covariant priors, Jeffreys and paradoxes
“If no information is available, π(α|M) must not deliver information about α.”
In a recent arXival apparently submitted to Bayesian Analysis, Giovanni Mana and Carlo Palmisano discuss of the choice of priors in metrology. Which reminded me of this meeting I attended at the Bureau des Poids et Mesures in Sèvres where similar debates took place, albeit being led by ferocious anti-Bayesians! Their reference prior appears to be the Jeffreys prior, because of its reparameterisation invariance.
“The relevance of the Jeffreys rule in metrology and in expressing uncertainties in measurements resides in the metric invariance.”
This, along with a second order approximation to the Kullback-Leibler divergence, is indeed one reason for advocating the use of a Jeffreys prior. I at first found it surprising that the (usually improper) prior is used in a marginal likelihood, as it cannot be normalised. A source of much debate [and of our alternative proposal].
“To make a meaningful posterior distribution and uncertainty assessment, the prior density must be covariant; that is, the prior distributions of different parameterizations must be obtained by transformations of variables. Furthermore, it is necessary that the prior densities are proper.”
The above quote is quite interesting both in that the notion of covariant is used rather than invariant or equivariant. And in that properness is indicated as a requirement. (Even more surprising is the noun associated with covariant, since it clashes with the usual notion of covariance!) They conclude that the marginal associated with an improper prior is null because the normalising constant of the prior is infinite.
“…the posterior probability of a selected model must not be null; therefore, improper priors are not allowed.”
Maybe not so surprisingly given this stance on improper priors, the authors cover a collection of “paradoxes” in their final and longest section: most of which makes little sense to me. First, they point out that the reference priors of Berger, Bernardo and Sun (2015) are not invariant, but this should not come as a surprise given that they focus on parameters of interest versus nuisance parameters. The second issue pointed out by the authors is that under Jeffreys’ prior, the posterior distribution of a given normal mean for n observations is a t with n degrees of freedom while it is a t with n-1 degrees of freedom from a frequentist perspective. This is not such a paradox since both distributions work in different spaces. Further, unless I am confused, this is one of the marginalisation paradoxes, which more straightforward explanation is that marginalisation is not meaningful for improper priors. A third paradox relates to a contingency table with a large number of cells, in that the posterior mean of a cell probability goes as the number of cells goes to infinity. (In this case, Jeffreys’ prior is proper.) Again not much of a bummer, there is simply not enough information in the data when faced with a infinite number of parameters. Paradox #4 is the Stein paradox, when estimating the squared norm of a normal mean. Jeffreys’ prior then leads to a constant bias that increases with the dimension of the vector. Definitely a bad point for Jeffreys’ prior, except that there is no Bayes estimator in such a case, the Bayes risk being infinite. Using a renormalised loss function solves the issue, rather than introducing as in the paper uniform priors on intervals, which require hyperpriors without being particularly compelling. The fifth paradox is the Neyman-Scott problem, with again the Jeffreys prior the culprit since the estimator of the variance is inconsistent. By a multiplicative factor of 2. Another stone in Jeffreys’ garden [of forking paths!]. The authors consider that the prior gives zero weight to any interval not containing zero, as if it was a proper probability distribution. And “solve” the problem by avoid zero altogether, which requires of course to specify a lower bound on the variance. And then introducing another (improper) Jeffreys prior on that bound… The last and final paradox mentioned in this paper is one of the marginalisation paradoxes, with a bizarre explanation that since the mean and variance μ and σ are not independent a posteriori, “the information delivered by x̄ should not be neglected”.