wolffd@0: function [pdag, G] = learn_struct_pdag_ic_star(cond_indep, n, k, varargin)
wolffd@0: % LEARN_STRUCT_PDAG_IC_STAR Learn a partially oriented DAG (pattern) with latent 
wolffd@0: % variables using the IC* algorithm
wolffd@0: % P = learn_struct_pdag_ic_star(cond_indep, n, k, ...)
wolffd@0: %
wolffd@0: % n is the number of nodes.
wolffd@0: % k is an optional upper bound on the fan-in (default: n)
wolffd@0: % cond_indep is a boolean function that will be called as follows:
wolffd@0: % feval(cond_indep, x, y, S, ...)
wolffd@0: % where x and y are nodes, and S is a set of nodes (positive integers),
wolffd@0: % and ... are any optional parameters passed to this function.
wolffd@0: %
wolffd@0: % The output P is an adjacency matrix, in which
wolffd@0: % P(i,j) = -1 if there is either a latent variable L such that i <-L-> j 
wolffd@0: % OR there is a directed edge from i->j.
wolffd@0: % P(i,j) = -2 if there is a marked directed i-*>j edge.
wolffd@0: % P(i,j) = P(j,i) = 1 if there is and undirected edge i--j
wolffd@0: % P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j.
wolffd@0: %
wolffd@0: % The IC* algorithm learns a latent structure associated with a set of observed 
wolffd@0: % variables. 
wolffd@0: % The latent structure revealed is the projection in which every latent variable is
wolffd@0: % 1) a root node
wolffd@0: % 2) linked to exactly two observed variables.
wolffd@0: % Latent variables in the projection are represented using a bidirectional graph, 
wolffd@0: % and thus remain implicit.
wolffd@0: %
wolffd@0: % See Pearl, "Causality: Models, Reasoning, and Inference", 2000, p52 for more details.
wolffd@0: % Written by Tamar Kushnir, 2000
wolffd@0: 
wolffd@0: sep = cell(n,n);
wolffd@0: ord = 0;
wolffd@0: done = 0;
wolffd@0: G = ones(n,n);
wolffd@0: G = setdiag(G,0);
wolffd@0: while ~done
wolffd@0:   done = 1;
wolffd@0:   [X,Y] = find(G); 
wolffd@0:   for i=1:length(X)
wolffd@0:     x = X(i); y = Y(i);
wolffd@0:     nbrs = mysetdiff(myunion(neighbors(G, x), neighbors(G,y)), [x y]);
wolffd@0:     if length(nbrs) >= ord & G(x,y) ~= 0
wolffd@0:       done = 0;
wolffd@0:       SS = subsets(nbrs, ord, ord); % all subsets of size ord
wolffd@0:       for si=1:length(SS)
wolffd@0: 	S = SS{si};
wolffd@0: 	if feval(cond_indep, x, y, S, varargin{:})  
wolffd@0: 	  G(x,y) = 0;
wolffd@0: 	  G(y,x) = 0;
wolffd@0: 	  sep{x,y} = myunion(sep{x,y}, S);
wolffd@0: 	  sep{y,x} = myunion(sep{y,x}, S);
wolffd@0: 	  break; % no need to check any more subsets 
wolffd@0: 	end
wolffd@0:       end
wolffd@0:     end
wolffd@0:   end
wolffd@0:   ord = ord + 1;
wolffd@0: end
wolffd@0: 
wolffd@0: % Create the minimal pattern,
wolffd@0: % i.e., the only directed edges are V structures.
wolffd@0: pdag = G;
wolffd@0: [X, Y] = find(G);
wolffd@0: % We want to generate all unique triples x,y,z
wolffd@0: % where y is a common neighbor to x and z
wolffd@0: for i=1:length(X)
wolffd@0:   x = X(i);
wolffd@0:   y = Y(i);
wolffd@0:   Z = find(G(y,:));
wolffd@0:   Z = mysetdiff(Z, x);
wolffd@0:   for z=Z(:)'
wolffd@0:     if G(x,z)==0 & ~ismember(y, sep{x,z}) & ~ismember(y, sep{z,x})
wolffd@0:       pdag(x,y) = -1; pdag(y,x) = 0;
wolffd@0:       pdag(z,y) = -1; pdag(y,z) = 0;
wolffd@0:     end
wolffd@0:   end
wolffd@0: end
wolffd@0: 
wolffd@0: % Convert the minimal pattern to a complete one using the following rules:
wolffd@0: % Rule 1:
wolffd@0: % if a and b are non-adjacent nodes with a common neighbor c,
wolffd@0: % if a->c and not b->c then c-*>b (marked arrow).
