Conditional independence ideals with hidden variables Fatemeh Mohammadi (IST Austria) Johannes Rauh (York University) June 10, 2015 Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 1 / 14
Conditional independence Consider three discrete random variables X 1 , X 2 , X 3 with finite ranges [ r 1 ] , [ r 2 ] , [ r 3 ] and the joint state space: X = [ r 1 ] × [ r 2 ] × [ r 3 ] . X 1 is (conditionally) independent of X 2 given X 3 P ( X 1 = x 1 , X 2 = x 2 | X 3 = x 3 ) = P ( X 1 = x 1 | X 3 = x 3 ) P ( X 2 = x 2 | X 3 = x 3 ) p x 1 x 2 x 3 = p x 1 + x 3 p + x 2 x 3 X 1 ⊥ ⊥ X 2 | X 3 Lemma The following equivalent conditions holds: X 1 ⊥ ⊥ X 2 | X 3 1 For each x 3 ∈ [ r 3 ] , the matrix ( p x 1 , x 2 , x 3 ) x 1 , x 2 has rank one. 2 p x 1 x 2 x 3 p x ′ 2 x 3 = p x 1 x ′ 2 x 3 p x ′ 3 1 x ′ 1 x 2 x 3 for all x 1 , x ′ 1 ∈ [ r 1 ] , x 2 , x ′ 2 ∈ [ r 2 ] , x 3 ∈ [ r 3 ] . Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 2 / 14
Conditional independence models Consider n random variables X 1 , . . . , X n , taking values in the finite sets [ r 1 ] , . . . , [ r n ] . For any A ⊆ [ n ] let X A be the random vector ( X i ) i ∈ A . For disjoint subsets A , B , C ⊂ [ n ] , a CI statements has the form X A ⊥ ⊥ X B | X C , or in short: A ⊥ ⊥ B | C Consider the joint distribution P of X 1 , . . . , X n as an n -tensor P = ( p x 1 ,..., x n ) x i ∈ [ r i ] . The statement A ⊥ ⊥ B | C says: Take the marginal over [ n ] \ ( A ∪ B ∪ C ) . 1 For any fixed value of X C take the slice with constant ( x k ) k ∈ C . 2 Flatten this slice to a matrix, with rows indexed by ( x i ) i ∈ A , columns 3 indexed by ( x j ) j ∈ B . The resulting matrix has rank one. Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 3 / 14
An example Short notation: X A ⊥ ⊥ X B | X C ⇐ ⇒ ( p x A x B x C + ) x A , x B has rank one for each x C . Example Let n = 5 and X i = { 0 , 1 } . The statement { 1 , 2 } ⊥ ⊥ { 3 } | { 4 } holds if and only if the two matrices p 00000 + p 00001 p 00100 + p 00101 p 00010 + p 00011 p 00110 + p 00111 p 01000 + p 01001 p 01100 + p 01101 p 01010 + p 01011 p 01110 + p 01111 , p 10000 + p 10001 p 10100 + p 10101 p 10010 + p 10011 p 10110 + p 10111 p 11000 + p 11001 p 11100 + p 11101 p 11010 + p 11011 p 11110 + p 11111 have rank one. Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 4 / 14
Saturated CI statements Definition A CI statement A ⊥ ⊥ B | C is saturated, if it involves all random variables, i.e. A ∪ B ∪ C = [ n ] . Observation: Saturated CI statements lead to binomial ideals. R = C [ p x 1 x 2 ··· x n : x 1 ∈ [ r 1 ] , . . . , x n ∈ [ r n ]] A ⊥ ⊥ B | C ⇐ ⇒ ( p x A x B x C + ) x A , x B has rank one for each x C I A ⊥ ⊥ B | C is generated by all 2-minors of ( p x A x B x C + ) x A , x B Theorem (Eisenbud, Sturmfels ’96) Binomial ideals have a binomial primary decomposition. Question: Describe the implications among a collection of CI statements (preferably in terms of prime components of I C ). � I C = I C i Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 5 / 14
Binomial edge ideals { X 0 ⊥ ⊥ X 1 | X 2 , X 0 ⊥ ⊥ X 2 | X 1 } [Fink] { X 0 ⊥ ⊥ X A | X [ n ] \ A : various subsets A } [HHHKR, Ay-Rauh] |X 0 | = 2: binomial edge ideals [HHHKR] |X 0 | > 2: generalized binomial edge ideals [Ay-Rauh] { X i ⊥ ⊥ X j | X [ n ] \{ i , j } : i < j } [Swanson-Taylor] Theorem [Fink] Let { X 0 ⊥ ⊥ X 1 | X 2 , X 0 ⊥ ⊥ X 2 | X 1 } with X 0 = [ 2 ] , X 1 = [ n 1 ] , X 2 = [ n 2 ] . Let P = ( p x 0 x 1 x 2 ) be a vanishing point of I C . Then I C is the binomial edge ideal of the bipartite graph G with vertex set [ n 1 ] ∪ [ n 2 ] and the edge set { ( x 1 , x 2 ) : p x 0 x 1 x 2 � = 0 for some x 0 } . 3 1 2 4 Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 6 / 14
Binomial edge ideals Definition Let G be a graph on the vertex set [ n ] and R = C [ p 1 x , p 2 x : x ∈ [ n ]] . The binomial edge ideal I G ⊂ R is generated by the binomials p 1 x p 2 y − p 1 y p 2 x for all edges { x , y } of G . Known facts about binomial edge ideals: radical ideal nice description for Gröbner bases of I G combinatorial description for primary decomposition. For W ⊂ [ n ] , let m W = � p 1 x , p 2 x : x ∈ W � then � p 1 x , p 2 x ) ∞ ( I G + m W ) : ( x �∈ W is a prime component of I G . Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 7 / 14
