How to combine maximum entropy principle and the likelihood (soft constraints)?

Suppose f(x) is a probability mass distribution. There are a number of statistics u_i=E_f[g_i(X)] and the corresponding estimates y_i. We only know y_i and the probability function p(y_i|u_i) (e.g., a Gaussian or Laplace distribution).

If we know u_i exactly, then we can seek for a distribution f that maximizes the entropy subject to the constrains E_f[g_i(X)]=u_i. But we only know the estimates y_i, so I don't know what should be maximized.

I suppose a reasonable objective function might be to maximize

Q(f)=H_f(X)+ln(p({y_i}|f))=-E_f[ln(f(x))]+ln(p({y_i}|{u_i(f)}))

, the sum of the entropy of X and the log-likelihood. But I don't know how to justify it.

Another problem is that it is hard to optimize for Q(f). f(x) should be in the form of

f(x;w)=1/Z(w)*exp(\sum_i g_i(x)*w_i)

I want to use (stochastic) gradient descend algorithm, so I tried to differentiate H_f(x) w.r.t. w_i, and obtain

dH_f(X)/dw_i=-sum_x[ln(f(x;w))*f(x;w)*[g_i(x)-u_i(f;w)]]

=u_i(f;w)*E[ln(f(X;w))|w]-E[ln(f(X;w))*g_i(X)|w]

The term u_i(f;w) can be simply estimated by Monte Carlo approach. But I don't know how to deal with ln(f(X;w))=\sum_i[g_i(x)*w_i]-ln(Z(w)), because the term ln(Z(w)) is hard to estimate.

A compromise is to drop the entropy term in the Q(f), and optimize only for the likelihood p({y_i}|w). But this solution seems not so elegant.