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

## Synopsis

• Usage:
localMarkov G
• Inputs:
• G, or
• Outputs:
• a list, whose entries are triples $\{A,B,C\}$ representing local Markov conditional independence statements of the form $A$ is independent of B given C'' that hold for G.

## 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}}, {{b, c}, {e}, {d, a}}, {{a, e}, {c}, {d, b}}, ------------------------------------------------------------------------ {{b}, {d, e}, {c, a}}, {{a, b}, {d}, {c, e}}} 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, 2, 3}, {7}, {4, 5, 6}}| +------------------------+------------------------+---------------------------+ |{{1, 4, 6}, {5}, {2, 3}}|{{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.