wolffd@0: % Rule 2:
wolffd@0: % if a and b are adjacent and there is a directed path (marked links) from a to b
wolffd@0: % then a->b (add arrowhead).
wolffd@0: %Pearl (2000)
wolffd@0: 
wolffd@0: arrowin = [-1 -2 2];
wolffd@0: old_pdag = zeros(n);
wolffd@0: iter = 0;
wolffd@0: while ~isequal(pdag, old_pdag)
wolffd@0:   iter = iter + 1;
wolffd@0:   old_pdag = pdag;
wolffd@0:   % rule 1
wolffd@0:   [X, Y] = find(pdag);
wolffd@0:   for i=1:length(X)
wolffd@0:     x = X(i);
wolffd@0:     y = Y(i);
wolffd@0:     Z = find(pdag(y,:));
wolffd@0:     Z = mysetdiff(Z, x);
wolffd@0:     for z=Z(:)'
wolffd@0:       if G(x,z)==0 & ismember(pdag(x,y),arrowin) & ~ismember(pdag(z,y),arrowin)
wolffd@0:         pdag(y,z) = -2; pdag(z,y) = 0;
wolffd@0:       end
wolffd@0:     end
wolffd@0:   end
wolffd@0:   % rule 2
wolffd@0:   [X, Y] = find(G); 
wolffd@0:   %check all adjacent nodes because if pdag(x,y) = -1 
wolffd@0:   %and pdag(y,x) = 0 there could still be an bidirected edge between x & y.
wolffd@0:   for i=1:length(X)
wolffd@0:     x = X(i);
wolffd@0:     y = Y(i);
wolffd@0:     if ~ismember(pdag(x,y), arrowin) %x->y doesn't exist yet
wolffd@0:       %find marked path from x to y
wolffd@0:       add_arrow = marked_path(x,y,pdag);
wolffd@0:       if add_arrow 
wolffd@0:         if pdag(y,x)==-1 %bidirected edge
wolffd@0:           pdag(x,y) = 2; pdag(y,x) = 2;
wolffd@0:         else
wolffd@0:           pdag(x,y) = -1;pdag(y,x) = 0;
wolffd@0:         end
wolffd@0:       end
wolffd@0:     end
wolffd@0:   end
wolffd@0: end
wolffd@0: 
wolffd@0: 
wolffd@0: %%%%%%%%%%%%%
wolffd@0: 
wolffd@0: function t = marked_path(x,y,L)
wolffd@0: % MARKED_PATH is a boolean function which returns 1 if a marked path 
wolffd@0: % between nodes x and y exists in the partially directed latent structure L.
wolffd@0: %
wolffd@0: % t = marked_path(x,y,L)
wolffd@0: %
wolffd@0: % x and y are the starting and ending nodes in the path, respectively.
wolffd@0: % L is a latent structure (partially directed graph with possible latent variables).
wolffd@0: %
wolffd@0: % Rule 2 of IC* algorithm (see Pearl, 2000)
wolffd@0: 
wolffd@0: t=0;
wolffd@0: 
wolffd@0: %find set of marked links from x
wolffd@0: marked = find(L(x,:)==-2);
wolffd@0: if ismember(y,marked)
wolffd@0:   t=1; %marked path found
wolffd@0: else
wolffd@0:   for m=marked(:)'
wolffd@0:     t = marked_path(m,y,L);
wolffd@0:     if t==1
wolffd@0:       break; %stop when marked path found
wolffd@0:     end
wolffd@0:   end
wolffd@0: end