Introduction to Graphical Modelling

Abstract

The aim of this chapter is twofold. In the first part (Sections 12.1, 12.2 and 12.3) we will provide a brief overview of the mathematical and statistical foundations of graphical models, along with their fundamental properties, estimation and basic inference procedures. In particular we will develop Markov networks (also known as Markov random fields) and Bayesian networks, which are the subjects of most past and current literature on graphical models. In the second part (Section 12.4) we will review some applications of graphical models in systems biology. Graphical Structures and Random VariablesGraphical models are a class of statistical models which combine the rigour of a probabilistic approach with the intuitive representation of relationships given by graphs. They are composed by two parts:1. a set X = {X 1 , X 2 , . . . , X p } of random variables describing the quantities of interest. The statistical distribution of X is called the global distribution of the data, while the components it factorises into are called local distributions.2. a graph G = (V, E) in which each vertex v ∈ V, also called a node, is associated with one of the random variables in X (they are usually referred to interchangeably). Edges e ∈ E, also called links, are used to express the dependence structure of the data (the set of dependence relationships among Graph representations meet our earlier requirements of explicitness, saliency, and stability. The links in the graph permit us to express directly and quantitatively the dependence relationships, and the graph topology displays these relationships explicitly and preserves them, under any assignment of numerical parameters.The nature of the link outlined above between the dependence structure of the data and its graphical representation is given again by Pearl (1988) in terms of conditional independence (denoted with ⊥ ⊥ P ) and graphical separation (denoted with ⊥ ⊥ G ).Definition 12.1.1 A graph G is a dependency map (or D-map) of the probabilistic dependence structure P of X if there is a one-to-one correspondence between the random variables in X and the nodes V of G, such that for all disjoint subsets A, B, C of X we have Similarly, G is an independency map (or I-map) of P ifG is said to be a perfect map of P if it is both a D-map and an I-map, that is (12.3) and in this case P is said to be isomorphic to G.Note that this definition does not depend on a particular characterisation of graphical separation, and therefore on the type of graph used in the graphical model. In fact both Markov networks (Whittaker 1990) and Bayesian networks (Pearl 1988), which are by far the two most common classes of graphical models treated in literature, are defined as minimal I-maps even though the former use undirected graphs an the latter use directed acyclic graphs. Minimality requires that, if the dependence structure P of X can be expressed by multiple graphs, we must use the one with the minimum number of edges; if any further edge is removed then the graph is no longer an I-map of P...

A network perspective of engaging patients in specialist and chronic illness care: The 2014 International Health Policy Survey

“…To understand the overall relationships between the variables in the 2014 IHPS, we used a probabilistic graphical model, known as Bayesian networks for the analysis. [ 13 , 19 , 20 ] Bayesian network analysis adopted Bayesian statistics for network analysis. [ 19 ] In Bayesian networks, variables were presented as nodes and the arcs were used to present the relationships between variables in terms of conditional probabilities.…”

“…[ 19 ] In Bayesian networks, variables were presented as nodes and the arcs were used to present the relationships between variables in terms of conditional probabilities. [ 13 , 19 , 20 ] Bayesian statistics built on Bayes’ theorem that described the relationships between prior probabilities and posterior probabilities based on given events were used to propagate information between nodes or variables. [ 19 ] In other words, the probability distribution of one variable was related to its parent variable.…”

Bayesian immunological model development from the literature: Example investigation of recent thymic emigrants

“…The selected form of the model should also facilitate visual display (Tufte, 2001) so that results can be more readily conveyed to the target audience. The rise in graphical models (e.g., Scutari and Strimmer, 2008) may due to the fact that these are quite flexible in terms of be the functional forms that they can accommodate but they also lend themselves readily to visual display that aids interpretation.…”

Optimizing Hospital-wide Patient Scheduling

“…Pearl [160] and Wasserman [231] provide further theoretical properties of Bayesian networks and other probabilistic graphical models. For an overview of statistical graphical models with applications in systems biology, see Nagarajan et al [144] as well as Scutari and Strimmer [200]. The Markov blanket of a vertex v 2 V, denoted by MB.v/, is a minimal subset of vertices containing vertex v, its direct parents and direct children as well as all direct parents of the children of v. The Markov blanket of vertex v contains all the variables needed to predict the value of that variable, since v is conditionally independent of all other variables given its Markov blanket.…”