Related papers: The Base Measure Problem and its Solution
We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data (quantities of interest). The solution, given as a probability measure, is…
The histogram method is a powerful non-parametric approach for estimating the probability density function of a continuous variable. But the construction of a histogram, compared to the parametric approaches, demands a large number of…
Probabilistic programming and statistical computing are vibrant areas in the development of the Julia programming language, but the underlying infrastructure dramatically predates recent developments. The goal of MeasureTheory.jl is to…
We consider the problem of fitting a probability density function when it is constrained to have a given number of modal intervals. We propose a dynamic programming approach to solving this problem numerically. When this number is not…
Over the last decade, a series of applied mathematics papers have explored a type of inverse problem--called by a variety of names including "inverse sensitivity", "pushforward based inference", "consistent Bayesian inference", or…
This paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second-order statistics. It is an inverse problem admitting multiple solutions, and selection of a…
Information theory is built on probability measures and by definition a probability measure has total mass 1. Probability measures are used to model uncertainty, and one may ask how important it is that the total mass is one. We claim that…
This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for…
Consider the problem of estimating the $\gamma$-level set $G^*_{\gamma}=\{x:f(x)\geq\gamma\}$ of an unknown $d$-dimensional density function $f$ based on $n$ independent observations $X_1,...,X_n$ from the density. This problem has been…
In classical density (or density-functional) estimation, it is standard to assume that the underlying distribution has a density with respect to the Lebesgue measure. However, when the data distribution is a mixture of continuous and…
Parameter identification problems are formulated in a probabilistic language, where the randomness reflects the uncertainty about the knowledge of the true values. This setting allows conceptually easily to incorporate new information, e.g.…
Probability density estimation is a core problem of statistics and signal processing. Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely…
This paper develops an inconsistency measure on conditional probabilistic knowledge bases. The measure is based on fundamental principles for inconsistency measures and thus provides a solid theoretical framework for the treatment of…
The Dirichlet process mixture model and more general mixtures based on discrete random probability measures have been shown to be flexible and accurate models for density estimation and clustering. The goal of this paper is to illustrate…
We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term,…
Univariate and multivariate normal probability distributions are widely used when modeling decisions under uncertainty. Computing the performance of such models requires integrating these distributions over specific domains, which can vary…
Estimating the ratio of two probability densities from a finite number of observations is a central machine learning problem. A common approach is to construct estimators using binary classifiers that distinguish observations from the two…
We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on…
A method is discussed that allows combining sets of differential or inclusive measurements. It is assumed that at least one measurement was obtained with simultaneously fitting a set of nuisance parameters, representing sources of…
A total set of states for which we have no resolution of the identity (a `pre-basis'), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes…