Harold Jeffreys’ default Bayes factor [for psychologists]

“One of Jeffreys’ goals was to create default Bayes factors by using prior distributions that obeyed a series of general desiderata.”

The paper Harold Jeffreys’s default Bayes factor hypothesis tests: explanation, extension, and application in Psychology by Alexander Ly, Josine Verhagen, and Eric-Jan Wagenmakers is both a survey and a reinterpretation cum explanation of Harold Jeffreys‘ views on testing. At about the same time, I received a copy from Alexander and a copy from the journal it had been submitted to! This work starts with a short historical entry on Jeffreys’ work and career, which includes four of his principles, quoted verbatim from the paper:

1. “scientific progress depends primarily on induction”;
2. “in order to formalize induction one requires a logic of partial belief” [enters the Bayesian paradigm];
3. “scientific hypotheses can be assigned prior plausibility in accordance with their complexity” [a.k.a., Occam’s razor];
4. “classical “Fisherian” p-values are inadequate for the purpose of hypothesis testing”.

“The choice of π(σ)  therefore irrelevant for the Bayes factor as long as we use the same weighting function in both models”

A very relevant point made by the authors is that Jeffreys only considered embedded or nested hypotheses, a fact that allows for having common parameters between models and hence some form of reference prior. Even though (a) I dislike the notion of “common” parameters and (b) I do not think it is entirely legit (I was going to write proper!) from a mathematical viewpoint to use the same (improper) prior on both sides, as discussed in our Statistical Science paper. And in our most recent alternative proposal. The most delicate issue however is to derive a reference prior on the parameter of interest, which is fixed under the null and unknown under the alternative. Hence preventing the use of improper priors. Jeffreys tried to calibrate the corresponding prior by imposing asymptotic consistency under the alternative. And exact indeterminacy under “completely uninformative” data. Unfortunately, this is not a well-defined notion. In the normal example, the authors recall and follow the proposal of Jeffreys to use an improper prior π(σ)∝1/σ on the nuisance parameter and argue in his defence the quote above. I find this argument quite weak because suddenly the prior on σ becomes a weighting function... A notion foreign to the Bayesian cosmology. If we use an improper prior for π(σ), the marginal likelihood on the data is no longer a probability density and I do not buy the argument that one should use the same measure with the same constant both on σ alone [for the nested hypothesis] and on the σ part of (μ,σ) [for the nesting hypothesis]. We are considering two spaces with different dimensions and hence orthogonal measures. This quote thus sounds more like wishful thinking than like a justification. Similarly, the assumption of independence between δ=μ/σ and σ does not make sense for σ-finite measures. Note that the authors later point out that (a) the posterior on σ varies between models despite using the same data [which shows that the parameter σ is far from common to both models!] and (b) the [testing] Cauchy prior on δ is only useful for the testing part and should be replaced with another [estimation] prior when the model has been selected. Which may end up as a backfiring argument about this default choice.

“Each updated weighting function should be interpreted as a posterior in estimating σ within their own context, the model.”

The re-derivation of Jeffreys’ conclusion that a Cauchy prior should be used on δ=μ/σ makes it clear that this choice only proceeds from an imperative of fat tails in the prior, without solving the calibration of the Cauchy scale. (Given the now-available modern computing tools, it would be nice to see the impact of this scale γ on the numerical value of the Bayes factor.) And maybe it also proceeds from a “hidden agenda” to achieve a Bayes factor that solely depends on the t statistic. Although this does not sound like a compelling reason to me, since the t statistic is not sufficient in this setting.

In a differently interesting way, the authors mention the Savage-Dickey ratio (p.16) as a way to represent the Bayes factor for nested models, without necessarily perceiving the mathematical difficulty with this ratio that we pointed out a few years ago. For instance, in the psychology example processed in the paper, the test is between δ=0 and δ≥0; however, if I set π(δ=0)=0 under the alternative prior, which should not matter [from a measure-theoretic perspective where the density is uniquely defined almost everywhere], the Savage-Dickey representation of the Bayes factor returns zero, instead of 9.18!

“In general, the fact that different priors result in different Bayes factors should not come as a surprise.”

The second example detailed in the paper is the test for a zero Gaussian correlation. This is a sort of “ideal case” in that the parameter of interest is between -1 and 1, hence makes the choice of a uniform U(-1,1) easy or easier to argue. Furthermore, the setting is also “ideal” in that the Bayes factor simplifies down into a marginal over the sample correlation only, under the usual Jeffreys priors on means and variances. So we have a second case where the frequentist statistic behind the frequentist test[ing procedure] is also the single (and insufficient) part of the data used in the Bayesian test[ing procedure]. Once again, we are in a setting where Bayesian and frequentist answers are in one-to-one correspondence (at least for a fixed sample size). And where the Bayes factor allows for a closed form through hypergeometric functions. Even in the one-sided case. (This is a result obtained by the authors, not by Jeffreys who, as the proper physicist he was, obtained approximations that are remarkably accurate!)

“The fact that the Bayes factor is independent of the intention with which the data have been collected is of considerable practical importance.”

The authors have a side argument in this section in favour of the Bayes factor against the p-value, namely that the “Bayes factor does not depend on the sampling plan” (p.29), but I find this fairly weak (or tongue in cheek) as the Bayes factor does depend on the sampling distribution imposed on top of the data. It appears that the argument is mostly used to defend sequential testing.

“The Bayes factor (…) balances the tension between parsimony and goodness of fit, (…) against overfitting the data.”

In fine, I liked very much this re-reading of Jeffreys’ approach to testing, maybe the more because I now think we should get away from it! I am not certain it will help in convincing psychologists to adopt Bayes factors for assessing their experiments as it may instead frighten them away. And it does not bring an answer to the vexing issue of the relevance of point null hypotheses. But it constitutes a lucid and innovative of the major advance represented by Jeffreys’ formalisation of Bayesian testing.