I am going over Hinton's paper [http://www.cs.toronto.edu/~fritz/absps/tr00-004.pdf ] and am having a difficulty linking his eq2 to eq4. The issue is with the second RHS term of eq 4. It's not difficult to see that the second RHS term in eq2 is the expectation of the derivative over distribution $P$. He later approximates $P$ with samples from MCMC's stationary distribution, denoted by $Q^{\infty}$, hence the second RHS term of eq4 is basically the expectation of the derivative under the stationary distribution. But as he mentions right above eq4, this is supposed to be the averaged over the data distribution, so I don't know where the average over the data distribution is gone. I was expecting to see $E[ E[\frac{\partial\log P_m(c|\theta_m)}{\partial\theta_m}]_{Q^\infty} ]_{Q^0}$, instead of $E[\frac{\partial\log P_m(c|\theta_m)}{\partial\theta_m}]_{Q^\infty}$
Notation: $E[f(x)]_{P}$ denotes expectation of $f(x)$ under distribution $P$.
I agree with you that the explanation is a bit unclear. I hope I got it right. The second RHS term in equation 2 is already a constant with respect to the "data distribution". That is
$ \sum_c p(c| \theta_1, \dots, \theta_m) \frac{\partial \log f_m(c| \theta_m)}{\partial \theta_m} = E_{Q^\infty}[\frac{\partial \log f_m(X| \theta_m)}{\partial \theta_m}]$.
There is no stochasticity left in the term above. Hence taking its expectation returns the same term. I guess that's why they dropped the expectation with respect to $Q^0$ sign.