Brittleness of Bayesian inference under finite information in a continuous world

Abstract

We derive, in the classical framework of Bayesian sensitivity analysis, optimal lower and upper bounds on posterior values obtained from Bayesian models that exactly capture an arbitrarily large number of finite-dimensional marginals of the datagenerating distribution and/or that are as close as desired to the data-generating distribution in the Prokhorov or total variation metrics; these bounds show that such models may still make the largest possible prediction error after conditioning on an arbitrarily large number of sample data measured at finite precision. These results are obtained through the development of a reduction calculus for optimization problems over measures on spaces of measures. We use this calculus to investigate the mechanisms that generate brittleness/robustness and, in particular, we observe that learning and robustness are antagonistic properties. It is now well understood that the numerical resolution of PDEs requires the satisfaction of specific stability conditions. Is there a missing stability condition for using Bayesian inference in a continuous world under finite information?Contents 2010 Mathematics Subject Classification: 62A01, 62E20, 62F12, 62F15, 62G20, 62G35.

On the Brittleness of Bayesian Inference

“…This paper summarizes recent results [46,47] that give conditions under which Bayesian inference appears to be nonrobust in the most extreme fashion, in the sense that arbitrarily small changes of the prior and model class lead to arbitrarily large changes of the posterior value of a quantity of interest. We call this extreme nonrobustness "brittleness," and it can be visualized as the smooth dependence of the value of the quantity of interest on the prior breaking into a fine patchwork, in which nearby priors are associated to diametrically opposed posterior values.…”

“…Using the notation of Definition 1, and Π α defined above in terms of the TV or Prokhorov metric, the Brittleness Theorem 6.4 of [47] then reads as follows.…”

“…To study this problem, we introduce a representation space Q (e.g., prototypically, R k ) and a mapping Ψ : A → Q from the subset A ⊆ M(X ) into Q, which can be 4 All results of this article and those in [46,47,48] require some mild technical measure-theoretic and topological assumptions. For example, here it is sufficient if P(Θ) is a Borel subset of a Polish space (a separable completely metrizable space).…”

“…Consequently, when Θ is Polish, X is Polish, and P is injective and measurable with respect to the weak topology, it then follows from Suslin's Theorem that P(Θ) is a Borel subset of the Polish space M(X ). For a thorough investigation of such matters, illustrating the benefits of Polish spaces as the foundation for the framework, see [47].…”

“…Hence, although Bayesian inference is robust in situations where the distributions of all but finitely many generalized moments of the data-generating distribution μ † are known, Theorem 2 suggests that it is brittle when the distributions of only finitely many generalized moments of μ † are known, while infinitely many remain unknown. As an example, it is instructive to observe how Theorem 2, applied to Example 3 in [47,Ex. 4.16], shows that if the data-generating measure has some nonatomic component, then when the number of samples n is large enough and δ small enough, the optimal bounds on posterior values of Φ(μ) = E μ [X], given the distribution Q defined on its moments, are 0 and 1.…”