function z = condEntropy (x, y)
% Compute conditional entropy z=H(x|y) of two discrete variables x and y.
% Input:
% x, y: two integer vector of the same length
% Output:
% z: conditional entropy z=H(x|y)
assert(numel(x) == numel(y));
n = numel(x);
x = reshape(x,1,n);
y = reshape(y,1,n);
l = min(min(x),min(y));
x = x-l+1;
y = y-l+1;
k = max(max(x),max(y));
idx = 1:n;
Mx = sparse(idx,x,1,n,k,n);
My = sparse(idx,y,1,n,k,n);
Pxy = nonzeros(Mx'*My/n); %joint distribution of x and y
Hxy = -dot(Pxy,log2(Pxy));
Py = nonzeros(mean(My,1));
Hy = -dot(Py,log2(Py));
% conditional entropy H(x|y)
z = Hxy-Hy;
z = max(0,z);
