This article discusses the dual interpretation of the Jeffreys-Lindley paradox associated with Bayesian posterior probabilities and Bayes factors, both as a differentiation between frequentist and Bayesian statistics and as a pointer to the difficulty of using improper priors while testing. I stress the considerable impact of this paradox on the foundations of both classical and Bayesian statistics. While assessing existing resolutions of the paradox, I focus on a critical viewpoint of the paradox discussed by Spanos in Philosophy of Science.

This paper is an extended review of the book Error and Inference, edited by Deborah Mayo and Aris Spanos, about their frequentist and philosophical perspective on testing of hypothesis and on the criticisms of alternatives like the Bayesian approach.

Since the choice of a particular loss function strongly influences the resulting inference, it seems necessary to rely on “intrinsic” losses when no information is available about the utility function of the decision-maker, rather than to call for classical losses like the squared error loss. Since this setting is quite similar to the derivation of noninformative priors in Bayesian analysis, we first recall the conditions of this derivation and deduce from these conditions some requirements on the intrinsic losses. It then (...) appears that these loss functions should only depend on the sampling distribution and that they should be independent of the parameterization of the distribution. The resulting estimators are therefore transformation equivariant. We study the properties of two natural intrinsic losses, namely entropy and Hellinger losses, and show that they can be expressed in closed form for exponential families. Moreover, the entropy loss also provides analytic expressions of Bayes estimators under conjugate priors; the derivation of Bayes estimators associated with the Hellinger loss is more cumbersome, as shown in Poisson and Gamma cases, while leading to similar estimators. (shrink)