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Loss-based clustering methods, such as k-means and its variants, are standard tools for finding groups in data. However, the lack of quantification of uncertainty in the estimated clusters is a disadvantage. Model-based clustering based on…
Bayesian model comparison (BMC) offers a principled probabilistic approach to study and rank competing models. In standard BMC, we construct a discrete probability distribution over the set of possible models, conditional on the observed…
Recently the quantum Bayesian prediction problem was formulated by Tanaka and Komaki (2005). It is shown that Bayesian predictive density operators are the best predictive density operators when we evaluate them by using the averaged…
We study frequentist risk properties of predictive density estimators for mean mixtures of multivariate normal distributions, involving an unknown location parameter $\theta \in \mathbb{R}^d$, and which include multivariate skew normal…
A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory…
We introduce categories of extended Gaussian maps and Gaussian relations which unify Gaussian probability distributions with relational nondeterminism in the form of linear relations. Both have crucial and well-understood applications in…
Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical…
The statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation is studied in the Bayesian framework. In practice, one often considers only…
We generalize L\'evy's lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix $\rho$ on a…
Recently quantum prediction problem was proposed in the Bayesian framework. It is shown that Bayesian predictive density operators are the best predictive density operators when we evaluate them by using the average relative entropy based…
The law of likelihood underlies a general framework, known as the likelihood paradigm, for representing and interpreting statistical evidence. As stated, the law applies only to simple hypotheses, and there have been reservations about…
A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction, disjunction, inverse, and conditional fallacies, as well as unpacking effects and partitioning effects. Quantum…
The notion of density of a finite set is discussed. We proof a general theorem of set theory which refines Bose-Einstein distribution.
Bayesian clustering methods have the widely touted advantage of providing a probabilistic characterization of uncertainty in clustering through the posterior distribution. An amazing variety of priors and likelihoods have been proposed for…
Bayesian inference provides a principled probabilistic framework for quantifying uncertainty by updating beliefs based on prior knowledge and observed data through Bayes' theorem. In Bayesian deep learning, neural network weights 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…
The key distinguishing property of a Bayesian approach is marginalization, rather than using a single setting of weights. Bayesian marginalization can particularly improve the accuracy and calibration of modern deep neural networks, which…
Bayesian filtering serves as the mainstream framework of state estimation in dynamic systems. Its standard version utilizes total probability rule and Bayes' law alternatively, where how to define and compute conditional probability is…
In solving Bayesian inverse problems, it is often desirable to use a common density parameterization to denote the prior and posterior. Typically we seek a density from the same family as the prior which closely approximates the true…
Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow…