arXiv subjects

Raphaël Trésor

Publications and source records attributed to Raphaël Trésor.

3 recordsLinked to original sources

arXiv2026

A resolution of the Borel-Kolmogorov Paradox via the Maximum Entropy Principle

Bayesian updating routinely conditions on events of prior probability zero, such as an exact observation of a continuous parameter or a parameter confined to a submanifold. Two equally natural parametrizations of the same event can then return different posteriors, which is the Borel--Kolmogorov paradox. We add a metric to the probabilistic model and define the posterior given a closed null set as the limit of Maximum Entropy solutions under a vanishing constraint on the distance to that set. We give a sufficient condition for existence and prove that, given this metric, the posterior is unique and invariant under measure-preserving isometries. It recovers the textbook Bayes formulas where these apply unambiguously, on positive-measure sets and in Euclidean models. Each stage of the limit has an explicit density that depends only on the distance to the conditioning set, so the posterior can be computed with standard tools for Kullback--Leibler minimization. The construction does not remove the choice of a metric but makes it an explicit modelling input. In a controlled experiment where the geometry of the measurement is known, we show that a Bayesian analysis ignoring this geometry can yield uncalibrated posteriors. On real palaeomagnetic data, where the geometry is disputed, we show how candidate metrics can be compared within the same Bayesian framework.

math.ST↗

arXiv2026

Reformulation Invariance and the Axiomatic Foundations of Inference

Maximum entropy, Bayesian updating, and exponential-family estimation are all instances of a common inference principle: selecting the measure or distribution that minimizes a divergence subject to the available constraints. Which divergence to use is usually decided by analytic convenience, by empirical performance, or by a set of axioms chosen to single it out, leaving open a basic question: why one divergence and not another? We answer it from a single requirement: an inference method should return the same answer whenever the same problem is presented in an equivalent form, for instance, after simply renaming its parts. This requirement alone forces inference to be the minimisation of a classical divergence, and each further reformulation it must respect tightens the admissible family one notch, narrowing the broad f-divergences to the α-divergences and finally to the single Kullback-Leibler (KL) divergence. Mathematically, inference is recast from minimising a numerical functional to selecting a least element under a preorder on positive measures, a divergence being merely one numerical scale that reproduces that preorder. The reformulations are the morphisms of a category of inference problems, and the invariance requirement says the inference operator is a covariant functor into the category of statistical models of Cencov, mirroring his characterisation of the Fisher metric. The representation is proved on finite spaces and lifted to general measurable spaces by an elementary closure, covering discrete and continuous spaces alike. Earlier axiomatisations, such as those of Shore-Johnson and Csiszar, postulate their consistency axioms directly and only on finite alphabets; here the axioms follow from reformulation invariance alone.

math.ST↗

arXiv2026

A Probabilistic Generative Model for Spectral Speech Enhancement

Speech enhancement in hearing aids remains a difficult task in nonstationary acoustic environments, mainly because current signal processing algorithms rely on fixed, manually tuned parameters that cannot adapt in situ to different users or listening contexts. This paper introduces a unified modular framework that formulates signal processing, learning, and personalization as Bayesian inference with explicit uncertainty tracking. The proposed framework replaces ad hoc algorithm design with a single probabilistic generative model that continuously adapts to changing acoustic conditions and user preferences. It extends spectral subtraction with principled mechanisms for in-situ personalization and adaptation to acoustic context. The system is implemented as an interconnected probabilistic state-space model, and inference is performed via variational message passing in the \texttt{RxInfer.jl} probabilistic programming environment, enabling real-time Bayesian processing under hearing-aid constraints. Proof-of-concept experiments on the \emph{VoiceBank+DEMAND} corpus show competitive speech quality and noise reduction with 85 effective parameters. The framework provides an interpretable, data-efficient foundation for uncertainty-aware, adaptive hearing-aid processing and points toward devices that learn continuously through probabilistic inference.

cs.SD↗