The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

/pdf/graphical-models-exponential-families-and-variational-3i6dwlfta1.pdf

Graphical Models, Exponential Families, and Variational Inference

This paper reviews the literature on Bayesian experimental design. A unified view of this topic is presented, based on a decision-theoretic approach. This framework casts criteria from the Bayesian literature of design as part of a single coherent approach. The decision-theoretic structure incorporates both linear and nonlinear design problems and it suggests possible new directions to the experimental design problem, motivated by the use of new utility functions. We show that, in some special cases of linear design problems, Bayesian solutions change in a sensible way when the prior distribution and the utility function are modified to allow for the specific structure of the experiment. The decision-theoretic approach also gives a mathematical justification for selecting the appropriate optimality criterion.

/pdf/bayesian-experimental-design-a-review-1z4pqf25ac.pdf

Bayesian Experimental Design: A Review

The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].

The maximum clique problem

Motivation: In recent years, there have been various efforts to overcome the limitations of standard clustering approaches for the analysis of gene expression data by grouping genes and samples simultaneously. The underlying concept, which is often referred to as biclustering, allows to identify sets of genes sharing compatible expression patterns across subsets of samples, and its usefulness has been demonstrated for different organisms and datasets. Several biclustering methods have been proposed in the literature; however, it is not clear how the different techniques compare with each other with respect to the biological relevance of the clusters as well as with other characteristics such as robustness and sensitivity to noise. Accordingly, no guidelines concerning the choice of the biclustering method are currently available.

Results: First, this paper provides a methodology for comparing and validating biclustering methods that includes a simple binary reference model. Although this model captures the essential features of most biclustering approaches, it is still simple enough to exactly determine all optimal groupings; to this end, we propose a fast divide-and-conquer algorithm (Bimax). Second, we evaluate the performance of five salient biclustering algorithms together with the reference model and a hierarchical clustering method on various synthetic and real datasets for Saccharomyces cerevisiae and Arabidopsis thaliana. The comparison reveals that (1) biclustering in general has advantages over a conventional hierarchical clustering approach, (2) there are considerable performance differences between the tested methods and (3) already the simple reference model delivers relevant patterns within all considered settings.

Availability: The datasets used, the outcomes of the biclustering algorithms and the Bimax implementation for the reference model are available at http://www.tik.ee.ethz.ch/sop/bimax

Contact: bleuler@tik.ee.ethz.ch

Supplementary information: Supplementary data are available at http://www.tik.ee.ethz.ch/sop/bimax

/pdf/a-systematic-comparison-and-evaluation-of-biclustering-cjtlekwmk8.pdf

A systematic comparison and evaluation of biclustering methods for gene expression data

Pseudo-boolean optimization

The quadratic knapsack (QK) model naturally arises in a variety of problems in operations research, statistics and combinatorics. Some “upper planes” for the QK problem are derived, and their different uses in a branch-and-bound scheme for solving such a problem are discussed. Some theoretical results concerning the class of all upper planes, as well as extensive computational experience, are reported.

Quadratic knapsack problems

This paper deals with the following graph partitioning problem. Consider a connected graph with n nodes, p of which are centers, while the remaining ones are units. For each unit-center pair there is a fixed service cost and the goal is to find a partition into connected components such that each component contains only one center and the total service cost is minimum. This problem is known to be NP-hard on general graphs, and here we show that it remains such even if the service cost is monotone and the graph is bipartite. However, in this paper we derive some polynomial time algorithms for trees. For this class of graphs we provide several reformulations of the problem as integer linear programs proving the integrality of the corresponding polyhedra. As a consequence, the tree partitioning problem can be solved in polynomial time either by linear programming or by suitable convex nondifferentiable optimization algorithms. Moreover, we develop a dynamic programming algorithm, whose recursion is based on sequences of minimum weight closure problems, which solves the problem on trees in O(np) time. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

Polynomial algorithms for partitioning a tree into single-center subtrees to minimize flat service costs

Consensus algorithms for the generation of all maximal bicliques

The Political Districting problem has been studied since the 60’s and many different models and techniques have been proposed with the aim of preventing districts’ manipulation which may favor some specific political party (gerrymandering). A variety of Political Districting models and procedures was provided in the Operations Research literature, based on single- or multiple-objective optimization. Starting from the forerunning papers published in the 60’s, this article reviews some selected optimization models and algorithms for Political Districting which gave rise to the main lines of research on this topic in the Operations Research literature of the last five decades.

Political Districting: from classical models to recent approaches

Preface Part I. Classification and Evaluation of Electoral Systems. 1. The four phases of an electoral process 2. A unified description of electoral systems 3. Performance of an electoral system Part II. Designing Electoral Systems. 4. No electoral system is perfect 5. Basic properties for electoral formulas 6. Integer optimization approach 7. Rewarded and punished parties 8. Mixed electoral systems Part III. Designing Electoral Districts. 9. Traps in electoral district plans 10. Criteria for political districting 11. Indicators for political districting 12. Optimization Models Part IV. The Process of Electoral Reform: A Retrospective Critical View of a Political Scientist. 13. A difficult crossroad 14. The planning and politics of political reform Part V. A Short Guide to the Literature.

Evaluation and Optimization of Electoral Systems

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