Specialty › Bayesian

Linear regression (Bayes factor)

Bayesian linear regression with the JZS prior reports the Bayes factor of the full model against intercept-only, plus posterior summaries for each coefficient.

What is Linear regression (Bayes factor)?

Frequentist regression gives p-values for null hypotheses; Bayesian regression gives a Bayes factor (BF) quantifying the relative evidence for the full model vs the intercept-only null, plus posterior distributions on each coefficient. BF₁₀ = 5 means data are 5× more likely under the full model than the null; BF₁₀ = 0.1 means 10× more likely under the null. Unlike p-values, BFs can express evidence *for* the null.

The JZS (Jeffreys-Zellner-Siow) prior is the default: scaled Cauchy on standardised effect sizes with r-scale parameter controlling vagueness. Medium (r = sqrt(2)/4) is the standard default; wide and ultrawide are progressively more diffuse and lead to smaller BFs (Lindley's paradox: vaguer prior penalises the full model).

Jeffreys' evidence scale: BF₁₀ 1–3 anecdotal, 3–10 moderate, 10–30 strong, 30–100 very strong, > 100 extreme. Report BF + the prior used + posterior CIs — never just 'significant'.

When should I use Linear regression (Bayes factor)?

When you want evidence framing rather than null-hypothesis testing.
When you might genuinely want to support the null (BF < 1/3).
When you want to update prior beliefs with the current data — though here we use the default JZS prior, not user-specified priors.

What data does it need?

Response + predictors + r-scale prior (medium / wide / ultrawide).

What does it report?

BF₁₀ + BF₀₁ + log₁₀ BF + evidence-strength label; posterior median + 95% CrI per coefficient.

How do I interpret the result?

BF < 1/3 supports the null; 1/3 < BF < 3 is inconclusive (data don't distinguish models); BF > 3 supports the alternative.

References

Rouder, Morey, Speckman & Province (2012). Default Bayes factors for ANOVA designs. JMP 56(5).