Bayes Factor Test
Bayes Factor Hypothesis Test · Also known as: bayes factor, BF10, Bayesian hypothesis test, Bayes Faktörü — Hipotez Testi
The Bayes factor test, formalised by Harold Jeffreys in 1961, is a Bayesian method for comparing two competing hypotheses. Rather than returning a binary reject/retain verdict, it produces a continuous ratio BF₁₀ that quantifies how much more (or less) probable the data are under the alternative hypothesis H₁ than under the null hypothesis H₀.
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When to use it
Use the Bayes factor test when you want to quantify evidence for a null hypothesis as well as for the alternative — a capability absent from classical significance testing. It is appropriate for continuous, categorical, or binary outcome data, requires at least five observations, and is valid without assuming normality, though the chosen prior scale must be justified and sensitivity-checked. Avoid it when the sample is too small for the prior not to dominate the posterior, or when a frequentist significance threshold is the required reporting convention.
Strengths & limitations
- Quantifies evidence for both the null and the alternative hypothesis on a continuous scale, not just a binary verdict.
- Does not require a minimum p-value threshold; a BF₁₀ below 1 constitutes positive evidence for H₀.
- Applicable to continuous, categorical, and binary data without a strict normality requirement.
- Communicates graded certainty through Jeffreys' interpretive scale, aiding transparent reporting.
- The Bayes factor is sensitive to the prior distribution; different priors yield different BF values, so a sensitivity analysis is always required.
- With very small samples (n < 5), the posterior is dominated by the prior and BF values are unreliable.
- When normality is violated and the prior scale dominates, a frequentist t-test is more reliable.
- Computationally more involved than a p-value, requiring integration over the prior-weighted parameter space.
Frequently asked
How is BF₁₀ different from a p-value?
A p-value measures only how surprising the data are if H₀ were true; it says nothing about H₁. BF₁₀ is a ratio of how probable the data are under H₁ versus under H₀, so it weighs both hypotheses simultaneously. A p-value cannot provide positive evidence for a null hypothesis, whereas a BF₁₀ below 1 does exactly that.
What does BF₁₀ = 5 mean in practice?
It means the observed data are five times more probable under H₁ than under H₀. On Jeffreys' scale this is moderate evidence for H₁ — informative but not conclusive. The corresponding posterior odds depend on your prior odds for the two hypotheses.
Why does the prior matter so much, and what prior should I use?
The Bayes factor is computed by averaging the likelihood over the prior. A different prior shape or scale can substantially change BF₁₀, so the choice must be justified. The default Cauchy prior with scale r ≈ 0.707 (medium in most software) is widely used for effect-size parameters because it reflects genuine uncertainty across a broad range of effect sizes. Always report the prior and run a sensitivity analysis across at least two or three alternative scales.
Can I compute a Bayes factor for correlations or contingency tables, not just t-tests?
Yes. The Bayes factor framework extends to correlations (correlationBF), contingency tables (contingencyTableBF), regression models, and ANOVA designs, all available in the BayesFactor R package and related tools. The interpretation of BF₁₀ remains the same regardless of the test form.
Sources
- Jeffreys, H. (1961). Theory of Probability (3rd ed.). Clarendon Press / Oxford University Press. ISBN: 978-0198503682
- Kass, R. E. & Raftery, A. E. (1995). Bayes Factors. Journal of the American Statistical Association, 90(430), 773–795. DOI: 10.1080/01621459.1995.10476572 ↗
How to cite this page
ScholarGate. (2026, June 1). Bayes Factor Hypothesis Test. ScholarGate. https://scholargate.app/en/bayesian/bayes-factor-test
Which method?
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