Related papers: A Novel Bayes' Theorem for Upper Probabilities
The conventional postulate for the probabilistic interpretation of quantum mechanics is asymmetric in preparation and measurement, making retrodiction reliant on inference by use of Bayes' theorem. Here, a more fundamental symmetric…
Bayesian sequence prediction is a simple technique for predicting future symbols sampled from an unknown measure on infinite sequences over a countable alphabet. While strong bounds on the expected cumulative error are known, there are only…
We obtain assumption-free, non-asymptotic, uniform bounds on the product of the height and the width of uniformly random trees with a given degree sequence, conditioned Bienaym\'e trees and simply generated trees. We show that for a tree of…
Bayesian probability theory is used as a framework to develop a formalism for the scientific method based on principles of inductive reasoning. The formalism allows for precise definitions of the key concepts in theories of physics and also…
We develop a new framework of uncertainty variables to model uncertainty. An uncertainty variable is characterized by an uncertainty set, in which its realization is bound to lie, while the conditional uncertainty is characterized by a set…
Bayesian neural networks (BNNs) have recently gained popularity due to their ability to quantify model uncertainty. However, specifying a prior for BNNs that captures relevant domain knowledge is often extremely challenging. In this work,…
Given a set of probability measures $\mathcal{P}$ representing an agent's knowledge on the elements of a sigma-algebra $\mathcal{F}$, we can compute upper and lower bounds for the probability of any event $A\in\mathcal{F}$ of interest. A…
Gaussian processes have become a promising tool for various safety-critical settings, since the posterior variance can be used to directly estimate the model error and quantify risk. However, state-of-the-art techniques for safety-critical…
The proposed approach extends the confidence posterior distribution to the semi-parametric empirical Bayes setting. Whereas the Bayesian posterior is defined in terms of a prior distribution conditional on the observed data, the confidence…
Uncertainty quantification is essential when dealing with ill-conditioned inverse problems due to the inherent nonuniqueness of the solution. Bayesian approaches allow us to determine how likely an estimation of the unknown parameters is…
Bayesian synthetic likelihood is a widely used approach for conducting Bayesian analysis in complex models where evaluation of the likelihood is infeasible but simulation from the assumed model is tractable. We analyze the behaviour of the…
This paper develops a methodology for approximating the posterior first two moments of the posterior distribution in Bayesian inference. Partially specified probability models, which are defined only by specifying means and variances, are…
In Bayesian inference, making deductions about a parameter of interest requires one to sample from or compute an integral against a posterior distribution. A popular method to make these computations cheaper in high-dimensional settings is…
Bayesian hypothesis testing is investigated when the prior probabilities of the hypotheses, taken as a random vector, are quantized. Nearest neighbor and centroid conditions are derived using mean Bayes risk error as a distortion measure…
The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by…
The Bayesian posterior probability of the true state is stochastically dominated by that same posterior under the probability law of the true state. This generalizes to notions of "optimism" about posterior probabilities.
We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive…
Bayesian probability theory is used to analyze the oft-made assumption that humans are typical observers in the universe. Some theoretical calculations make the {\it selection fallacy} that we are randomly chosen from a class of objects by…
The well-known Bayes theorem assumes that a posterior distribution is a probability distribution. However, the posterior distribution may no longer be a probability distribution if an improper prior distribution (non-probability measure)…
This paper establishes Paley-Wiener perturbation theorems for probabilistic frames. The classical Paley-Wiener perturbation theorem shows that if a sequence is close to a basis in a Banach space, then this sequence is also a basis. Similar…