CI statements with hidden variables X 0 , X 1 : visible random variables, H : hidden random variable joint probability distribution: P = ( p i , x , h ) i ∈X 0 , x ∈X 1 , h ∈H . X 0 ⊥ ⊥ X 1 | H iff each slice P h := ( p i , x ) i ∈X 0 , x ∈X 1 has rank one. The marginal distribution of X 0 and X 1 is P X 0 , X 1 = � h ∈H P h . Therefore P X 0 , X 1 has rank at most |H| . Find all matrices P = ( p i , x ) i ∈X 0 , x ∈X 1 of non-negative rank at most |H| with the normalization condition � i , x p i , x = 1. The set of matrices of given non-negative rank at most r is a semi-algebraic set whose semi-algebraic condition is not known for general r . However, it is known that its Zariski closure equals the set of all rank r matrices, and it is described by the determinantal ideal of all ( r + 1 ) × ( r + 1 ) -minors of P. Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 8 / 14
CI statements with hidden variables Question Let C = { X 0 ⊥ ⊥ X 1 |{ X 2 , H 1 } , X 0 ⊥ ⊥ X 2 |{ X 1 , H 2 } } . Describe I C and its primary decomposition combinatorially. Theorem Let C = { X 0 ⊥ ⊥ X 1 |{ X 2 , H 1 } , X 0 ⊥ ⊥ X 2 |{ X 1 , H 2 } } with X 0 = [ d ] , X 1 = [ n 1 ] and X 2 = [ n 2 ] H 1 = [ r 1 ] and H 2 = [ r 2 ] ∆ n 1 , 0 = � � { ( i , 1 ) , ( i , 2 ) , . . . , ( i , n 2 ) } : i ∈ [ n 1 ] � � ∆ 0 , n 2 = { ( 1 , j ) , ( 2 , j ) , . . . , ( n 1 , j ) } : j ∈ [ n 2 ] . Then I C = I ∆ , where ∆ is the union of the r 1 -skeleton of ∆ n 1 , 0 and the r 2 -skeleton of ∆ 0 , n 2 , and all its prime components can be read from subcomplexes of ∆ . Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 9 / 14
Example of CI statements with hidden variables C = { X 0 ⊥ ⊥ X 1 |{ X 2 , H 1 } , X 0 ⊥ ⊥ X 2 |{ X 1 , H 2 } } |X 0 | = 3, |X 1 | = 2, |X 2 | = 3, |H 1 | = 3 and |H 2 | = 2. ∆ = { 135 , 246 , 12 , 34 , 56 } 2 1 3 5 4 6 p 11 p 12 · · · p 16 P |X 0 |×|X 1 ||X 2 | = p 21 p 22 · · · p 26 p 31 p 32 · · · p 36 Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 10 / 14
The ideal I C and its prime components We take all maximal minors of the submatrices of P corresponding to ∆ = { 135 , 246 , 12 , 34 , 56 } : � p 11 p 13 p 15 � p 12 p 14 p 16 � p 11 p 12 � p 13 p 14 � p 15 p 16 � � � � � p 21 p 23 p 25 p 22 p 24 p 26 p 21 p 22 p 23 p 24 p 25 p 26 , , , , p 31 p 33 p 35 p 32 p 34 p 36 p 31 p 32 p 33 p 34 p 35 p 36 Then I C = I ∆ has seven minimal primes associated to the complexes: ∆ 1 , 4 = { 1 , 4 , 56 } , ∆ 1 , 6 = { 1 , 6 , 34 } , ∆ 2 , 3 = { 2 , 3 , 56 } , ∆ 2 , 5 = { 2 , 5 , 34 } , ∆ 3 , 6 = { 3 , 6 , 12 } , ∆ 4 , 5 = { 4 , 5 , 12 } , ∆ 0 = { 12 , 34 , 56 , 135 , 145 , 136 , 146 , 235 , 245 , 236 , 246 } . Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 11 / 14
CI statements with hidden variables X 0 , X 1 , . . . , X k : visible random variables H 1 , . . . , H l : hidden random variables C : a family of CI statements of the form X 0 ⊥ ⊥ X A | X B , where A , B ⊆ { X 1 , . . . , X k , H 1 , . . . , H l } are disjoint We are interested in the set P C of marginal distributions of X 0 , X 1 , . . . , X k of the set of those joint distributions of X 0 , X 1 , . . . , X k , H 1 , . . . , H l that satisfy the statements in C . Question Whether CI statements with hidden variables can be given an algebraic interpretation? What can we say about the ideal I C ? Is I C a radical ideal? Give a nice combinatorial primary decomposition for I C . Describe a Gröbner basis. Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 12 / 14
Prime components of I C Theorem [M.-Rauh] The minimal primes of I C are of the form � p 1 x , p 2 x , . . . , p dx ) ∞ ( I C + m W ) : ( x �∈ W where m W is the ideal associated to a subcomplex of ∆ . Theorem [EHHM 2013], [M. 2012], [M.-Rauh] The ideal I C and all its prime components can be read from a simplicial complex associated to C , i.e. these ideals are all determinantal facet ideals studied in [EHHM]. We computed some class of examples, which all are very nice: radical ideal, nice combinatorial primary decomposition. Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 13 / 14
References Ene, Herzog, Hibi, and Mohammadi: Determinantal facet ideals (Michigan Mathematical Journal, 2013) Mohammadi: Prime splittings of determinantal ideals (arXiv:1208.2930, 2012) Mohammadi & Rauh: Conditional independence ideals with hidden variables (in preparation) * ) ( ' & % $ # " ! Fatemeh Mohammadi, Johannes Rauh Conditional independence ideals June 10, 2015 14 / 14
Recommend
More recommend
