The Variational Bayesian framework has been widely used to approximate the Bayesian learning. In various applications, it has provided computational tractability and good generalization performance. In this paper, we discuss the Variational Bayesian learning of the mixture of exponential families and provide some additional theoretical support by deriving the asymptotic form of the stochastic complexity. The stochastic complexity, which corresponds to the minimum free energy and a lower bound of the marginal likelihood, is a key quantity for model selection. It also enables us to discuss the effect of hyperparameters and the accuracy of the Variational Bayesian approach as an approximation of the true Bayesian learning.

We show that linear generalizations of Rescorla-Wagner can perform Maximum Likelihood estimation of the parameters of all generative models for causal reasoning. Our approach involves augmenting variables to deal with conjunctions of causes, similar to the agumented model of Rescorla. Our results involve genericity assumptions on the distributions of causes. If these assumptions are violated, for example for the Cheng causal power theory, then we show that a linear Rescorla-Wagner can estimate the parameters of the model up to a nonlinear transformtion. Moreover, a nonlinear Rescorla-Wagner is able to estimate the parameters directly to within arbitrary accuracy.

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 study the problem of maximum entropy density estimation in the presence of known sample selection bias. We propose three bias cor- rection approaches. The first one takes advantage of unbiased sufficient statistics which can be obtained from biased samples. The second one es- timates the biased distribution and then factors the bias out. The third one approximates the second by only using samples from the sampling distri- bution.

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 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.13

The classical Bayes rule computes the posterior model probability from the prior probability and the data likelihood. We generalize this rule to the case when the prior is a density matrix (symmetric positive definite and trace one) and the data likelihood a covariance matrix. The classical Bayes rule is retained as the special case when the matrices are diagonal. In the classical setting, the calculation of the probability of the data is an expected likelihood, where the expectation is over the prior distribution. In the generalized setting, this is replaced by an expected variance calculation where the variance is computed along the eigenvectors of the prior density matrix and the expectation is over the eigenvalues of the density matrix (which form a probability vector).

We investigate the problem of automatically constructing efficient rep- resentations or basis functions for approximating value functions based on analyzing the structure and topology of the state space. In particu- lar, two novel approaches to value function approximation are explored based on automatically constructing basis functions on state spaces that can be represented as graphs or manifolds: one approach uses the eigen- functions of the Laplacian, in effect performing a global Fourier analysis on the graph; the second approach is based on diffusion wavelets, which generalize classical wavelets to graphs using multiscale dilations induced by powers of a diffusion operator or random walk on the graph. Together, these approaches form the foundation of a new generation of methods for solving large Markov decision processes, in which the underlying repre- sentation and policies are simultaneously learned.

Despite all the attention paid to variational methods based on sum-product message passing (loopy belief propagation, tree-reweighted sum-product), these methods are still bound to inference on a small set of probabilistic models. Mean field approximations have been applied to a broader set of problems, but the solutions are often poor. We propose a new class of conditionally-specified variational approximations based on mean field theory. While not usable on their own, combined with sequential Monte Carlo they produce guaranteed improvements over conventional mean field. Moreover, experiments on a well-studied problem-- inferring the stable configurations of the Ising spin glass--show that the solutions can be significantly better than those obtained using sum-product-based methods.

The ill-posed nature of the MEG/EEG source localization problem requires the incorporation of prior assumptions when choosing an appropriate solution out of an infinite set of candidates. Bayesian methods are useful in this capacity because they allow these assumptions to be explicitly quantified. Recently, a number of empirical Bayesian approaches have been proposed that attempt a form of model selection by using the data to guide the search for an appropriate prior. While seemingly quite different in many respects, we apply a unifying framework based on automatic relevance determination (ARD) that elucidates various attributes of these methods and suggests directions for improvement. We also derive theoretical properties of this methodology related to convergence, local minima, and localization bias and explore connections with established algorithms.