Related papers: Approximation with Independent Variables

Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over…

Approximate inference in probability models is a fundamental task in machine learning. Approximate inference provides powerful tools to Bayesian reasoning, decision making, and Bayesian deep learning. The main goal is to estimate the…

Given a sample of i.i.d. high-dimensional centered random vectors, we consider a problem of estimation of their covariance matrix $\Sigma$ with an additional assumption that $\Sigma$ can be represented as a sum of a few Kronecker products…

Let $X$ be a random vector valued in $\mathbb{R}^{m}$ such that $\|X\|_{2} \le 1$ almost surely. For every $k\ge 3$, we show that there exists a sigma algebra $\mathcal{F}$ generated by a partition of $\mathbb{R}^{m}$ into $k$ sets such…

A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The…

Let $X$ and $Y$ be two real-valued random variables. Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be independent identically distributed copies of $(X,Y)$. Suppose there are two players A and B. Player A has access to $X_{1},X_{2},\ldots$ and…

Spherical symmetry arguments are used to produce a general device to convert identities and inequalities for the $p$th absolute moments of real-valued random variables into the corresponding identities and inequalities for the $p$th moments…

Maximum Likelihood Estimation of continuous variable models can be very challenging in high dimensions, due to potentially complex probability distributions. The existence of multiple interdependencies among variables can make it very…

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a…

We show that, for two non-trivial random variables X and Y under a sublinear expectation space, if X is independent from Y and Y is independent from X, then X and Y must be maximally distributed.

We construct meta-intransitive systems of independent random variables of any finite order from basic tuple of random variables which generalize intransitive dice. Under this construction, the equality of some linear functional is…

For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…

Consider a random matrix $H:\mathbb{R}^n\longrightarrow\mathbb{R}^m$. Let $D\geq2$ and let $\{W_l\}_{l=1}^{p}$ be a set of $k$-dimensional affine subspaces of $\mathbb{R}^n$. We ask what is the probability that for all $1\leq l\leq p$ and…

Bergsma (2006) proposed a covariance $\kappa$(X,Y) between random variables X and Y. He derived their asymptotic distributions under the null hypothesis of independence between X and Y. The non-null (dependent) case does not seem to have…

A copula of continuous random variables $X$ and $Y$ is called an \emph{implicit dependence copula} if there exist functions $\alpha$ and $\beta$ such that $\alpha(X) = \beta(Y)$ almost surely, which is equivalent to $C$ being factorizable…

Statistical inference in high-dimensional settings is challenging when standard unregularized methods are employed. In this work, we focus on the case of multiple correlated proportions for which we develop a Bayesian inference framework.…

We reveal the phenomenon that ``naive'' multivariate local polynomial regression can adapt to local smooth lower dimensional structure in the sense that it achieves the optimal convergence rate for nonparametric estimation of regression…

Let X_1, X_2,..., X_n be a sequence of independent random variables, let M be a rearrangement invariant space on the underlying probability space, and let N be a symmetric sequence space. This paper gives an approximate formula for the…

The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples…

The inferential models (IM) framework provides prior-free, frequency-calibrated, posterior probabilistic inference. The key is the use of random sets to predict unobservable auxiliary variables connected to the observable data and unknown…