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Standard formulations of supervenience typically treat higher level properties as point valued facts strictly fixed by underlying base states. However, in many scientific domains, from statistical mechanics to machine learning, basal…

We obtain the posterior distribution of a random process conditioned on observing the empirical frequencies of a finite sample path. We find under a rather broad assumption on the "dependence structure" of the process, {\em c.f.}…

Probabilistic Graphical Models (PGMs) encode conditional dependencies among random variables using a graph -nodes for variables, links for dependencies- and factorize the joint distribution into lower-dimensional components. This makes PGMs…

The basic concepts of category theory are developed and examples of them are presented to illustrate them using measurement theory and probability theory tools. Motivated by Perrone's workarXiv:1912.10642 where notes on category theory are…

We tackle the problem of conditioning probabilistic programs on distributions of observable variables. Probabilistic programs are usually conditioned on samples from the joint data distribution, which we refer to as deterministic…

A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of…

This paper begins with a description of methods for estimating image probability density functions that reflects the observation that such data is usually constrained to lie in restricted regions of the high-dimensional image space-not…

The probability folder of Mathlib, Lean's mathematical library, makes a heavy use of Markov kernels. We present their definition and properties and describe the formalization of the disintegration theorem for Markov kernels. That theorem is…

This paper deals with the scenario approach to robust optimization. This relies on a random sampling of the possibly infinite number of constraints induced by uncertainties in the parameters of an optimization problem. Solving the resulting…

An "element-free" probability distribution is what remains of a probability distribution after we forget the elements to which the probabilities were assigned. These objects naturally arise in Bayesian statistics, in situations where…

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov…

We consider a general statistical estimation problem wherein binary labels across different observations are not independent conditioned on their feature vectors, but dependent, capturing settings where e.g. these observations are collected…

The problem of comparing concepts of dependence in general rough sets with those in probability theory had been initiated by the present author in some of her recent papers. This problem relates to the identification of the limitations of…

We consider the task of modeling a dependent sequence of random partitions. It is well-known that a random measure in Bayesian nonparametrics induces a distribution over random partitions. The community has therefore assumed that the best…

We develop a new framework of uncertainty variables to model uncertainty. An uncertainty variable is characterized by an uncertainty set, in which its realization is bound to lie, while the conditional uncertainty is characterized by a set…

From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization…

Categorical random variables are a common staple in machine learning methods and other applications across disciplines. Many times, correlation within categorical predictors exists, and has been noted to have an effect on various algorithm…

We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a…

The title refers to the Free Will Theorem by Conway and Kochen whose flashy formulation is: if experimenters possess free will, then so do particles. In more modest terms, the theorem says that individual pairs of spacelike separated…

Given a set of several inputs into a system (e.g., independent variables characterizing stimuli) and a set of several stochastically non-independent outputs (e.g., random variables describing different aspects of responses), how can one…