I am trying to derive the EM-algorithm of mixtures of negative binomial distribution $Neg\;Bin(r,p)$. I have the updating equations for updating the E-step as well as $p$ and the mixing coefficients $\phi_j$ s.
However, when I try to derive the M-step for $r$, I got stuck with an equation that involves the digamma function:
$$\frac{dQ}{dr} = \sum_{i=1}^m w^{(i)} \left( \psi(x^{(i)} + r) - \psi(r) + log(p) \right) = 0$$
Note that $Q$ is conditional expectation of log-likelihood and $w^{(i)}$ comes form the E-step, and hence can be treated as a constant. And $\psi$ is the digamma function. The superscript of $w$ and $x$ are just the index of data.
Does anyone have any idea how to solve the above equation for $r$ mathematically or computationally?