Statistical relational learning models combine the power of first-order logic, the de facto tool for handling relational structure, with that of probabilistic graphical models, the de facto tool for handling uncertainty. Lifted probabilistic inference algorithms for them have been the subject of much recent research. The main idea in these algorithms is to improve the speed, accuracy and scalability of existing graphical models' inference algorithms by exploiting symmetry in the first-order representation. In this paper, we consider blocked Gibbs sampling, an advanced variation of the classic Gibbs sampling algorithm and lift it to the first-order level. We propose to achieve this by partitioning the first-order atoms in the relational model into a set of disjoint clusters such that exact lifted inference is polynomial in each cluster given an assignment to all other atoms not in the cluster. We propose an approach for constructing such clusters and determining their complexity and show how it can be used to trade accuracy with computational complexity in a principled manner. Our experimental evaluation shows that lifted Gibbs sampling is superior to the propositional algorithm in terms of accuracy and convergence.

Statistical decision theory axiomatically assumes that the relative desirability of different options that humans perceive is well described by assigning them option-specific scalar utility functions. However, this assumption is refuted by observed human behavior, including studies wherein preferences have been shown to change systematically simply through variation in the set of choice options presented. In this paper, we show that interpreting desirability as a relative comparison between available options at any particular decision instance results in a rational theory of value-inference that explains heretofore intractable violations of rational choice behavior in human subjects. Complementarily, we also characterize the conditions under which a rational agent selecting optimal options indicated by dynamic value inference in our framework will behave identically to one whose preferences are encoded using a static ordinal utility function.

We consider unsupervised estimation of mixtures of discrete graphical models, where the class variable is hidden and each mixture component can have a potentially different Markov graph structure and parameters over the observed variables. We propose a novel method for estimating the mixture components with provable guarantees. Our output is a tree-mixture model which serves as a good approximation to the underlying graphical model mixture. The sample and computational requirements for our method scale as $\poly(p, r)$, for an $r$-component mixture of $p$-variate graphical models, for a wide class of models which includes tree mixtures and mixtures over bounded degree graphs.

Bayesian inference provides a unifying framework for addressing problems in machine learning, artificial intelligence, and robotics, as well as the problems facing the human mind. Unfortunately, exact Bayesian inference is intractable in all but the simplest models. Therefore minds and machines have to approximate Bayesian inference. Approximate inference algorithms can achieve a wide range of time-accuracy tradeoffs, but what is the optimal tradeoff? We investigate time-accuracy tradeoffs using the Metropolis-Hastings algorithm as a metaphor for the mind's inference algorithm(s). We find that reasonably accurate decisions are possible long before the Markov chain has converged to the posterior distribution, i.e. during the period known as burn-in. Therefore the strategy that is optimal subject to the mind's bounded processing speed and opportunity costs may perform so few iterations that the resulting samples are biased towards the initial value. The resulting cognitive process model provides a rational basis for the anchoring-and-adjustment heuristic. The model's quantitative predictions are tested against published data on anchoring in numerical estimation tasks. Our theoretical and empirical results suggest that the anchoring bias is consistent with approximate Bayesian inference.

Factor analysis models effectively summarise the covariance structure of high dimensional data,but the solutions are typically hard to interpret. This motivates attempting to find a disjoint partition, i.e. a simple clustering, of observed variables into highly correlated subsets. We introduce a Bayesian nonparametric approach to this problem, and demonstrate advantages over heuristic methods proposed to date. Our Dirichlet process variable clustering (DPVC) model can discover blockdiagonal covariancestructures in data. We evaluate our method on both synthetic and gene expression analysis problems.

Links between probabilistic and non-probabilistic learning algorithms can arise by performing small-variance asymptotics, i.e., letting the variance of particular distributions in a graphical model go to zero. For instance, in the context of clustering, such an approach yields precise connections between the k-means and EM algorithms. In this paper, we explore small-variance asymptotics for exponential family Dirichlet process (DP) and hierarchical Dirichlet process (HDP) mixture models. Utilizing connections between exponential family distributions and Bregman divergences, we derive novel clustering algorithms from the asymptotic limit of the DP and HDP mixtures that feature the scalability of existing hard clustering methods as well as the flexibility of Bayesian nonparametric models. We focus on special cases of our analysis for discrete-data problems, including topic modeling, and we demonstrate the utility of our results by applying variants of our algorithms to problems arising in vision and document analysis.

We present a dynamic nonlinear generative model for visual motion based on a latent representation of binary-gated Gaussian variables. Trained on sequences of images, the model learns to represent different movement directions in different variables. We use an online approximate-inference scheme that can be mapped to the dynamics of networks of neurons. Probed with drifting grating stimuli and moving bars of light, neurons in the model show patterns of responses analogous to those of direction-selective simple cells in primary visual cortex. Most model neurons also show speed tuning and respond equally well to a range of motion directions and speeds aligned to the constraint line of their respective preferred speed. We show how these computations are enabled by a specific pattern of recurrent connections learned by the model.

A common challenge for Bayesian models of perception is the fact that the two fundamental Bayesian components, the prior distribution and the likelihood function, areformally unconstrained. Here we argue that a neural system that emulates Bayesian inference is naturally constrained by the way it represents sensory information inpopulations of neurons. More specifically, we show that an efficient coding principle creates a direct link between prior and likelihood based on the underlying stimulus distribution. The resulting Bayesian estimates can show biases awayfrom the peaks of the prior distribution, a behavior seemingly at odds with the traditional view of Bayesian estimation, yet one that has been reported in human perception. We demonstrate that our framework correctly accounts for the repulsive biases previously reported for the perception of visual orientation, and show that the predicted tuning characteristics of the model neurons match the reported orientation tuning properties of neurons in primary visual cortex. Our results suggest that efficient coding is a promising hypothesis in constraining Bayesianmodels of perceptual inference.

The variational Bayesian (VB) approach is one of the best tractable approximations to the Bayesian estimation, and it was demonstrated to perform well in many applications. However, its good performance was not fully understood theoretically. For example, VB sometimes produces a sparse solution, which is regarded as a practical advantage of VB, but such sparsity is hardly observed in the rigorous Bayesian estimation. In this paper, we focus on probabilistic PCA and give more theoretical insight into the empirical success of VB. More specifically, for the situation where the noise variance is unknown, we derive a sufficient condition for perfect recovery of the true PCA dimensionality in the large-scale limit when the size of an observed matrix goes to infinity. In our analysis, we obtain bounds for a noise variance estimator and simple closed-form solutions for other parameters, which themselves are actually very useful for better implementation of VB-PCA.

We present a new model based on Gaussian processes (GPs) for learning pairwise preferences expressed by multiple users. Inference is simplified by using a \emph{preference kernel} for GPs which allows us to combine supervised GP learning of user preferences with unsupervised dimensionality reduction for multi-user systems. The model not only exploits collaborative information from the shared structure in user behavior, but may also incorporate user features if they are available. Approximate inference is implemented using a combination of expectation propagation and variational Bayes. Finally, we present an efficient active learning strategy for querying preferences. The proposed technique performs favorably on real-world data against state-of-the-art multi-user preference learning algorithms.