis is therefore part of a reproducible comparison.1
Bayes factors can be combined across independent datasets by multiplication, or more safely by adding their logarithms. Sequential updating is valid when the same models and coherent priors are retained, but repeatedly searching over analyses and then presenting only a favorable comparison can distort interpretation. In hierarchical and high-dimensional models, marginal-likelihood estimation may be especially difficult; predictive criteria such as cross-validation answer a related but different question and should not be silently substituted for a Bayes factor.
BF₁₂ denotes evidence for M₁ relative to M₂; reversing the order gives BF₂₁ = 1 / BF₁₂.
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