The Gibbs sampler is one of the most popular algorithms for inference in statistical models. In this paper, we introduce a herding variant of this algorithm, called herded Gibbs, that is entirely deterministic. We prove that herded Gibbs has an $O(1/T)$ convergence rate for models with independent variables and for fully connected probabilistic graphical models. Herded Gibbs is shown to outperform Gibbs in the tasks of image denoising with MRFs and named entity recognition with CRFs. However, the convergence for herded Gibbs for sparsely connected probabilistic graphical models is still an open problem.

We present a variational Bayesian method of joint image reconstruction and point spread function (PSF) estimation when the PSF of the imaging device is only partially known. To solve this semi-blind deconvolution problem, prior distributions are specified for the PSF and the 3D image. Joint image reconstruction and PSF estimation is then performed within a Bayesian framework, using a variational algorithm to estimate the posterior distribution. The image prior distribution imposes an explicit atomic measure that corresponds to image sparsity. Importantly, the proposed Bayesian deconvolution algorithm does not require hand tuning. Simulation results clearly demonstrate that the semi-blind deconvolution algorithm compares favorably with previous Markov chain Monte Carlo (MCMC) version of myopic sparse reconstruction. It significantly outperforms mismatched non-blind algorithms that rely on the assumption of the perfect knowledge of the PSF. The algorithm is illustrated on real data from magnetic resonance force microscopy (MRFM).

Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost and it is difficult to design nonstationary GP priors in practice. In this paper, we propose a sparse Gaussian process model, EigenGP, based on the Karhunen-Loeve (KL) expansion of a GP prior. We use the Nystrom approximation to obtain data dependent eigenfunctions and select these eigenfunctions by evidence maximization. This selection reduces the number of eigenfunctions in our model and provides a nonstationary covariance function. To handle nonlinear likelihoods, we develop an efficient expectation propagation (EP) inference algorithm, and couple it with expectation maximization for eigenfunction selection. Because the eigenfunctions of a Gaussian kernel are associated with clusters of samples - including both the labeled and unlabeled - selecting relevant eigenfunctions enables EigenGP to conduct semi-supervised learning. Our experimental results demonstrate improved predictive performance of EigenGP over alternative state-of-the-art sparse GP and semisupervised learning methods for regression, classification, and semisupervised classification.

This paper presents a Bayesian framework for assessing the adequacy of a model without the necessity of explicitly enumerating a specific alternate model. A test statistic is developed for tracking the performance of the model across repeated problem instances. Asymptotic methods are used to derive an approximate distribution for the test statistic. When the model is rejected, the individual components of the test statistic can be used to guide search for an alternate model.

In the probabilistic approach to uncertainty management the input knowledge is usually represented by means of some probability distributions. In this paper we assume that the input knowledge is given by two discrete conditional probability distributions, represented by two stochastic matrices P and Q. The consistency of the knowledge base is analyzed. Coherence conditions and explicit formulas for the extension to marginal distributions are obtained in some special cases.

Bayesian networks have been used extensively in diagnostic tasks such as medicine, where they represent the dependency relations between a set of symptoms and a set of diseases. A criticism of this type of knowledge representation is that it is restricted to this kind of task, and that it cannot cope with the knowledge required in other artificial intelligence applications. For example, in computer vision, we require the ability to model complex knowledge, including temporal and relational factors. In this paper we extend Bayesian networks to model relational and temporal knowledge for high-level vision. These extended networks have a simple structure which permits us to propagate probability efficiently. We have applied them to the domain of endoscopy, illustrating how the general modelling principles can be used in specific cases.

The analysis of decision making under uncertainty is closely related to the analysis of probabilistic inference. Indeed, much of the research into efficient methods for probabilistic inference in expert systems has been motivated by the fundamental normative arguments of decision theory. In this paper we show how the developments underlying those efficient methods can be applied immediately to decision problems. In addition to general approaches which need know nothing about the actual probabilistic inference method, we suggest some simple modifications to the clustering family of algorithms in order to efficiently incorporate decision making capabilities.

Valuation-based system (VBS) provides a general framework for representing knowledge and drawing inferences under uncertainty. Recent studies have shown that the semantics of VBS can represent and solve Bayesian decision problems (Shenoy, 1991a). The purpose of this paper is to propose a decision calculus for Dempster-Shafer (D-S) theory in the framework of VBS. The proposed calculus uses a weighting factor whose role is similar to the probabilistic interpretation of an assumption that disambiguates decision problems represented with belief functions (Strat 1990). It will be shown that with the presented calculus, if the decision problems are represented in the valuation network properly, we can solve the problems by using fusion algorithm (Shenoy 1991a). It will also be shown the presented decision calculus can be reduced to the calculus for Bayesian probability theory when probabilities, instead of belief functions, are given.

We discuss problems for convex Bayesian decision making and uncertainty representation. These include the inability to accommodate various natural and useful constraints and the possibility of an analog of the classical Dutch Book being made against an agent behaving in accordance with convex Bayesian prescriptions. A more general set-based Bayesianism may be as tractable and would avoid the difficulties we raise.

This paper presents a new approach for computing posterior probabilities in Bayesian nets, which sidesteps the triangulation problem. The current state of art is the clique tree propagation approach. When the underlying graph of a Bayesian net is triangulated, this approach arranges its cliques into a tree and computes posterior probabilities by appropriately passing around messages in that tree. The computation in each clique is simply direct marginalization. When the underlying graph is not triangulated, one has to first triangulated it by adding edges. Referred to as the triangulation problem, the problem of finding an optimal or even a ?good? triangulation proves to be difficult. In this paper, we propose to first decompose a Bayesian net into smaller components by making use of Tarjan's algorithm for decomposing an undirected graph at all its minimal complete separators. Then, the components are arranged into a tree and posterior probabilities are computed by appropriately passing around messages in that tree. The computation in each component is carried out by repeating the whole procedure from the beginning. Thus the triangulation problem is sidestepped.