Related papers: A categorical foundation for Bayesian probability
In this note we consider the stability of posterior measures occuring in Bayesian inference w.r.t. perturbations of the prior measure and the log-likelihood function. This extends the well-posedness analysis of Bayesian inverse problems. In…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
We consider the task of forecasting an infinite sequence of future observations based on some number of past observations, where the probability measure generating the observations is "suspected" to satisfy one or more of a set of…
This report introduces general ideas and some basic methods of the Bayesian probability theory applied to physics measurements. Our aim is to make the reader familiar, through examples rather than rigorous formalism, with concepts such as:…
Let $S$ be a Polish space and $(X_n:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_n(\cdot)=P(X_{n+1}\in \cdot\mid X_1,\...,X_n)$ be the predictive measure and $\alpha$ a random probability measure on $S$ such…
Bayesian tests on the symmetry of the generalized von Mises model for planar directions (Gatto and Jammalamadaka, 2007) are introduced. The generalized von Mises distribution is a flexible model that can be axially symmetric or asymmetric,…
Let $X \subset \mathbb{R}^N$ be a Borel set, $\mu$ a Borel probability measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in \mathbb{N}$ greater than the (Hausdorff) dimension of $X$ and assume that the set of…
Prediction is a central task of statistics and machine learning, yet many inferential settings provide only partial information, typically in the form of moment constraints or estimating equations. We develop a finite, fully Bayesian…
In many domains, we are interested in analyzing the structure of the underlying distribution, e.g., whether one variable is a direct parent of the other. Bayesian model-selection attempts to find the MAP model and use its structure to…
In sequential design strategies, common in geostatistics and Bayesian optimization, the selection of a new observation point $X_{n+1}$ of a random function $\mathbf f$ is informed by past data, captured by the filtration $\mathcal…
Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are…
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…
Empirical likelihood is a popular nonparametric statistical tool that does not require any distributional assumptions. In this paper, we explore the possibility of conducting variable selection via Bayesian empirical likelihood. We show…
Decomposable models and Bayesian networks can be defined as sequences of oligo-dimensional probability measures connected with operators of composition. The preliminary results suggest that the probabilistic models allowing for effective…
The proposal and study of dependent prior processes has been a major research focus in the recent Bayesian nonparametric literature. In this paper, we introduce a flexible class of dependent nonparametric priors, investigate their…
Let $(\Omega,\mathcal{F})$ be a standard Borel space and $\mathcal{P}(\mathcal{F})$ the collection of all probability measures on $\mathcal{F}$. Let $E\subset\Omega\times\Omega$ be a measurable equivalence relation, that is,…
If the prior probability distributions of all possible hypothetical true means and all possible observed means of a continuous variable are conditional on the universal set of all numbers (i.e., before the nature of a study is known and a…
Given a compact metric space $X$ and a probability measure in the $\sigma-$algebra of Borel subsets of $X$, we will establish a dominated convergence theorem for ultralimits of sequences of integrable maps and apply it to deduce a…
Predicting the winner of an election is of importance to multiple stakeholders. To formulate the problem, we consider an independent sequence of categorical data with a finite number of possible outcomes in each. The data is assumed to be…
Loss-based updating, including generalized Bayes, Gibbs, and quasi-posteriors, replaces likelihoods by a user-chosen loss and produces a posterior-like distribution via exponential tilt. We give a decision-theoretic characterization that…