Bayesfactor

Bayes factor is a statistic used in Bayesian inference to quantify the strength of evidence provided by the data in favor of one hypothesis relative to another. It is defined as BF10 = p(D | H1) / p(D | H0), the ratio of the data's marginal likelihood under the alternative hypothesis to that under the null. The reciprocal BF01 = 1/BF10 is the evidence for H0.

Calculation requires specifying priors on model parameters; the marginal likelihood integrates over parameters with their priors. In simple models closed-form results may exist; otherwise, numerical methods such as Laplace approximations, bridge sampling, thermodynamic integration, or MCMC-based approaches are used. Improper priors can make marginal likelihood undefined, so careful prior choice is important.

Interpretation: BF10 > 1 indicates evidence in favor of H1; BF10 < 1 favors H0. There is no universal cut-off; researchers often use qualitative scales (Jeffreys scale) to label evidence as 'anecdotal', 'moderate', 'strong', or 'very strong', but these are rough guidelines and depend on the context and priors.

Compared with p-values, Bayes factors measure evidence for models rather than long-run error control; they incorporate prior information and allow direct comparison of competing hypotheses. They can be sensitive to prior choices and require reporting of priors and sensitivity analyses.

Applications include testing means, correlations, regression coefficients, or comparing non-nested models in psychology, neuroscience, and social sciences. Bayes factors are widely used in Bayesian hypothesis testing and model selection.