Conditional independence statements

ci_stmt(I::Vector{Int}, J::Vector{Int}, K::Vector{Int}; symmetric=true, disjoint=true)

A conditional independence statement asserting that I is independent of J given K. These parameters are lists of names of random variables. The sets I and J must be disjoint as this package cannot yet deal with functional dependencies.

If symmetric is true, CI statements are assumed to be symmetric in their I and J components. The constructor then reorders the arguments to make the I field lexicographically smaller than the J to ensure that comparisons and hashing respect the symmetry.

If disjoint is set to true, the constructor also removes elements in the intersection of I and K from I (and symmetrically removes the intersection of J and K from J).

As all three fields are sets, each of them may be deduplicated and sorted to ensure consistent comparison and hashing.

Examples

julia> ci_stmt([1], [2], [3,4])
[1 _||_ 2 | {3, 4}]

julia> ci_stmt([1], [2, 3], [4, 5])
[1 _||_ {2, 3} | {4, 5}]

 CI"I...,J...|K..."

A literal syntax for denoting CI statements is provided for cases in which all variables consist of a single digit. If I and J only consist of a single element, then even the comma may be omitted. Once the three sets are extracted, ci_stmt is called with default values for its options.

Examples

julia> CI"12|3"
[1 _||_ 2 | 3]

julia> CI"1,23|5424"
[1 _||_ 3 | {2, 4, 5}]

ci_statements(random_variables::Vector{Int})

Return a list of all elementary CI statements over a given set of variables. A CIStmt(I, J, K) is elementary if both I and J have only one element.

As a consequence of the semigraphoid properties, these statements are enough to describe the entire CI structure of a probability distribution.

Examples

julia> ci_statements([1, 2, 3, 4])
24-element Vector{CIStmt}:
 [1 _||_ 2 | {}]
 [1 _||_ 2 | 3]
 [1 _||_ 2 | 4]
 [1 _||_ 2 | {3, 4}]
 [1 _||_ 3 | {}]
 [1 _||_ 3 | 2]
 [1 _||_ 3 | 4]
 [1 _||_ 3 | {2, 4}]
 [1 _||_ 4 | {}]
 [1 _||_ 4 | 2]
 ⋮
 [2 _||_ 3 | {1, 4}]
 [2 _||_ 4 | {}]
 [2 _||_ 4 | 1]
 [2 _||_ 4 | 3]
 [2 _||_ 4 | {1, 3}]
 [3 _||_ 4 | {}]
 [3 _||_ 4 | 1]
 [3 _||_ 4 | 2]
 [3 _||_ 4 | {1, 2}]

ci_statements(R::MarkovRing)

Return all the CIStmt objects which can be formed on the random_variables(R).

julia> R = markov_ring(2, 2, 2, 2)
Markov ring over rational field for 4 random variables and states (2, 2, 2, 2)

julia> ci_statements(R)
24-element Vector{CIStmt}:
 [1 _||_ 2 | {}]
 [1 _||_ 2 | 3]
 [1 _||_ 2 | 4]
 [1 _||_ 2 | {3, 4}]
 [1 _||_ 3 | {}]
 [1 _||_ 3 | 2]
 [1 _||_ 3 | 4]
 [1 _||_ 3 | {2, 4}]
 [1 _||_ 4 | {}]
 [1 _||_ 4 | 2]
 ⋮
 [2 _||_ 3 | {1, 4}]
 [2 _||_ 4 | {}]
 [2 _||_ 4 | 1]
 [2 _||_ 4 | 3]
 [2 _||_ 4 | {1, 3}]
 [3 _||_ 4 | {}]
 [3 _||_ 4 | 1]
 [3 _||_ 4 | 2]
 [3 _||_ 4 | {1, 2}]

ci_statements(R::GaussianRing)

Return all the CIStmt objects which can be formed on the random_variables(R).

julia> R = gaussian_ring(3)
GaussianRing over Rational field in 6 variables
s[1, 1], s[1, 2], s[1, 3], s[2, 2], s[2, 3], s[3, 3]

julia> ci_statements(R)
6-element Vector{CIStmt}:
 [1 _||_ 2 | {}]
 [1 _||_ 2 | 3]
 [1 _||_ 3 | {}]
 [1 _||_ 3 | 2]
 [2 _||_ 3 | {}]
 [2 _||_ 3 | 1]

ci_ideal(R::MarkovRing, stmts)::MPolyIdeal

Return the ideal for the conditional independence statements given by stmts.

Examples

julia> R = markov_ring(2, 2, 2)
Markov ring over rational field for 3 random variables and states (2, 2, 2)

julia> ci_ideal(R, [CI"1,3|2", CI"2,3|1"])
Ideal generated by
  p[1, 1, 1]*p[2, 1, 2] - p[2, 1, 1]*p[1, 1, 2]
  p[1, 2, 1]*p[2, 2, 2] - p[2, 2, 1]*p[1, 2, 2]
  p[1, 1, 1]*p[1, 2, 2] - p[1, 2, 1]*p[1, 1, 2]
  p[2, 1, 1]*p[2, 2, 2] - p[2, 2, 1]*p[2, 1, 2]

ci_ideal(R::GaussianRing, stmts::Vector{CIStmt})

Return the ideal for the conditional independence statements given by stmts.

Examples

julia> R = gaussian_ring(3)
GaussianRing over Rational field in 6 variables
s[1, 1], s[1, 2], s[1, 3], s[2, 2], s[2, 3], s[3, 3]

julia> ci_ideal(R, [CI"1,2|", CI"1,2|3"])
Ideal generated by
  s[1, 2]
  s[1, 2]*s[3, 3] - s[1, 3]*s[2, 3]

make_elementary(stmt::CIStmt; semigaussoid=false)

Convert a CIStmt into an equivalent list of CIStmts all of which are elementary. The default operation assumes the semigraphoid axioms and converts $[I \mathrel{⫫} J \mid K]$ into the list consisting of $[i \mathrel{⫫} j \mid L]$ for all $i \in I$, $j \in J$ and $L$ in the interval $K \subseteq L \subseteq (I \cup J \cup K) \setminus \{i,j\}$.

If semigaussoid is true, the stronger semigaussoid axioms are assumed and L in the above procedure does not range in the interval above K but is always fixed to K. Semigaussoids are also known as compositional graphoids.

Examples

julia> make_elementary(CI"12,34|56")
16-element Vector{CIStmt}:
 [1 _||_ 3 | {5, 6}]
 [1 _||_ 3 | {5, 6, 2}]
 [1 _||_ 3 | {5, 6, 4}]
 [1 _||_ 3 | {5, 6, 2, 4}]
 [1 _||_ 4 | {5, 6}]
 [1 _||_ 4 | {5, 6, 2}]
 [1 _||_ 4 | {5, 6, 3}]
 [1 _||_ 4 | {5, 6, 2, 3}]
 [2 _||_ 3 | {5, 6}]
 [2 _||_ 3 | {5, 6, 1}]
 [2 _||_ 3 | {5, 6, 4}]
 [2 _||_ 3 | {5, 6, 1, 4}]
 [2 _||_ 4 | {5, 6}]
 [2 _||_ 4 | {5, 6, 1}]
 [2 _||_ 4 | {5, 6, 3}]
 [2 _||_ 4 | {5, 6, 1, 3}]

julia> make_elementary(CI"12,34|56"; semigaussoid=true)
4-element Vector{CIStmt}:
 [1 _||_ 3 | {5, 6}]
 [1 _||_ 4 | {5, 6}]
 [2 _||_ 3 | {5, 6}]
 [2 _||_ 4 | {5, 6}]