### Foundations of compositional models: structural properties

R. Jiroušek, V. Kratochvíl
2014 International Journal of General Systems
The paper is a follow-up of [R.J.: Foundations of compositional model theory. IJGS, 40(2011): 623-678], where basic properties of compositional models, as one of the approaches to multidimensional probability distributions representation and processing, were introduced. In fact, it is an algebraic alternative to graphical models, which does not use graphs to represent conditional independence statements. Here, these statements are encoded in a sequence of distributions to which an operator of
more » ... mposition -the key element of this theory -is applied in order to assemble a multidimensional model from its low-dimensional parts. In this paper, we show a way to read conditional independence relations, and to solve related topics, above all the socalled equivalence problem, i.e. the problem of recognizing whether two different structures induce the same system of conditional independence relations. Downloaded by [Vaclav Kratochvil] at 02:06 24 September 2014 2 R. Jiroušek and V. Kratochvíl parts. More precisely, a sequence of sets of variables -which will be called a model structure in this paper -plays the same role as a graph in the case of graphical modelling. Recall that several ways to read conditional independence statements have been designed in the area of graphical models. In the case of undirected graphs, conditional independence relations can be uncovered using a graph separation criterion. In the case of acyclic directed graphs, the so-called d-separation criterion designed by Pearl is often used (Pearl 1988 ). An alternative test for d-separation was devised in Lauritzen et al. (1990) . It is based on the notion of moralized ancestral graphs. In Section 3.2 of this paper, we show a way to read conditional independence relations for compositional models. A related topic is the so-called equivalence problem, i.e. the problem of recognizing whether two different structures induce the same system of conditional independence statements. For Bayesian networks, the problem was solved by Verma and Pearl (1991) ; two acyclic directed graphs induce the same independence structure if they have the same adjacencies and immoralities (the latter are special induced subgraphs). Later, a unique representation of a class of equivalent graphs was found, the so-called essential graph (Andersson, Madigan, and Perlman 1997). In the present paper, both these problems are solved for the compositional models in Section 3.3. Having two different structures inducing the same system of conditional independence statements, it may be of special importance to have an easy way to transform one structure into the other using some elementary operations. This issue was treated in a case of directed graphs in Chickering (1995) by legal arrow reversal. To solve this problem for compositional models, we introduce a special system of operations in Section 4. Section 5, incorporated into the text at the suggestion of the anonymous reviewer, describes the relationship between the compositional and graphical approaches to multidimensional probability distribution representation. In the last two sections of this paper, we show how the structural properties are manifested in the properties of the multidimensional probability distributions represented in the form of compositional models. Basic notions and notation In this paper, we deal with a finite system of finite-valued variables {u, v, x, . . .}, sets of which will be denoted by upper-case Roman characters such as K , U, V, W and Z , with possible indices. Ordered sequences of variable sets will be denoted by calligraphic characters like P = (U, W, Z , V ), P = (K 1 , K 2 , K 3 , K 4 , K 5 ), or, P = (K 1 , K 3 , K 5 , K 4 , K 2 ). Notice that here P = P because P is a reordering of P . Symbol |P| denotes the number of sets in the sequence, i.e. for the previously introduced sequences |P| = 4 and |P | = |P | = 5. Lower-case Greek characters will denote probability distributions, e.g. π(K ) will denote a probability distribution defined for variables from K . Its marginal distribution for variables from U ⊂ K will be denoted by either simply π(U ), or π ↓U . For U = ∅, π ↓∅ = 1. Conditional independence One of the most important notions of this paper, a concept of conditional independence, generalizes the well-known independence of variables. Definition 2.1 Consider a probability distribution π(K ) and three disjoint subsets U, V, Z ⊆ K such that both U, V = ∅. We say that groups of variables U and V are conditionally independent given Z for probability distribution π (in symbols U ⊥ ⊥V |Z [π ]) if