Last week I posted a manuscript on arXiv entitled On decision-theoretic justifications for Bayesian hypothesis testing through credible sets. A few days later, a discussion of it appeared on Xi’ans’ Og. I’ve read papers and books by Christian Robert with great interest and have been a follower of his “Og” for quite some time, and so was honoured and excited when he chose to blog about my work. I posted a comment to his blog post, but for some reason or other it has not yet appeared on the site. I figured that I’d share my thoughts on his comments here on my own blog for the time being.

The main goal of the paper was to discuss decision-theoretic justifications for testing the point-null hypothesis Θ_{0}={θ_{0}} against the alternative Θ_{1}={θ: θ≠θ_{0}} using credible sets. In this test procedure, Θ_{0} is rejected if θ_{0 }is not in the credible set. This is not the standard solution to the problem, but certainly not uncommon (I list several examples in the introduction to the paper). Tests of composite hypotheses are also discussed.

Judging from his blog post, Xi’an is not exactly in love with the manuscript. (Hmph! What does he know about Bayesian decision theory anyway? It’s not like he wrote the book on… oh, wait.) To some extent however, I think that his criticism is due to a misunderstanding.

Before we get to the misunderstanding though: Xi’an starts out by saying that he doesn’t like point-null hypothesis testing, so the prior probability that he would like it was perhaps not that great. I’m not crazy about point-null hypotheses either, but the fact remains that they are used a lot in practice and that there are situations where they are very natural. Xi’an himself gives a few such examples in Section 5.2.4 of The Bayesian Choice, as do Berger and Delampady (1987).

What is not all that natural, however, is the standard Bayesian solution to point-null hypothesis testing. It requires a prior with a mass on θ_{0}, which seems like a very artificial construct to me. Apart from leading to such complications as Lindley’s paradox, it leads to very partial priors. Casella and Berger (1987, Section 4) give an example where the seemingly impartial prior probabilities P(θ_{0})=1/2 and P(Θ_{1})=1/2 actually yield a test with strong bias towards the null hypothesis. One therefore has to be extremely careful when applying the standard tests of point-null hypotheses, and carefully think about what the point-mass really means and how it affects the conclusions.

Tests based on credible sets, on the other hand, allows us to use a nice continuous prior for θ. It can, unlike the prior used in the standard solution, be non-informative. As for informative priors, it is often easier to construct a continuous prior based on expert opinion than it is to construct a mixed prior.

Theorem 2 of my paper presents a weighted 0-1-type loss function that leads to the acceptance region being the central (symmetric) credible interval. The prior distribution is assumed to be continuous, with no point-mass in θ_{0}. The loss is constructed using directional conclusions, meaning that when θ_{0} is rejected, it is rejected in favour of either {θ: θ<θ_{0}} or {θ: θ>θ_{0}}, instead of simply being rejected in favour of {θ: θ≠θ_{0}}. Indeed, this is how credible and confidence intervals are used in practice: if θ_{0 }is smaller than all values in the interval, then θ_{0 }is rejected and we conclude that θ>θ_{0}. The theorem shows that tests based on central intervals can be viewed as a solution to the directional three-decision problem – a solution that does not require a point-mass for the null hypothesis. I therefore do not agree with Xi’an’s comment that “[tests using credible sets] cannot bypass the introduction of a prior mass on Θ_{0}“. While a test traditionally only has one way to reject the null hypothesis, allowing two different directions in which Θ_{0 }can be rejected seems perfectly reasonable for the point-null problem.

Regarding this test, Xi’an writes that it “essentially [is] a composition of two one-sided tests, […], so even at this face-value level, I do not find the result that convincing”. But any (?) two-sided test can be said to be a composition of two one-sided tests (and therefore implicitly includes a directional conclusion), so I’m not sure why he regards it as a reason to remain unconvinced about the validity of the result.

As for the misunderstanding, Theorem 3 of the paper deals with one-sided hypothesis tests. It was not meant as an attempt to solve the problem of testing point-null hypotheses, but rather to show how credible sets can be used to test composite hypotheses – as was Theorem 4. Xi’an’s main criticism of the paper seems to be that the tests in Theorems 3 and 4 fail for point-null hypotheses, but they were never meant to be used for such hypotheses in the first place. After reading his comments, I realized that this might not have been perfectly clear in the first draft of the paper. In particular, the abstract seemed to imply that the paper only dealt with point-null hypotheses, which is not the case. In the submitted version (not yet uploaded to arXiv), I’ve tried to make the fact that both point-null and composite hypotheses are studied clearer.

There are certainly reasons to question the use of credible sets for testing, chief among them being that the evidence against Θ_{0 }is evaluated in a roundabout way. On the other hand, credible sets are reasonably easy to compute and tend to have favourable properties in frequentist analysis. It seems to me that a statistician that would like to use a method that is reasonable both in Bayesian and frequentist inference would want to consider tests based on credible sets.

xianblog

Sorry about your lost comments Måns, I eventually found them in the spam box! Along with 296 others, which explains why I never check that box…!

Måns

I know but too well how annoying spam comments can be! Although I find it quite amusing that my comment was marked as being spam.

xianblog

Just a naïve question: what happens when you accept the null hypothesis? If you keep a Bayesian perspective, you need to put a prior under the null, however this prior was not used in choosing between the null and the alternative…

Måns

To my mind, that's the beauty of it all! All the prior mass (and hence also the posterior mass) is on {θ: θ>θ_0} and {θ: θ<θ_0}. This seems reasonable for most point-null problems, where we don't _really_ believe that θ=θ_0. The value θ_0 is not given special weight by putting a point-mass on it.

If the evidence is non-conclusive, we're not sure whether θ>θ_0 or θ<θ_0, and so "believe in" Θ_0 for the time being. In a sense, accepting Θ_0 corresponds to a "no decision"-decision: of course θ is not exactly θ_0, but given the data at hand we can't say whether it is greater or less than θ_0. This seems to me like perhaps the only reasonable way to interpret most point-null problems.

Måns

This is reflected in the loss function in Theorem 2: falsely accepting Θ_0 is associated with a (much) lower loss than is falsely accepting either of the alternatives. (There are many situations in which such a loss would be applicable.) If we accept Θ_0 then our loss is non-zero with probability 1. But sometimes it is better to accept a certain loss than to risk a higher loss.

Måns

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