Probabilistic graphical models (PGMs) have become a popular tool for computational analysis of biological data in a variety of domains. But, what exactly are they and how do they work? How can we use PGMs to discover patterns that are biologically relevant? And to what extent can PGMs help us formulate new hypotheses that are testable at the bench? This note sketches out some answers and illustrates the main ideas behind the statistical approach to biological pattern discovery.

The Bayesian framework is a well-studied and successful framework for inductive reasoning, which includes hypothesis testing and confirmation, parameter estimation, sequence prediction, classification, and regression. But standard statistical guidelines for choosing the model class and prior are not always available or fail, in particular in complex situations. Solomonoff completed the Bayesian framework by providing a rigorous, unique, formal, and universal choice for the model class and the prior. We discuss in breadth how and in which sense universal (non-i.i.d.) sequence prediction solves various (philosophical) problems of traditional Bayesian sequence prediction. We show that Solomonoff's model possesses many desirable properties: Strong total and weak instantaneous bounds, and in contrast to most classical continuous prior densities has no zero p(oste)rior problem, i.e. can confirm universal hypotheses, is reparametrization and regrouping invariant, and avoids the old-evidence and updating problem. It even performs well (actually better) in non-computable environments.

This paper proposes a new neuro-rough model for modelling the risk of HIV from demographic data. The model is formulated using Bayesian framework and trained using Markov Chain Monte Carlo method and Metropolis criterion. When the model was tested to estimate the risk of HIV infection given the demographic data it was found to give the accuracy of 62% as opposed to 58% obtained from a Bayesian formulated rough set model trained using Markov chain Monte Carlo method and 62% obtained from a Bayesian formulated multi-layered perceptron (MLP) model trained using hybrid Monte. The proposed model is able to combine the accuracy of the Bayesian MLP model and the transparency of Bayesian rough set model. Keywords: Neuro-rough model, multi-layered perceptron, Bayesian, HIV modelling Introduction The role of machine learning is to be able to make predictions given a set of inputs.

Numerous formalisms and dedicated algorithms have been designed in the last decades to model and solve decision making problems. Some formalisms, such as constraint networks, can express "simple" decision problems, while others are designed to take into account uncertainties, unfeasible decisions, and utilities. Even in a single formalism, several variants are often proposed to model different types of uncertainty (probability, possibility...) or utility (additive or not). In this article, we introduce an algebraic graphical model that encompasses a large number of such formalisms: (1) we first adapt previous structures from Friedman, Chu and Halpern for representing uncertainty, utility, and expected utility in order to deal with generic forms of sequential decision making; (2) on these structures, we then introduce composite graphical models that express information via variables linked by "local" functions, thanks to conditional independence; (3) on these graphical models, we finally define a simple class of queries which can represent various scenarios in terms of observabilities and controllabilities. A natural decision-tree semantics for such queries is completed by an equivalent operational semantics, which induces generic algorithms. The proposed framework, called the Plausibility-Feasibility-Utility (PFU) framework, not only provides a better understanding of the links between existing formalisms, but it also covers yet unpublished frameworks (such as possibilistic influence diagrams) and unifies formalisms such as quantified boolean formulas and influence diagrams. Our backtrack and variable elimination generic algorithms are a first step towards unified algorithms.

This dissertation presents several new methods of supervised and unsupervised learning of word sense disambiguation models. The supervised methods focus on performing model searches through a space of probabilistic models, and the unsupervised methods rely on the use of Gibbs Sampling and the Expectation Maximization (EM) algorithm. In both the supervised and unsupervised case, the Naive Bayesian model is found to perform well. An explanation for this success is presented in terms of learning rates and bias-variance decompositions.

We consider the problem of estimating the parameters of a Gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse. Our approach is to solve a maximum likelihood problem with an added l_1-norm penalty term. The problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes. We present two new algorithms for solving problems with at least a thousand nodes in the Gaussian case. Our first algorithm uses block coordinate descent, and can be interpreted as recursive l_1-norm penalized regression. Our second algorithm, based on Nesterov's first order method, yields a complexity estimate with a better dependence on problem size than existing interior point methods. Using a log determinant relaxation of the log partition function (Wainwright & Jordan (2006)), we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case. We test our algorithms on synthetic data, as well as on gene expression and senate voting records data.

Conformal prediction uses past experience to determine precise levels of confidence in new predictions. Given an error probability $\epsilon$, together with a method that makes a prediction $\hat{y}$ of a label $y$, it produces a set of labels, typically containing $\hat{y}$, that also contains $y$ with probability $1-\epsilon$. Conformal prediction can be applied to any method for producing $\hat{y}$: a nearest-neighbor method, a support-vector machine, ridge regression, etc. Conformal prediction is designed for an on-line setting in which labels are predicted successively, each one being revealed before the next is predicted. The most novel and valuable feature of conformal prediction is that if the successive examples are sampled independently from the same distribution, then the successive predictions will be right $1-\epsilon$ of the time, even though they are based on an accumulating dataset rather than on independent datasets. In addition to the model under which successive examples are sampled independently, other on-line compression models can also use conformal prediction. The widely used Gaussian linear model is one of these. This tutorial presents a self-contained account of the theory of conformal prediction and works through several numerical examples. A more comprehensive treatment of the topic is provided in "Algorithmic Learning in a Random World", by Vladimir Vovk, Alex Gammerman, and Glenn Shafer (Springer, 2005).

Observations consisting of measurements on relationships for pairs of objects arise in many settings, such as protein interaction and gene regulatory networks, collections of author-recipient email, and social networks. Analyzing such data with probabilisic models can be delicate because the simple exchangeability assumptions underlying many boilerplate models no longer hold. In this paper, we describe a latent variable model of such data called the mixed membership stochastic blockmodel. This model extends blockmodels for relational data to ones which capture mixed membership latent relational structure, thus providing an object-specific low-dimensional representation. We develop a general variational inference algorithm for fast approximate posterior inference. We explore applications to social and protein interaction networks.

A discriminant function f: X R yields a classifier sgn(f(x)) { 1, 1} that represents our guess of the label Y of a future observation X and we err if the margin y · f(x) 0. Since observations x for which the conditional probability η(x) P{Y 1 X x} (1) is close to 1/2 are difficult to classify, we introduce a reject option for classifiers, by allowing for a third decision, R (reject), expressing doubt. We built in the reject option by using a threshold value 0 τ 1 as follows. Given a discriminant function f: X R, we report sgn(f(x)) { 1,1} if f(x) τ, but we withhold decision if f(x) τ and report R. We assume that the cost of making a wrong decision is 1 and the cost of utilizing the reject option is d 0. The appropriate risk function is then E[l(Yf(X))] P{Yf(X) τ} dP{ Yf(X) τ} (2) Research is supported in part by NSF grant DMS 0706829 155 M. Wegkamp/Lasso type classifiers with a reject option 156

This paper proposes an approach to training rough set models using Bayesian framework trained using Markov Chain Monte Carlo (MCMC) method. The prior probabilities are constructed from the prior knowledge that good rough set models have fewer rules. Markov Chain Monte Carlo sampling is conducted through sampling in the rough set granule space and Metropolis algorithm is used as an acceptance criteria. The proposed method is tested to estimate the risk of HIV given demographic data. The results obtained shows that the proposed approach is able to achieve an average accuracy of 58% with the accuracy varying up to 66%. In addition the Bayesian rough set give the probabilities of the estimated HIV status as well as the linguistic rules describing how the demographic parameters drive the risk of HIV.