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Probability metrics constitute an important tool in probability theory and statistics \cite{DKS91}, \cite{R91}, \cite{Z83} as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with…
Complex scientific models where the likelihood cannot be evaluated present a challenge for statistical inference. Over the past two decades, a wide range of algorithms have been proposed for learning parameters in computationally feasible…
Random percolation can be fully interpreted as a confining pure gauge theory. With numerical high-precision measurements of Polyakov-Polyakov correlators at finite temperature, we could well observe the presence of shape effects due to…
Probabilistic metrology attempts to improve parameter estimation by occasionally reporting an excellent estimate and the rest of the time either guessing or doing nothing at all. Here we show that probabilistic metrology can never improve…
We explore the interplay between random and deterministic phenomena using a representation of uncertainty based on the measure-theoretic concept of outer measure. The meaning of the analogues of different probabilistic concepts is…
Consider the following prediction problem. Assume that there is a block box that produces bits according to some unknown computable distribution on the binary tree. We know first $n$ bits $x_1 x_2 \ldots x_n$. We want to know the…
Information theoretic quantities play an important role in various settings in machine learning, including causality testing, structure inference in graphical models, time-series problems, feature selection as well as in providing privacy…
Consider universal data compression: the length $l(x^n)$ of sequence $x^n\in A^n$ with finite alphabet $A$ and length $n$ satisfies Kraft's inequality over $A^n$, and $-\frac{1}{n}\log \frac{P^n(x^n)}{Q^n(x^n)}$ almost surely converges to…
We use the concept of excursions for the prediction of random variables without any moment existence assumptions. To do so, an excursion metric on the space of random variables is defined which appears to be a kind of a weighted…
We present a detailed motivation for and definition of the contextual values of an observable, which were introduced by Dressel et al. [Phys. Rev. Lett. 104 240401 (2010)]. The theory of contextual values extends the well-established theory…
We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
We consider the problem of approximating the stationary distribution of an ergodic Markov chain given a set of sampled transitions. Classical simulation-based approaches assume access to the underlying process so that trajectories of…
This paper investigates universal polar coding schemes. In particular, a notion of ordering (called convolutional path) is introduced between probability distributions to determine when a polar compression (or communication) scheme designed…
We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant 0}$ with measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily…
We study the expectation-maximization (EM) algorithm for general latent-variable models under (i) distributional misspecification and (ii) nonidentifiability induced by a group action. We formulate EM on the quotient parameter space and…
We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov's system of axioms. We show that this new measure verifies the usual…
A burgeoning paradigm in algorithm design is the field of algorithms with predictions, in which algorithms can take advantage of a possibly-imperfect prediction of some aspect of the problem. While much work has focused on using predictions…
The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We…
We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here "random" means that the coefficients of the polynomials defining the complete intersections are sampled uniformly…
Aleatoric uncertainty quantification seeks for distributional knowledge of random responses, which is important for reliability analysis and robustness improvement in machine learning applications. Previous research on aleatoric uncertainty…