Markov Independence (Continued)

Transcription

1 Markov Independence (Continued) As an Undirected Graph Basic idea: Each variable V is represented as a vertex in an undirected graph G = (V(G), E(G)), with set of vertices V(G) and set of edges E(G) the independence relation is encoded as the absence of edges; a missing edge between vertices and X 2 indicates that and X 2 are (conditionally) independent 1 2 Global Markov Property Separation Let G = (V(G), E(G)) be an undirected graph, and let X, Y, Z V be sets of vertices in G. The set Z separates X and Y, denoted as X G Y Z if every path from a vertex in X to a vertex in Y contains at least one vertex in Z Example Consider the following undirected graph G: X 7 X 2 X 5 Remarks X 6 This criterion is known as the global Markov property or separation criterion for undirected graphs Note that G indicates that the independence relation is defined in terms of G (cf. P ) If there are no paths between two vertices X and Y, then {X} G {Y } 3 { } G {, X 6 } {X 2 } { } G {X 6 } {X 2, X 5 } { } G {X 6 } {, X 2,, X 5 } { } G {X 5 } { }, as the path X 2 X 5 does not contain {, X 5, X 6 } G {X 7 } 4

2 D-map and I-map Let V be a set of variables and let be an independence relation defined on V. Let G = (V(G), E(G)) be an undirected graph, then for each X, Y, Z V: G is called an undirected dependence map, D-map for short, if Examples D-maps Let V = {, X 2,, } be a set of variables, and consider the independence relation, defined by { } { } {X 2, } {X 2 } { } {, } The following undirected graphs are examples of D-maps: X Y Z X G Y Z G is called an undirected independence map, I-map for short, if X G Y Z X Y Z G is called an undirected perfect map, or P-map for short, if G is both a D-map and an I-map, or, equivalently X Y Z X G Y Z 5 6 D-maps and I-maps Again Let be an independence relation. An D- maps and I-maps are limited in expressiveness in the following sense: Examples of I-maps Let V = {, X 2,, } be a set of variables, and consider the independence relation : { } { } {X 2, } {X 2 } { } {, } The following undirected graphs are examples of I-maps: A pair of neighbour vertices in a D-map for are dependent. However, not all dependent variables are neighbours A pair of non-neighbour variables in an I- map for corresponds to independent variables, but not each pair of independent variables in an I-map are non-neighbours 7 (So, what is the P-map?) 8

3 Obvious Properties Lemma For each independence relation there exists an undirected D-map. Proof: The undirected graph G = (V, ) is a D-map for Expressiveness: Directed vs Undirected Graphs Directed graphs are more subtle when it comes to expressing independence information than undirected graphs Lemma For each independence relation there exists an undirected I-map. vs Proof: The undirected graph G = (V, V V) is an I-map for 9 10 d-separation: 3 Situations Example Blockage A chain k (= path in undirected underlying graph) in an acyclic directed graph G = (V(G), A(G)) can be blocked: Diverging X 2 X 5 X 2 blocks (d-separates) and : { } { } {X 2 } X 6 Serial X 2 blocks (d-separates) and : { } { } {X 2 } Converging X 2 d-connects and : { } { } {X 2 } (same holds for successors of X 2 ); furthermore: { } { } 11 X 7 The chain, X 2, X 5 from to X 5 is blocked by {X 2 } The chain, X 2, X 5, X 6 from to X 6 is blocked by {X 5 }, and also by {X 2 } and {X 2, X 5 } The chain,, X 6, X 5 from to X 5 is blocked by { } and {, X 6 }, but not by {X 6 } 12

4 Directed Global Markov Property = d-separation Let G = (V(G), A(G)) be an acyclic directed graph, and let X, Y, Z V be sets of vertices in G. The set Z d-separates X and Y, denoted as X d G Y Z if every chain from a vertex in X to a vertex in Y is blocked by Z There is also a Local Markov Property Let G = (V(G), A(G)) be an acyclic, directed graph, then the following local Markov property holds: {X i } d G ν(x i) π(x i ) with ν(x i ) non-descendants of vertex X i, and π(x i ) set of parents ν( X i ) Remarks This criterion is known as the global Markov property or d-separation criterion for acyclic directed graphs Note that d G indicates that the independence relation is defined in terms of G (cf. P ) X i π( ) X i ν( X i ) Directed D-map and I-map Let V be a set of variables and let be an independence relation defined on V. Let G = (V(G), A(G)) be an acyclic directed graph, then for each X, Y, Z V: G is called a directed dependence map, D- map for short, if Examples Directed I-maps Consider the following independence relation : { } {X 2 } {, X 2 } { } { } and the following directed I-maps for : X 2 X 2 X Y Z X d G Y Z G is called a directed independence map, I-map for short, if X d G Y Z X Y Z G is called a directed perfect map, or P- map for short, if G is both a D-map and an I-map, or, equivalently X Y Z X d G Y Z 15 X 2 X 2 16

5 Minimal directed I-map In the context of Bayesian networks, we are interested in I-maps that contain as few arcs as possible (makes probability tables smaller), i.e. minimal directed I-maps Let V be a set of variables, and let G = V(G), A(G)) be an acyclic directed graph. G is called a minimal directed I-maps for, if G is a directed I-map for, and none of the subgraphs of G is a directed I-map for G Example 17