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.

I read a paper here , which primarily deals with mixtures of gamma distribution. And it uses gradient ascent algorithm to estimate $\alpha$. If I follow the paper's approach to estimate $r$, then I will have the following gradient ascent algorithm:

$$r^{(t+1)} = r^{(t)} + \frac{a_t}{n} \frac{dQ}{dr}$$ I am not sure if this is the right way to go about this problem.

Does anyone what I can do solve for $r$ ?


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