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GraphicalModels :: localMarkov

localMarkov -- local Markov statements for a graph or a directed graph

Synopsis

Description

Given an undirected graph $G$, a local Markov statement is of the form \{$v$, non-neighbours($v$), neighbours($v$)\} . That is, every vertex $v$ of $G$ is independent of its non-neighbours given its neighbours.

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

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

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

o1 : Graph
i2 : localMarkov G

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

o2 : List

Given a directed acyclic graph $G$, local Markov statements are of the form \{$v$, nondescendents($v$) - parents($v$), parents($v$)\} . In other words, every vertex $v$ of $G$ is independent of its nondescendents (excluding parents) given its parents.

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

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

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

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

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

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

Ways to use localMarkov :

For the programmer

The object localMarkov is a method function.