Conditional independence statements

ci_stmt(I::Vector{<:VarName}, J::Vector{<:VarName}, K::Vector{<:VarName}; symmetric=true, semigraphoid=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 semigraphoid 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(["A"], ["B"], ["X"])
[A _||_ B | X]

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 variable names consist of a single character. 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.

Examples

julia> CI"AB|X"
[A _||_ B | X]

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

Base.:(==)(lhs::CIStmt, rhs::CIStmt)

Compares CIStmts for identity in all their three fields.

Base.hash(stmt:;CIStmt, h::UInt)

Computes the hash of a CIStmt.

ci_statements(random_variables::Vector{<:VarName})

Return a list of all elementary CI statements over a given set of variable names. 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(["A", "B", "X", "Y"])
24-element Vector{CIStmt}:
 [A _||_ Y | {}]
 [A _||_ Y | B]
 [A _||_ Y | X]
 [A _||_ Y | {B, X}]
 [B _||_ Y | {}]
 [B _||_ Y | A]
 [B _||_ Y | X]
 [B _||_ Y | {A, X}]
 [X _||_ Y | {}]
 [X _||_ Y | A]
 ⋮
 [A _||_ X | {B, Y}]
 [B _||_ X | {}]
 [B _||_ X | A]
 [B _||_ X | Y]
 [B _||_ X | {A, Y}]
 [A _||_ B | {}]
 [A _||_ B | X]
 [A _||_ B | Y]
 [A _||_ B | {X, Y}]

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}]