This makes Gaussian process regression too slow for large datasets. In this paper, we present a fast approximation method, based on kd-trees, that significantly reduces both the prediction and the training times of Gaussian process regression.

The misjudgement of tilt in images lies at the heart of entertaining visual illusions and rigorous perceptual psychophysics. A wealth of findings has attracted many mechanistic models, but few clear computational principles. We adopt a Bayesian approach to perceptual tilt estimation, showing how a smoothness prior offers a powerful way of addressing much confusing data. In particular, we faithfully model recent results showing that confidence in estimation can be systematically affected by the same aspects of images that affect bias. Confidence is central to Bayesian modeling approaches, and is applicable in many other perceptual domains. Perceptual anomalies and illusions, such as the misjudgements of motion and tilt evident in so many psychophysical experiments, have intrigued researchers for decades.

We present a model of edge and region grouping using a conditional random field built over a scale-invariant representation of images to integrate multiple cues. Our model includes potentials that capture low-level similarity, mid-level curvilinear continuity and high-level object shape. Maximum likelihood parameters for the model are learned from human labeled groundtruth on a large collection of horse images using belief propagation. Using held out test data, we quantify the information gained by incorporating generic mid-level cues and high-level shape.

We present a family of approximation techniques for probabilistic graphical models, based on the use of graphical preconditioners developed in the scientific computing literature. Our framework yields rigorous upper and lower bounds on event probabilities and the log partition function of undirected graphical models, using non-iterative procedures that have low time complexity. As in mean field approaches, the approximations are built upon tractable subgraphs; however, we recast the problem of optimizing the tractable distribution parameters and approximate inference in terms of the well-studied linear systems problem of obtaining a good matrix preconditioner. Experiments are presented that compare the new approximation schemes to variational methods.

We discuss a method for obtaining a subject's a priori beliefs from his/her behavior in a psychophysics context, under the assumption that the behavior is (nearly) optimal from a Bayesian perspective. The method is nonparametric in the sense that we do not assume that the prior belongs to any fixed class of distributions (e.g., Gaussian). Despite this increased generality, the method is relatively simple to implement, being based in the simplest case on a linear programming algorithm, and more generally on a straightforward maximum likelihood or maximum a posteriori formulation, which turns out to be a convex optimization problem (with no non-global local maxima) in many important cases. In addition, we develop methods for analyzing the uncertainty of these estimates. We demonstrate the accuracy of the method in a simple simulated coin-flipping setting; in particular, the method is able to precisely track the evolution of the subject's posterior distribution as more and more data are observed. We close by briefly discussing an interesting connection to recent models of neural population coding.

We consider criteria for variational representations of non-Gaussian latent variables, and derive variational EM algorithms in general form. We establish a general equivalence among convex bounding methods, evidence based methods, and ensemble learning/Variational Bayes methods, which has previously been demonstrated only for particular cases.

Humans make optimal perceptual decisions in noisy and ambiguous conditions. Computations underlying such optimal behavior have been shown to rely on probabilistic inference according to generative models whose structure is usually taken to be known a priori. We argue that Bayesian model selection is ideal for inferring similar and even more complex model structures from experience. We find in experiments that humans learn subtle statistical properties of visual scenes in a completely unsupervised manner. We show that these findings are well captured by Bayesian model learning within a class of models that seek to explain observed variables by independent hidden causes.

The problem of computing a resample estimate for the reconstruction error in PCA is reformulated as an inference problem with the help of the replica method. Using the expectation consistent (EC) approximation, the intractable inference problem can be solved efficiently using only two variational parameters. A perturbative correction to the result is computed and an alternative simplified derivation is also presented.

We propose a new Bayesian method for spatial cluster detection, the "Bayesian spatial scan statistic," and compare this method to the standard (frequentist) scan statistic approach. We demonstrate that the Bayesian statistic has several advantages over the frequentist approach, including increased power to detect clusters and (since randomization testing is unnecessary) much faster runtime. We evaluate the Bayesian and frequentist methods on the task of prospective disease surveillance: detecting spatial clusters of disease cases resulting from emerging disease outbreaks. We demonstrate that our Bayesian methods are successful in rapidly detecting outbreaks while keeping number of false positives low.

Nested sampling is a new Monte Carlo method by Skilling [1] intended for general Bayesian computation. Nested sampling provides a robust alternative to annealing-based methods for computing normalizing constants. It can also generate estimates of other quantities such as posterior expectations. The key technical requirement is an ability to draw samples uniformly from the prior subject to a constraint on the likelihood. We provide a demonstration with the Potts model, an undirected graphical model.