Eric-Jan Wagenmakers, Tom Lodewyckx, Himanshu Kuriyal and Raoul Grasman (2010), Cognitive Psychology 60(3):158–189 — DOI-10.1016/j.cogpsych.2009.12.001
Key takeaways
- The ratio. When H0 fixes a parameter φ at φ0 and H1 leaves it free, BF01 = p(φ = φ0 | D, H1) / p(φ = φ0 | H1): the posterior ordinate over the prior ordinate, both under the larger model (Eqs. 11–12). In the binomial example (9 of 10 correct, uniform prior) both routes give BF01 = 0.107.
- The condition. The nuisance parameters ψ must have p(ψ | φ → φ0, H1) = p(ψ | H0). Then they affect the Bayes factor only through the posterior of φ, and their own priors may be vague or improper. The prior on φ is the opposite case: its height at φ0 sits in the denominator, so doubling the width of a uniform prior roughly doubles BF01.
- Order restrictions come for free: truncating to φ < φ0 doubles the prior ordinate, and the posterior ordinate rises only as much as the data disagree with the restriction. A restriction the data already obey can at most double the evidence against the null.
- The limits, as the authors give them (§8): a one-dimensional density estimate at a point, unreliable when φ0 is in the posterior’s tail; nested models only; and the conditioning on a probability-zero event that makes the result depend on parameterisation (the Borel–Kolmogorov paradox), so a test of μ = 0 can differ from a test of μ/σ = 0.
Standing in the record
Filed on 2026-10-03 at the owner’s request, as the first of a batch on Bayesian model comparison. No anthology topic holds it: it is a tutorial on statistical evidence for experimental psychologists, and it carries no instruction for machine-learning practice.
It is the record’s readable statement of the Savage–Dickey density ratio. The original is Dickey’s 1971 paper (LIT-610), which the record has now read, and this tutorial’s account of it holds up. The authors credit the ratio to Dickey and Lientz (1970), who attributed it to Savage, and cite Dickey (1971) for the name. Dickey’s own text agrees: he calls the result “Savage’s Density Ratio” (his Theorem, Eqs. 3.8–3.9), credits its proof to Dickey and Lientz and its discovery to Savage’s unpublished notes, and states it for any likelihood continuous in the parameters, not only for normal models. Their continuity condition, that the nuisance prior under the alternative tends to the one under the null as φ → φ0, is Dickey’s condition (3.8) in the limit form he uses himself when he applies it (§4.1.2, §5.1.2). Dickey dates the Dickey and Lientz paper 1968 in his text but places it in the 1970 volume of the Annals, so the 1970 here is the publication year. What this tutorial builds on Dickey is the MCMC route, logspline density estimation at one point, order-restricted and hierarchical tests. Appendix A then derives the ratio in four lines from the continuity condition and Bayes’ rule, which is Dickey’s one-line proof written out.
Where it sits in the batch:
- Friston & Penny (LIT-620) show that the Savage–Dickey ratio is one special case of a more general identity: the evidence for any model formed from a full model by changing its prior is the full evidence times the posterior expectation of the prior ratio. Savage–Dickey is the case where the reduced prior is a point mass. Neither paper cites the other.
- Bayesian model reduction (LIT-614) calls itself a generalisation of the Savage–Dickey ratio “to any new prior”. It is the same identity as Friston & Penny’s, worked out for more distributions.
- MacKay (LIT-623) supplies the reason Bayes factors penalise complexity, the Occam factor. This tutorial gives the same argument in its §2.2.1, citing MacKay’s 2003 textbook.
The prior-sensitivity point is where the two halves of the batch meet. Here the prior ordinate at φ0 sets the Bayes factor directly. In MacKay the Occam factor is the ratio of posterior to prior accessible volume. For a single parameter whose posterior is a peak inside a flat prior, these are two readings of the same quantity.