Discrete random variables

markov_ring(rvs::Pair{<:VarName, <:AbstractArray}...; unknown::VarName="p", K::Field=QQ, cached=false)::MarkovRing
tensor_ring(rvs::Pair{<:VarName, <:AbstractArray}...; unknown::VarName="p", K::Field=QQ, cached=false)::MarkovRing

The polynomial ring whose unknowns are the entries of a probability tensor. rvs is a list of pairs X => Q where X is the name of a random variable and Q is the list of states it takes. The polynomial ring being constructed will have one variable for each element in the cartesian product of the Qs. It is a multivariate polynomial ring whose variables are named p[...] and whose coefficient field K is by default QQ. These settings can be changed via the optional arguments.

If cached is true, the internally generated polynomial ring will be cached.

The name tensor_ring is an alias for the constructor markov_ring because that is really what a MarkovRing is: the coordinate ring of tensors of a fixed format. The name MarkovRing is kept for compatibility in terminology with the Macaulay2 package GraphicalModels.

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

markov_ring(rvs::Pair{<:VarName, <:AbstractArray}...; unknown::VarName="p", K::Field=QQ, cached=false)::MarkovRing
tensor_ring(rvs::Pair{<:VarName, <:AbstractArray}...; unknown::VarName="p", K::Field=QQ, cached=false)::MarkovRing

The polynomial ring whose unknowns are the entries of a probability tensor. rvs is a list of pairs X => Q where X is the name of a random variable and Q is the list of states it takes. The polynomial ring being constructed will have one variable for each element in the cartesian product of the Qs. It is a multivariate polynomial ring whose variables are named p[...] and whose coefficient field K is by default QQ. These settings can be changed via the optional arguments.

If cached is true, the internally generated polynomial ring will be cached.

The name tensor_ring is an alias for the constructor markov_ring because that is really what a MarkovRing is: the coordinate ring of tensors of a fixed format. The name MarkovRing is kept for compatibility in terminology with the Macaulay2 package GraphicalModels.

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

ring(R::MarkovRing)

Return the multivariate polynomial ring inside R.

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

julia> ring(R)
Multivariate polynomial ring in 16 variables p[1, 1, 1, 1], p[2, 1, 1, 1], p[1, 2, 1, 1], p[2, 2, 1, 1], ..., p[2, 2, 2, 2]
  over rational field

random_variables(R::MarkovRing)

Return the list of random variables used to create the MarkovRing.

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

julia> random_variables(R)
4-element Vector{Union{Char, AbstractString, Symbol}}:
 "A"
 "B"
 "X"
 "Y"

ci_statements(R::MarkovRing)

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

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

julia> ci_statements(R)
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}]

unknowns(R::MarkovRing)

Return the generators of the polynomial ring.

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

julia> unknowns(R)
Dict{NTuple{4, Int64}, QQMPolyRingElem} with 16 entries:
  (2, 2, 2, 2) => p[2, 2, 2, 2]
  (2, 2, 2, 1) => p[2, 2, 2, 1]
  (1, 2, 1, 2) => p[1, 2, 1, 2]
  (1, 2, 1, 1) => p[1, 2, 1, 1]
  (2, 2, 1, 2) => p[2, 2, 1, 2]
  (2, 2, 1, 1) => p[2, 2, 1, 1]
  (1, 1, 2, 2) => p[1, 1, 2, 2]
  (1, 1, 2, 1) => p[1, 1, 2, 1]
  (2, 1, 2, 2) => p[2, 1, 2, 2]
  (2, 1, 2, 1) => p[2, 1, 2, 1]
  (1, 1, 1, 2) => p[1, 1, 1, 2]
  (1, 1, 1, 1) => p[1, 1, 1, 1]
  (1, 2, 2, 2) => p[1, 2, 2, 2]
  (1, 2, 2, 1) => p[1, 2, 2, 1]
  (2, 1, 1, 2) => p[2, 1, 1, 2]
  (2, 1, 1, 1) => p[2, 1, 1, 1]

gens(R::MarkovRing)

Alias for unknowns. Return generators of the polynomial ring.

find_random_variables(R::MarkovRing, K)

Given a subset K of random_variables(R) return its indices into the data structure R.random_variables.

find_state(R::MarkovRing, K, x)

Given a set of random variables K and a joint event x they can take, return the index vector y of x in the data structure R.state_spaces such that x[i] = R.state_spaces[J[i]][v[i]] where J = find_random_variables(R, K).

state_space(R::MarkovRing, K=random_variables(R))

Return all states that the random subvector indexed by K can attain in the ring R. The result is an Iterators.product iterator unless K has only one element in which case it is a vector.

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

julia> collect(state_space(R, ["A", "B"]))
2×2 Matrix{Tuple{Int64, Int64}}:
 (1, 1)  (1, 2)
 (2, 1)  (2, 2)

marginal(R::MarkovRing, K, x)

Return a marginal as a sum of unknowns from R. The argument K lists random variables which are fixed to the event x; all other random variables in R are summed over their respective state spaces.

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2, "Y" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2}, Y -> {1, 2} in 16 variables over Rational field

julia> marginal(R, ["A", "X"], [1,2])
p[1, 1, 2, 1] + p[1, 2, 2, 1] + p[1, 1, 2, 2] + p[1, 2, 2, 2]

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

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

Examples

julia> R = markov_ring("A" => 1:2, "B" => 1:2, "X" => 1:2)
MarkovRing for random variables A -> {1, 2}, B -> {1, 2}, X -> {1, 2} in 8 variables over Rational field

julia> ci_ideal(R, [CI"X,A|B", CI"X,B|A"])
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]