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GraphicalModels :: pairMarkov

pairMarkov -- pairwise Markov statements for a graph or a directed graph

Synopsis

Description

Given an undirected graph G, pairwise Markov statements are statements of the form {v, w, all other vertices} for each pair of non-adjacent vertices v and w of G.

For example, for the undirected 5-cycle graph G, that is, the graph on 5 vertices with edges a---b---c---d---e---a, we get the following pairwise Markov statements:

i1 : G = graph({{a,b},{b,c},{c,d},{d,e},{e,a}})

o1 = Graph{a => set {b, e}}
           b => set {a, c}
           c => set {b, d}
           d => set {c, e}
           e => set {a, d}

o1 : Graph
i2 : pairMarkov G

o2 = {{{b}, {d}, {c, a, e}}, {{a}, {d}, {b, c, e}}, {{c}, {e}, {b, d, a}},
     ------------------------------------------------------------------------
     {{a}, {c}, {b, d, e}}, {{b}, {e}, {c, d, a}}}

o2 : List

Given a directed graph G, pairwise Markov statements are statements of the form {v, w, nondescendents(G,v)-w} for each vertex v of G and each non-descendent vertex w of v. In other words, for every vertex v of G and each nondescendent w of v, this method returns the statement: v is independent of w given all other nondescendents.

For example, given the digraph D on 7 vertices with edges 1 →2, 1 →3, 2 →4, 2 →5, 3 →5, 3 →6, 4 →7, 5 →7, and 6→7, we get the following pairwise Markov statements:

i3 : D = digraph {{1,{2,3}}, {2,{4,5}}, {3,{5,6}}, {4,{7}}, {5,{7}},{6,{7}},{7,{}}}

o3 = Digraph{1 => set {2, 3}}
             2 => set {4, 5}
             3 => set {5, 6}
             4 => set {7}
             5 => set {7}
             6 => set {7}
             7 => set {}

o3 : Digraph
i4 : netList pack (3, pairMarkov D)

     +------------------------+---------------------------+---------------------------+
o4 = |{{2}, {6}, {1, 3}}      |{{4}, {6}, {1, 2, 3, 5}}   |{{2}, {6}, {1, 3, 4, 5}}   |
     +------------------------+---------------------------+---------------------------+
     |{{3}, {4}, {1, 2}}      |{{2}, {3}, {4, 1}}         |{{1}, {4}, {2, 3, 5, 6}}   |
     +------------------------+---------------------------+---------------------------+
     |{{4}, {5}, {1, 2, 3, 6}}|{{3}, {7}, {1, 2, 4, 5, 6}}|{{1}, {7}, {2, 3, 4, 5, 6}}|
     +------------------------+---------------------------+---------------------------+
     |{{2}, {3}, {1, 6}}      |{{1}, {6}, {2, 3, 4, 5}}   |{{5}, {6}, {1, 2, 3, 4}}   |
     +------------------------+---------------------------+---------------------------+
     |{{1}, {5}, {2, 3, 4, 6}}|{{3}, {4}, {1, 2, 5, 6}}   |{{2}, {7}, {1, 3, 4, 5, 6}}|
     +------------------------+---------------------------+---------------------------+

This method displays only non-redundant statements. In general, given a set S of conditional independent statements and a statement s, then we say that s is a a redundant statement if s can be obtained from the statements in S using the semigraphoid axioms of conditional independence: symmetry, decomposition, weak union, and contraction as described in Section 1.1 of Judea Pearl, Causality: models, reasoning, and inference, Cambridge University Press. We do not use the intersection axiom since it is only valid for strictly positive probability distributions.

See also

  • localMarkov -- local Markov statements for a graph or a directed graph
  • globalMarkov -- global Markov statements for a graph or a directed graph

Ways to use pairMarkov :

  • pairMarkov(Digraph)
  • pairMarkov(Graph)