# Conditional independence statements

Conditional independence (CI) statements over a ground set $N$ are triples of pairwise disjoint subsets $I, J, K \subseteq N$ denoted as $[I \mathrel{⫫} J \mid K]$. The ground set indexes objects under consideration and the CI statement asserts that once the objects in $K$ are "controlled" (conditioned on, in statistical language), the objects in $I$ reveal no information about (are independent of) the objects in $J$.

The functionality documented here deals with CI statements are combinatorial objects. Collections of CI statements are often used to state Markov properties of graphical models in statistics and are ultimately used to define ideals. Their interpretations as polynomial equations depend on the ambient ring (markov_ring or gaussian_ring).

ci_stmtMethod
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 symetrically 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}]
Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source
@CI_strMacro
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}]
Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

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

Compares CIStmts for identity in all their three fields.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

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

Computes the hash of a CIStmt.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source
ci_statementsMethod
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}]
Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source
make_elementaryMethod
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}]
Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source