We consider the problem of learning the structure of Ising models (pairwise binary Markov random fields) from i.i.d. samples. While several methods have been proposed to accomplish this task, their relative merits and limitations remain somewhat obscure. By analyzing a number of concrete examples, we show that low-complexity algorithms systematically fail when the Markov random field develops long-range correlations. More precisely, this phenomenon appears to be related to the Ising model phase transition (although it does not coincide with it).

Kernel supervised learning methods can be unified by utilizing the tools from regularization theory. The duality between regularization and prior leads to interpreting regularization methods in terms of maximum a posteriori estimation and has motivated Bayesian interpretations of kernel methods. In this paper we pursue a Bayesian interpretation of sparsity in the kernel setting by making use of a mixture of a point-mass distribution and prior that we refer to as ``Silverman's g-prior.'' We provide a theoretical analysis of the posterior consistency of a Bayesian model choice procedure based on this prior. We also establish the asymptotic relationship between this procedure and the Bayesian information criterion.

Multi-level hierarchical models provide an attractive framework for incorporating correlations induced in a response variable organized in a hierarchy. Model fitting is challenging, especially for hierarchies with large number of nodes. We provide a novel algorithm based on a multi-scale Kalman filter that is both scalable and easy to implement. For non-Gaussian responses, quadratic approximation to the log-likelihood results in biased estimates. We suggest a bootstrap strategy to correct such biases. Our method is illustrated through simulation studies and analyses of real world data sets in health care and online advertising.

Mixture of Gaussian processes models extended a single Gaussian process with ability of modeling multi-modal data and reduction of training complexity. Previous inference algorithms for these models are mostly based on Gibbs sampling, which can be very slow, particularly for large-scale data sets. We present a new generative mixture of experts model. Each expert is still a Gaussian process but is reformulated by a linear model. This breaks the dependency among training outputs and enables us to use a much faster variational Bayesian algorithm for training. Our gating network is more flexible than previous generative approaches as inputs for each expert are modeled by a Gaussian mixture model. The number of experts and number of Gaussian components for an expert are inferred automatically. A variety of tests show the advantages of our method.

In a variety of behavioral tasks, subjects exhibit an automatic and apparently sub-optimal sequential effect: they respond more rapidly and accurately to a stimulus if it reinforces a local pattern in stimulus history, such as a string of repetitions or alternations, compared to when it violates such a pattern. This is often the case even if the local trends arise by chance in the context of a randomized design, such that stimulus history has no predictive power. In this work, we use a normative Bayesian framework to examine the hypothesis that such idiosyncrasies may reflect the inadvertent engagement of fundamental mechanisms critical for adapting to changing statistics in the natural environment. We show that prior belief in non-stationarity can induce experimentally observed sequential effects in an otherwise Bayes-optimal algorithm. The Bayesian algorithm is shown to be well approximated by linear-exponential filtering of past observations, a feature also apparent in the behavioral data. We derive an explicit relationship between the parameters and computations of the exact Bayesian algorithm and those of the approximate linear-exponential filter. Since the latter is equivalent to a leaky-integration process, a commonly used model of neuronal dynamics underlying perceptual decision-making and trial-to-trial dependencies, our model provides a principled account of why such dynamics are useful. We also show that near-optimal tuning of the leaky-integration process is possible, using stochastic gradient descent based only on the noisy binary inputs. This is a proof of concept that not only can neurons implement near-optimal prediction based on standard neuronal dynamics, but that they can also learn to tune the processing parameters without explicitly representing probabilities.

Many interesting problems, including Bayesian network structure-search, can be cast in terms of finding the optimum value of a function over the space of graphs. However, this function is often expensive to compute exactly. We here present a method derived from the study of reproducing-kernel Hilbert spaces which takes advantage of the regular structure of the space of all graphs on a fixed number of nodes to obtain approximations to the desired function quickly and with reasonable accuracy. We then test this method on both a small testing set and a real-world Bayesian network; the results suggest that not only is this method reasonably accurate, but that the BDe score itself varies quadratically over the space of all graphs.

Many human interactions involve pieces of information being passed from one person to another, raising the question of how this process of information transmission is affected by the capacities of the agents involved. In the 1930s, Sir Frederic Bartlett explored the influence of memory biases in “serial reproduction” of information, in which one person’s reconstruction of a stimulus from memory becomes the stimulus seen by the next person. These experiments were done using relatively uncontrolled stimuli such as pictures and stories, but suggested that serial reproduction would transform information in a way that reflected the biases inherent in memory. We formally analyze serial reproduction using a Bayesian model of reconstruction from memory, giving a general result characterizing the effect of memory biases on information transmission. We then test the predictions of this account in two experiments using simple one-dimensional stimuli. Our results provide theoretical and empirical justification for the idea that serial reproduction reflects memory biases.

The synchronous brain activity measured via MEG (or EEG) can be interpreted as arising from a collection (possibly large) of current dipoles or sources located throughout the cortex. Estimating the number, location, and orientation of these sources remains a challenging task, one that is significantly compounded by the effects of source correlations and the presence of interference from spontaneous brain activity, sensor noise, and other artifacts. This paper derives an empirical Bayesian method for addressing each of these issues in a principled fashion. The resulting algorithm guarantees descent of a cost function uniquely designed to handle unknown orientations and arbitrary correlations. Robust interference suppression is also easily incorporated. In a restricted setting, the proposed method is shown to have theoretically zero bias estimating both the location and orientation of multi-component dipoles even in the presence of correlations, unlike a variety of existing Bayesian localization methods or common signal processing techniques such as beamforming and sLORETA. Empirical results on both simulated and real data sets verify the efficacy of this approach.

We propose a multiplicative approximation scheme (MAS) for inference problems in graphical models, which can be applied to various inference algorithms. The method uses $\epsilon$-decompositions which decompose functions used throughout the inference procedure into functions over smaller sets of variables with a known error $\epsilon$. MAS translates these local approximations into bounds on the accuracy of the results. We show how to optimize $\epsilon$-decompositions and provide a fast closed-form solution for an $L_2$ approximation. Applying MAS to the Variable Elimination inference algorithm, we introduce an algorithm we call DynaDecomp which is extremely fast in practice and provides guaranteed error bounds on the result. The superior accuracy and efficiency of DynaDecomp is demonstrated.

Sampling functions in Gaussian process (GP) models is challenging because of the highly correlated posterior distribution. We describe an efficient Markov chain Monte Carlo algorithm for sampling from the posterior process of the GP model. This algorithm uses control variables which are auxiliary function values that provide a low dimensional representation of the function. At each iteration, the algorithm proposes new values for the control variables and generates the function from the conditional GP prior. The control variable input locations are found by continuously minimizing an objective function. We demonstrate the algorithm on regression and classification problems and we use it to estimate the parameters of a differential equation model of gene regulation.