chapter  31
Graphical Modeling of Spatial Health Data
ByAdrian Dobra
Pages 20

Graphical models (Whittaker, 1990; Lauritzen, 1996) that encode multivariate independence and conditional independence relationships among observed variables X = (X1, . . . , Xp) have a widespread use in major scientific areas (e.g., biomedical and social sciences). In particular, a Gaussian graphical model (GGM) is obtained by setting off-diagonal elements of the precision matrix K = Σ−1 to zero of a p-dimensional multivariate normal model (Dempster, 1972). Employing a GGM instead of a multivariate normal model leads to a significant reduction in the number of parameters that need to be estimated if most elements of K are constrained to be zero and p is large. A pattern of zero constraints in K can be recorded as an undirected graph G = (V,E), where the set of vertices V = {1, 2, . . . , p} represent observed variables, while the set of edges E ⊂ V ×V link all the pairs of vertices that correspond to off-diagonal elements of K that have not been set to zero. The absence of an edge between Xv1 and Xv2 corresponds with the conditional independence of these two random variables given the rest and is denoted by Xv1 Xv2 | XV \{v1,v2} (Wermuth, 1976). This is called the pairwise Markov property relative to G, which in turn implies the local as well as the global Markov properties relative to G (Lauritzen, 1996). The local Markov property plays a key role since it gives the regression model induced by G on each variable Xv. More explicitly, consider the neighbors of v in G, that is, the set of vertices v′ ∈ V such that (v, v′) ∈ E. We denote this set by bdG(v). The local Markov property relative to G says that Xv XV \{{v}∪bdG(v)} | XbdG(v). This is precisely the statement we make when we drop the variables {Xv′ : v′ ∈ V \bdG(v)} from the regression of Xv on {Xv′ : v′ ∈ V \ {v}}.