Resolution of the Borel-Kolmogorov Paradox via the Maximum Entropy Principle

TL;DR

The paper resolves the Borel–Kolmogorov paradox by introducing a metric-based MaxEnt framework that extends conditional probability to null-measure events in standard Borel spaces. By linking relative-entropy minimization under distance-to-set constraints to a limiting posterior, it yields a unique Bayes-like update that depends on the underlying metric and topology, and it recovers classical Bayes’ rule when conditioning on sets of positive measure. The authors demonstrate the geometric nature of the paradox via a sphere example: with geodesic distance the MaxEnt posterior on great circles is uniform, while non-invariant distances yield metric-dependent posteriors, clarifying why multiple posteriors can arise. This framework provides a principled, transferable basis for Bayesian inference in metrizable spaces, and it unifies classical conditional probability with the Lebesgue-consistent Bayes update under MaxEnt, offering a coherent axiomatic interpretation.

Abstract

This paper presents a rigorous resolution of the Borel-Kolmogorov paradox using the Maximum Entropy Principle. We construct a metric-based framework for Bayesian inference that uniquely extends conditional probability to events of null measure. The results unify classical Bayes' rules and provide a robust foundation for Bayesian inference in metric spaces.

Figures (1)

• Figure 1: Measure-preserving Mercator projection. We map the sphere $\mathbb{S}^2$ (see Section \ref{['subsec:set-up']}) onto the domain $[-\frac{\pi}{2}, \frac{\pi}{2}] \times ]-\pi, \pi ]$ via the Mercator map defined in equation \ref{['eq:chart on sphere']}. When $\mathbb{S}^2$ is equipped with the metric $\widetilde{d}$, defined in equation \ref{['eq: non rotation invariant distance']}, the map \ref{['eq:chart on sphere']} onto the Euclidean metric space $[-\frac{\pi}{2}, \frac{\pi}{2}] \times ]-\pi, \pi ]$ is isometric. The distortion of both distance and area introduced by the map is illustrated by the transformation of the triangle (in red) in Figure \ref{['fig:sphere parametrisation']} into the triangles and rectangle (in red) shown in Figure \ref{['fig:Projection']}. The push-forward of the uniform measure $\nu$ (see equation \ref{['eq:area-formula']}) under the map is given by equation \ref{['eq: uniform measure on sphere']}. Although the original measure on $\mathbb{S}^2$ is uniform (Figure \ref{['fig:sphere parametrisation']}), the push-forward measure on the Euclidean space (Figure \ref{['fig:Projection']}), which preserves the geometric structure, is not uniform. These observations lead to two key conclusions. First, Geometric considerations are essential for a coherent definition of Bayesian inference. Second, the arguments used to define the Bayes posterior on every great circles of $(\mathbb{S}^2, \nu)$ as uniform are inherently tied to the geodesic distance $d_g$ (defined in Section \ref{['subsec:set-up']}) and cannot be derived solely from measure-theoretic principles (see Section \ref{['subsec:BK-Paradox-interpretation']}).

Theorems & Definitions (49)

• Definition 1: Relative Entropy or Kullback Leibler divergence
• Proposition 1: cover_elements_2001Theorem 2.6.3
• Definition 2: MaxEnt with Linear Constraints
• Definition 3: Support
• Theorem 1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof