Related papers: The numeraire e-variable and reverse information p…

Optimal transport has found numerous applications across data science, many of which require differentiating the optimal transport map with respect to the underlying probability densities in the Fr\'echet sense. In this work, we show that…

We introduce the concepts of max-closedness and numeraires of convex subsets in the nonnegative orthant of the topological vector space of all random variables built over a probability space, equipped with a topology consistent with…

Let $P\in\Z[n]$ with $P(0)=0$ and $\VE>0$. We show, using Fourier analytic techniques, that if $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and $A\subseteq\{1,\...,N\}$, then there must exist $n\in\N$ such that \[\frac{|A\cap…

Confidence sequences, anytime p-values (called p-processes in this paper), and e-processes all enable sequential inference for composite and nonparametric classes of distributions at arbitrary stopping times. Examining the literature, one…

The reciprocal of $e^{-x}$ has a power series about $0$ in which all coefficients are non-negative. Gessel [Reciprocals of exponential polynomials and permutation enumeration, Australas. J. Combin., 74, 2019] considered truncates of the…

We develop E-variables for testing whether two or more data streams come from the same source or not, and more generally, whether the difference between the sources is larger than some minimal effect size. These E-variables lead to exact,…

Approximate inference via information projection has been recently introduced as a general-purpose approach for efficient probabilistic inference given sparse variables. This manuscript goes beyond classical sparsity by proposing efficient…

Possible parameter values in a random sampling model are shown by definition to have uniform base-rate prior probabilities. This allows a frequentist posterior probability distribution to be calculated for such possible parameter values…

In many data analyses, each measurement may come with a simple yes/no correction; for example, belonging to one of two populations or being contaminated or not. Ignoring such binary effects may bias the results, while accounting for them…

Let $X$ be a random variable with finite second moment. We investigate the inequality: $P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}\ge P\{|Z|\le 1\}$, where $Z$ is a standard normal random variable. We prove that this inequality holds for many…

E-variables enable safe and anytime-valid inference, with log-optimal e-variables given by the likelihood ratio of the least favorable distributions (LFDs) when they exist in composite settings. While this unconstrained theory is well…

Probability integral transforms (PITs) and empirical $p$-values are widely used to assess the calibration of predictive distributions. While exact PIT values are uniformly distributed under correct model specification, practical…

We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$ without complex multiplication and with conductor $N$. For each positive integer $m$, the action of the absolute Galois group…

To alleviate the data requirement for training effective binary classifiers in binary classification, many weakly supervised learning settings have been proposed. Among them, some consider using pairwise but not pointwise labels, when…

We introduce the binary value principle which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional…

We consider a nonlinear polynomial regression model in which we wish to test the null hypothesis of structural stability in the regression parameters against the alternative of a break at an unknown time. We derive the extreme value…

Let cp(R) be the probability that two random elements of a finite ring R commute and zp(R) the probability that the product of two random elements in R is zero. We show that if cp(R)=e, then there exists a Lie-ideal D in the Lie-ring…

In astrophysical (inverse) regression problems it is an important task to decide whether a given parametric model describes the observational data sufficiently well or whether a non-parametric modelling becomes necessary. However, in…

In this paper, we address the question of information preservation in ill-posed, non-linear inverse problems, assuming that the measured data is close to a low-dimensional model set. We provide necessary and sufficient conditions for the…