Related papers: On Bayes' theorem for improper mixtures

In this paper, we show how a complete and exact Bayesian analysis of a parametric mixture model is possible in some cases when components of the mixture are taken from exponential families and when conjugate priors are used. This restricted…

In the usual Bayesian setting, a full probabilistic model is required to link the data and parameters, and the form of this model and the inference and prediction mechanisms are specified via de Finetti's representation. In general, such a…

After experimenting with a number of non-probabilistic methods for dealing with uncertainty many researchers reaffirm a preference for probability methods [1] [2], although this remains controversial. The importance of being able to form…

The problem of estimating cosmological parameters such as $\Omega$ from noisy or incomplete data is an example of an inverse problem and, as such, generally requires a probablistic approach. We adopt the Bayesian interpretation of…

Bayesian approaches to data analysis and machine learning are widespread and popular as they provide intuitive yet rigorous axioms for learning from data; see Bernardo and Smith (2004) and Bishop (2006). However, this rigour comes with a…

We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution $P_0$, which may not be in the support of the prior, we show that the posterior…

The Pitman-Yor process is a random probability distribution, that can be used as a prior distribution in a nonparametric Bayesian analysis. The process is of species sampling type and generates discrete distributions, which yield of the…

We describe a new method for evaluating Bayes factors. The key idea is to introduce a hypermodel in which the competing models are components of a mixture distribution. Inference for the mixing probabilities then yields estimates of the…

We study the convergence rates of empirical Bayes posterior distributions for nonparametric and high-dimensional inference. We show that as long as the hyperparameter set is discrete, the empirical Bayes posterior distribution induced by…

We consider the Bayesian analysis of a few complex, high-dimensional models and show that intuitive priors, which are not tailored to the fine details of the model and the estimated parameters, produce estimators which perform poorly in…

Sparse Bayesian learning models are typically used for prediction in datasets with significantly greater number of covariates than observations. Such models often take a reproducing kernel Hilbert space (RKHS) approach to carry out the task…

Many penalized maximum likelihood estimators correspond to posterior mode estimators under specific prior distributions. Appropriateness of a particular class of penalty functions can therefore be interpreted as the appropriateness of a…

Bayesian inference paradigms are regarded as powerful tools for solution of inverse problems. However, when applied to inverse problems in physical sciences, Bayesian formulations suffer from a number of inconsistencies that are often…

Bayesian parameter inference depends on a choice of prior probability distribution for the parameters in question. The prior which makes the posterior distribution maximally sensitive to data is called the Jeffreys prior, and it is…

Signal processing makes extensive use of point estimators and accompanying error bounds. These work well up until the likelihood function has two or more high peaks. When it is important for an estimator to remain reliable, it becomes…

Particle filtering is a Bayesian inference method and a fundamental tool in state estimation for dynamic systems, but its effectiveness is often limited by the constraints of the initial prior distribution, a phenomenon we define as the…

Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty.…

Recent decades have seen an interest in prediction problems for which Bayesian methodology has been used ubiquitously. Sampling from or approximating the posterior predictive distribution in a Bayesian model allows one to make inferential…

The main object of Bayesian statistical inference is the determination of posterior distributions. Sometimes these laws are given for quantities devoid of empirical value. This serious drawback vanishes when one confines oneself to…

In this paper we leverage on probability over Riemannian manifolds to rethink the interpretation of priors and posteriors in Bayesian inference. The main mindshift is to move away from the idea that "a prior distribution establishes a…