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A large number of human psychophysical results have been successfully explained in recent years using Bayesian models. However, the neural implementation of such mod-els remains largely unclear. In this paper, we show that a network architecture com-monly used to model the cerebral cortex can implement Bayesian inference for an arbi-trary hidden Markov model. We illustrate the approach using an orientation discrimi-nation task and a visual motion detection task. In the case of orientation discrimination, we show that the model network can infer the posterior distribution over orientations and correctly estimate stimulus orientation in the presence of significant noise. In the case of motion detection, we show that the resulting model network exhibits direction selectivity and correctly computes the posterior probabilities over motion direction and position. When used to solve the well-known random dots motion discrimination task, the model generates responses that mimic the activities of evidence-accumulating neu-rons in cortical areas LIP and FEF. The framework introduced in the paper posits a new interpretation of cortical activities in terms of log posterior probabilities of stimuli occurring in the natural world. 1 1

One of the major successes in computational biology has been the unification, using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences. Graphical models that have been applied towards these problems include hidden Markov models for annotation, tree models for phylogenetics, and pair hidden Markov models for alignment. A single algorithm, the sum-product algorithm, solves many of the inference problems associated with different statistical models. This paper introduces the polytope propagation algorithm for computing the Newton polytope of an observation from a graphical model. This algorithm is a geometric version of the sumproduct algorithm and is used to analyze the parametric behavior of maximum a posteriori inference calculations for graphical models. 1 Inference with Graphical Models for Biological Sequence Analysis This paper develops a new algorithm for graphical models based on the mathematical foundation for statistical models proposed in [18]. Its relevance for computational biology can be summarized as follows: (a) Graphical models are a unifying statistical framework for biological sequence analysis. (b) Parametric inference is important for obtaining biologically meaningful results.

We propose a statistical formulation for 2-D human pose estimation from single images. The human body configuration is modeled by a Markov network and the estimation problem is to infer pose parameters from image cues such as appearance, shape, edge, and color. From a set of hand labeled images, we accumulate prior knowledge of 2-D body shapes by learning their low-dimensional representations for inference of pose parameters. A data driven belief propagation Monte Carlo algorithm, utilizing importance sampling functions built from bottom-up visual cues, is proposed for efficient probabilistic inference. Contrasted to the few sequential statistical formulations in the literature, our algorithm integrates both top-down as well as bottom-up reasoning mechanisms, and can carry out the inference tasks in parallel. Experimental results demonstrate the potency and effectiveness of the proposed algorithm in estimating 2-D human pose from single images. 1.

This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. The question addressed here is how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. A key role is played by the Newton polytope of a statistical model. Our results are applied to the hidden Markov model and to the general Markov model on a binary tree. 1 Algebraic Statistics, Tropical Geometry, and Inference This paper presents a unified mathematical framework for probabilistic inference with statistical models, such as graphical models. Our approach is summarized as follows: (a) Statistical models are algebraic varieties. (b) Every algebraic variety can be tropicalized. (c) Tropicalized statistical models are fundamental for parametric inference. By a statistical model we mean a family of joint probability distributions for a collection of discrete

Phylogenetic hidden Markov models, or phylo-HMMs, are probabilistic models that consider not only the way substitutions occur through evolutionary history at each site of a genome, but also the way this process changes from one site to the next. By treating molecular evolution as a combination of two Markov processes—one that operates in the dimension of space (along a genome) and one that operates in the dimension of time (along the branches of a phylogenetic tree)—these models allow aspects of both sequence structure and sequence evolution to be captured. Moreover, as we will discuss, they permit key computations to be performed exactly and efficiently. Phylo-HMMs allow evolutionary information to be brought to bear on a wide variety of problems of sequence “segmentation, ” such as gene prediction and the identification of conserved elements. Phylo-HMMs were first proposed as a way of improving phylogenetic models that allow for variation among sites in the rate of substitution [8, 52]. Soon afterward, they were adapted for the problem of secondary structure

Abstract — This paper presents a 3D object representation framework. We develop a hierarchical model based on probabilistic correspondences and probabilistic relations between 3D visual features. Features at the bottom of the hierarchy are bound to local observations. Pairs of features that present strong geometric correlation are iteratively grouped into higher-level meta-features that encode probabilistic relative spatial relationships between their children. The model is instantiated by propagating evidence up and down the hierarchy using a Belief Propagation algorithm, which infers the pose of high-level features from local evidence and reinforces local evidence from globally consistent knowledge. We demonstrate how to use our framework to estimate the pose of a known object in an unknown scene, and provide a quantitative performance evaluation on synthetic data. I.

Abstract. This paper presents a probabilistic representation for 3D objects, and details the mechanism of inferring the pose of real-world objects from vision. Our object model has the form of a hierarchy of increasingly expressive 3D features, and probabilistically represents 3D relations between these. Features at the bottom of the hierarchy are bound to local perceptions; while we currently only use visual features, our method can in principle incorporate features from diverse modalities within a coherent framework. Model instances are detected using a Nonparametric Belief Propagation algorithm which propagates evidence through the hierarchy to infer globally consistent poses for every feature of the model. Belief updates are managed by an importance-sampling mechanism that is critical for efficient and precise propagation. We conclude with a series of pose estimation experiments on real objects, along with quantitative performance evaluation. Key words: Computer vision, 3D object representation, pose estimation, Nonparametric Belief Propagation. 1

Side-chain prediction is an important subtask in the protein-folding problem. We show that finding a minimal energy side-chain configuration is equivalent to performing inference in an undirected graphical model. The graphical model is relatively sparse yet has many cycles. We used this equivalence to assess the performance of approximate inference algorithms in a real-world setting. Specifically, we were interested in two questions: (1) which approximate inference algorithms give superior performance and (2) how does this performance compare to the state-of-the-art in computational biology. We looked at three tasks in side-chain graphical models — finding the minimal energy configuration, finding the M best configurations and approximating the free energy and conformational entropy. In all three subtasks we found that belief propagation gave the best results among the approximate inference algorithms and in many cases it outperformed the state-of-the-art in algorithms developed in the computational biology field. 1 1

Abstract. We present biRNA, a novel algorithm for prediction of binding sites between two RNAs based on minimization of binding free energy. Similar to RNAup approach [29], we assume the binding free energy is the sum of accessibility and the interaction free energies. Our algorithm maintains tractability and speed and also has two important advantages over previous similar approaches: 1) it is able to predict multiple simultaneous binding sites and 2) it computes a more accurate interaction free energy by considering both intramolecular and intermolecular base pairing. Moreover, biRNA can handle crossing interactions as well as hairpins interacting in a zigzag fashion. To deal with simultaneous accessibility of binding sites, our algorithm models their joint probability of being unpaired. Since computing the exact joint probability distribution is intractable, we approximate the joint probability by a polynomially representable graphical model namely a Chow-Liu tree-structured Markov Random Field. Experimental results show that biRNA outperforms RNAup and also support the accuracy of our approach. Our proposed Bayesian approximation of the Boltzmann joint probability distribution provides a powerful, novel framework that can also be utilized in other applications. 1