Given an $n$-vertex weighted tree with structural diameter $S$ and a subset of $m$ vertices, we present a technique to compute a corresponding $m \times m$ Gram matrix of the pseudoinverse of the graph Laplacian in $O(n+ m^2 + m S)$ time. We discuss the application of this technique to fast label prediction on a generic graph. We approximate the graph with a spanning tree and then we predict with the kernel perceptron. We address the approximation of the graph with either a minimum spanning tree or a shortest path tree. The fast computation of the pseudoinverse enables us to address prediction problems on large graphs. To this end we present experiments on two web-spam classification tasks, one of which includes a graph with 400,000 nodes and more than 10,000,000 edges. The results indicate that the accuracy of our technique is competitive with previous methods using the full graph information.

The nested Chinese restaurant process (nCRP) is a powerful nonparametric Bayesian model for learning tree-based hierarchies from data. Since its posterior distribution is intractable, current inference methods have all relied on MCMC sampling. In this paper, we develop an alternative inference technique based on variational methods. To employ variational methods, we derive a tree-based stick-breaking construction of the nCRP mixture model, and a novel variational algorithm that efficiently explores a posterior over a large set of combinatorial structures. We demonstrate the use of this approach for text and hand written digits modeling, where we show we can adapt the nCRP to continuous data as well.

The goal of perception is to infer the hidden states in the hierarchical process by which sensory data are generated. Human behavior is consistent with the optimal statistical solution to this problem in many tasks, including cue combination and orientation detection. Understanding the neural mechanisms underlying this behavior is of particular importance, since probabilistic computations are notoriously challenging. Here we propose a simple mechanism for Bayesian inference which involves averaging over a few feature detection neurons which fire at a rate determined by their similarity to a sensory stimulus. This mechanism is based on a Monte Carlo method known as importance sampling, commonly used in computer science and statistics. Moreover, a simple extension to recursive importance sampling can be used to perform hierarchical Bayesian inference. We identify a scheme for implementing importance sampling with spiking neurons, and show that this scheme can account for human behavior in cue combination and oblique effect.

The Indian Buffet Process is a Bayesian nonparametric approach that models objects as arising from an infinite number of latent factors. Here we extend the latent factor model framework to two or more unbounded layers of latent factors. From a generative perspective, each layer defines a conditional \emph{factorial} prior distribution over the binary latent variables of the layer below via a noisy-or mechanism. We explore the properties of the model with two empirical studies, one digit recognition task and one music tag data experiment.

In many domains, data are distributed among datasets that share only some variables; otherrecorded variables may occur in only one dataset. While there are asymptotically correct, informative algorithms for discovering causal relationships froma single dataset, even with missing values and hidden variables, there have been no such reliable procedures for distributed data with overlapping variables. Wepresent a novel, asymptotically correct procedure that discovers a minimal equivalence class of causal DAG structures using local independence information fromdistributed data of this form and evaluate its performance using synthetic and real-world data against causal discovery algorithms for single datasets and applying Structural EM, a heuristic DAG structure learning procedure for data with missing values, to the concatenated data.

Across a wide range of cognitive tasks, recent experience influences behavior. For example, when individuals repeatedly perform a simple two-alternative forcedchoice task(2AFC), response latencies vary dramatically based on the immediately preceding trial sequence. These sequential effects have been interpreted as adaptation to the statistical structure of an uncertain, changing environment (e.g., Jones and Sieck, 2003; Mozer, Kinoshita, and Shettel, 2007; Yu and Cohen, 2008).The Dynamic Belief Model (DBM) (Yu and Cohen, 2008) explains sequential effects in 2AFC tasks as a rational consequence of a dynamic internal representation that tracks second-order statistics of the trial sequence (repetition rates) and predicts whether the upcoming trial will be a repetition or an alternation ofthe previous trial. Experimental results suggest that first-order statistics (base rates) also influence sequential effects. We propose a model that learns both first-and second-order sequence properties, each according to the basic principles ofthe DBM but under a unified inferential framework. This model, the Dynamic BeliefMixture Model (DBM2), obtains precise, parsimonious fits to data. Furthermore, the model predicts dissociations in behavioral (Maloney, Martello, Sahm, and Spillmann, 2005) and electrophysiological studies (Jentzsch and Sommer, 2002),supporting the psychological and neurobiological reality of its two components.

Empirical studies have documented cases of belief polarization, where two people withopposing prior beliefs both strengthen their beliefs after observing the same evidence. Belief polarization is frequently offered as evidence of human irrationality, but we demonstrate that this phenomenon is consistent with a fully Bayesian approach to belief revision. Simulation results indicate that belief polarization isnot only possible but relatively common within the set of Bayesian models that we consider. Suppose that Carol has requested a promotion at her company and has received a score of 50 on an aptitude test. Alice, one of the company's managers, began with a high opinion of Carol and became even more confident of her abilities after seeing her test score.

There has been a clear distinction between induction or training time and diagnosis time active information acquisition. While active learning during induction focuses on acquiring data that promises to provide the best classification model, the goal at diagnosis time focuses completely on next features to observe about the test case at hand in order to make better predictions about the case. We introduce a model and inferential methods that breaks this distinction. The methods can be used to extend case libraries under a budget but, more fundamentally, provide a framework for guiding agents to collect data under scarce resources, focused by diagnostic challenges. This extension to active learning leads to a new class of policies for real-time diagnosis, where recommended information-gathering sequences include actions that simultaneously seek new data for the case at hand and for cases in the training set.

Research in animal learning and behavioral neuroscience has distinguished between two forms of action control: a habit-based form, which relies on stored action values, and a goal-directed form, which forecasts and compares action outcomes based on a model of the environment. While habit-based control has been the subject of extensive computational research, the computational principles underlying goal-directed control in animals have so far received less attention. In the present paper, we advance a computational framework for goal-directed control in animals and humans. We take three empirically motivated points as founding premises: (1) Neurons in dorsolateral prefrontal cortex represent action policies, (2) Neurons in orbitofrontal cortex represent rewards, and (3) Neural computation, across domains, can be appropriately understood as performing structured probabilistic inference. On a purely computational level, the resulting account relates closely to previous work using Bayesian inference to solve Markov decision problems, but extends this work by introducing a new algorithm, which provably converges on optimal plans. On a cognitive and neuroscientific level, the theory provides a unifying framework for several different forms of goal-directed action selection, placing emphasis on a novel form, within which orbitofrontal reward representations directly drive policy selection.

We propose Dirichlet-Bernoulli Alignment (DBA), a generative model for corpora in which each pattern (e.g., a document) contains a set of instances (e.g., paragraphs in the document) and belongs to multiple classes. By casting predefined classes as latent Dirichlet variables (i.e., instance level labels), and modeling the multi-label of each pattern as Bernoulli variables conditioned on the weighted empirical average of topic assignments, DBA automatically aligns the latent topics discovered from data to human-defined classes. DBA is useful for both pattern classification and instance disambiguation, which are tested on text classification and named entity disambiguation for web search queries respectively.