Confusion in parameter estimation of conditional random field

Question

I have a certain confusion while taking the derivative of the log likelihood of the conditional random field. As given in this paper http://people.cs.umass.edu/~mccallum/papers/crf-tutorial.pdf

I mean while calculating the gradient with respect to a parameter $$ \lambda_k $$

the derivate of the logZ term bring this term. $$ p(y,y'|x^{(i)}) $$

I am not sure how this term appeared while calculating the gradient. It is equation 1.22 of page 12 in the paper http://people.cs.umass.edu/~mccallum/papers/crf-tutorial.pdf

Can anyone please provide some insights?

Answer 1

I agree it's puzling; an insight comes from another McCallum-Sutton paper:

In the likelihood, inference is needed to compute the partition function Z(x(i)), which is a sum over all possible labellings. In the derivatives, inference is required to compute the marginal distributions p(y, y' | x(i)).

Remember that Z(x) is defined as a sum over Y of functions of the form f(y, y', x), so its derivative "brings out again" the y, y' arguments.