bayes-factor

A Bayes factor is a ratio of marginal likelihoods that quantifies the evidence provided by observed data D in favor of one statistical model or hypothesis H₁ over another H₀. Given data and models M₁, M₀, the Bayes factor BF₁₀ = p(D|M₁) / p(D|M₀), where each marginal likelihood integrates the likelihood over its prior: p(D|Mᵢ) = ∫ p(D|θᵢ, Mᵢ) p(θᵢ|Mᵢ) dθᵢ. Unlike the frequentist likelihood ratio, the Bayes factor automatically penalizes model complexity through prior integration — the 'Occam factor' — and produces a continuous measure of evidential strength rather than a binary accept/reject decision. Interpretation follows calibrated scales (e.g., Jeffreys' scale): BF₁₀ > 1 supports H₁, BF₁₀ < 1 supports H₀, with thresholds marking 'barely worth mentioning' through 'decisive' evidence. The concept was formalized by Harold Jeffreys (1939, 1961) and remains central to Bayesian model comparison. [formal: factor-bayesianus | substrate: mind | horizon: a life | explicit: yes | epoch: 0.01]