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LIT-610: The Weighted Likelihood Ratio, Linear Hypotheses on Normal Location Parameters

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StatusActive — read in full on 2026-10-03 (NOTE-508) from an owner-supplied JSTOR scan, via OCR checked against the page images; worth reading as the original general statement of the Savage–Dickey ratio: for a sharp hypothesis η(θ) = η_H with positive prior mass, the Bayes factor is the posterior-to-prior density ratio of η at η_H under the alternative (3.9), exactly, if the null's nuisance prior is the alternative's prior conditioned at η_H (3.8), stated for any likelihood continuous in θ before the normal-theory applications.
Version2
Tagsmodel-comparison, probabilistic-modeling
Extended byLIT-609 — Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method, LIT-620 — Post hoc Bayesian model selection
Cited as “paper” byNOTE-508 — The Weighted Likelihood Ratio, Linear Hypotheses on Normal Location Parameters
Cited as “source” byTHEORY-081 — The evidence of any model formed from a full model by changing only its prior is the full evidence times the full posterior expectation of the prior ratio, so the Savage–Dickey ratio is its point-mass case and choosing which parameters to keep is evidence maximisation over the prior

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James M. Dickey (1971), The Annals of Mathematical Statistics 42(1):204–223 — DOI-10.1214/aoms/1177693507

Key takeaways

  • The theorem (§3, (3.8)–(3.9)). Reparameterise θ as (η, ζ) and test the sharp hypothesis H: η = η_H, given positive prior mass, against H̄: η ≠ η_H. The prior is a point mass on H inside a continuous prior f ((3.2), (3.5)). If the nuisance prior under H is the alternative’s prior conditioned at η_H, g(ζ) = f(η_H, ζ)/∫ f(η_H, ζ) dζ (3.8), then the weighted likelihood ratio, the Bayes factor for H, is L_D(H) = P′(η_H | H̄, D)/P′(η_H | H̄) (3.9), exactly. The proof is one substitution of (3.8) into the null’s marginal likelihood (3.6).
  • It is general. The likelihood is any φ(D | θ) continuous in θ, η can be a vector, and H any smooth sharp hypothesis. Normal location parameters are the applications of §§4–5, not a condition of the theorem.
  • The condition, and where continuity comes in. Dickey makes f meaningful on H by defining densities as elementary derivatives, limits of probability over volume on shrinking balls (3.3)–(3.4), and in the applications states (3.8) as a limit: the null’s nuisance prior is “the limiting distribution” of the alternative’s as the tested parameter goes to its null value (§4.1.2, §5.1.2). That is the continuity condition Wagenmakers et al. (LIT-609) write down, though Dickey never calls it continuity: he uses the word only for the likelihood (§2) and for the utility version of the theorem (3.9*).
  • The attribution is Dickey’s own. He calls the result “Savage’s Density Ratio”, says Savage found it as an exact general formula in unpublished 1963 notes, and credits the proof to Dickey and Lientz. Approximate forms go back to Jeffreys (1948), Lindley (1961) and Savage (1959, 1961).
  • Closed forms from conjugate priors. A normal mean with known variance gives L_D(H) = (n1/n0)^½ exp(½σ⁻²Q) (4.6)–(4.7), which goes to infinity as the alternative’s prior gets vague (n0 → 0). Unknown variance gives a ratio of Student-t ordinates (4.19). A linear hypothesis C_Hβ = η_H in the linear normal model gives (5.24), a Bayesian replacement for the F test built from prior and posterior “sums of squares between”, and with σ² unknown an F-density ordinate in the stable-estimation limit (5.45). A new conjugate family (4.23) lets the mean and variance be independent a priori, at the price of Behrens–Fisher densities.
  • The hedges. Savage’s stable-estimation approximation needs only one prior ordinate, but “should not form the basis of a formal decision procedure without an evaluation of the quality of the approximation” (§1) and is unlikely to be realistic with many parameters (§3). Positive mass on a sharp hypothesis is “proposed as a good approximation to many prior opinions” (§3), not defended. There are no numerical examples.

Standing in the record

Filed on 2026-10-03 at the owner’s request, in the model-comparison batch, and read the same day from a JSTOR scan the owner supplied (NOTE-508). No anthology topic holds it, and it carries no instruction for machine-learning practice.

It is the original of the result the record reads in Wagenmakers et al. (LIT-609) and in Friston & Penny (LIT-620), and it agrees with both. The question this entry used to leave open is settled: Dickey states the ratio for general sharp hypotheses, with any continuous likelihood, so when Friston & Penny and Bayesian model reduction (LIT-614) say they generalise “the Savage–Dickey ratio”, what they generalise is the reduced prior, from a point mass to any narrower prior, not the setting. The point-mass clause of THEORY-081 now cites it directly.

Two details of the paper’s own citations are worth knowing before quoting it. It dates Dickey & Lientz 1968 in the text, but its reference list places that paper in Ann. Math. Statist. 41, the 1970 volume, which is the year LIT-609 gives. And the invariance result it cites as Dickey & Lientz (1969) has no entry in its reference list.

Read it for §3 (pp. 207–210). The rest is conjugate algebra for normal models, the Behrens–Fisher cases and a decision-theoretic generalisation to utilities ((2.14), (3.9*)).

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