Related papers: Pointwise defined version of conditional expectati…
When performing Bayesian inference, we frequently need to work with conditional probability densities. For example, the posterior function is the conditional density of the parameters given the data. Some might worry that conditional…
We formulate simple equivalent conditions for the validity of Bayes' formula for conditional densities. We show that for any random variables X and Y (with values in arbitrary measurable spaces), the following are equivalent: 1. X and Y…
In this paper, we examine the distribution and convergence properties of the estimation error $W = X - \hat{X}(Y)$, where $\hat{X}(Y)$ is the Bayesian estimator of a random variable $X$ from a noisy observation $Y = X +\sigma Z$ where…
The concept of conditional expectation is important in applications of probability and statistics in many areas such as reliability engineering, economy, finance, and actuarial sciences due to its property of being the best predictor of a…
We study the conditional distribution of low-dimensional projections from high-dimensional data, where the conditioning is on other low-dimensional projections. To fix ideas, consider a random d-vector Z that has a Lebesgue density and that…
Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we…
In this paper, we show that the conditional expectation of a random variable with finite second moment given a $\sigma$-algebra is the unique critical point of an energy functional in Hilbert space $L^2$. Then, we extend by density the…
The problem of nonparametric estimation of the conditional density of a response, given a vector of explanatory variables, is classical and of prominent importance in many prediction problems since the conditional density provides a more…
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…
We study the interpretability of conditional probability estimates for binary classification under the agnostic setting or scenario. Under the agnostic setting, conditional probability estimates do not necessarily reflect the true…
We develop a new method for generating prediction sets that combines the flexibility of conformal methods with an estimate of the conditional distribution $P_{Y \mid X}$. Existing methods, such as conformalized quantile regression and…
Consider a positive random variable of interest Y depending on a covariate X, and a random observation time T independent of Y given X. Assume that the only knowledge available about Y is its current status at time T: \delta = 1_{Y \leq T}.…
Measuring conditional dependencies among the variables of a network is of great interest to many disciplines. This paper studies some shortcomings of the existing dependency measures in detecting direct causal influences or their lack of…
In this paper we consider the problem of estimating $f$, the conditional density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$ in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$ is a…
The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In…
In this paper, we investigate the distributions of random couples $(X,Y)$ with $X$ real-valued such that any non-negative integrable random variable $f(X)$ can be represented as a conditional expectation, $f(X)=\mathbb{E}[g(Y)|X]$, for some…
The definition of the conditional probability is very important in the theory of the probability. This definition is based on the fact, that random events can be simultaneously measurable. This paper deal with the problem of conditioning…
A quantum probability measure is a function on a sigma-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but whose values are positive operators acting on a complex…
A necessary and sufficient condition on a sequence $\{\mathfrak{A}_n\}_{n\in \mathbb{N}}$ of $\sigma$-subalgebras that assures convergence almost every where of conditional expectations is given.
We consider the Bayesian analysis of models in which the unknown distribution of the outcomes is specified up to a set of conditional moment restrictions. The nonparametric exponentially tilted empirical likelihood function is constructed…