The Zero-inflated Negative Binomial (ZINB) model adresses both zero inflation and overdispersion in the data. It can be expressed as:

\begin{equation*}
 X_j \sim (1-\pi_j)\; \mathds{1}_{w_j=0} + \pi_j\; \operatorname{NegativeBinomial}(\mu_j, r)
\end{equation*}

where the $w_{j}$ are indivicator variables such that $w_{j} = x_{j} = 0$ with probability $1 - π_{j}$ and $w_{j} = 1$ with probability $π_{j}$ . We typically model the $w_{j}$ using a logistic model.

Sample

We can use the method described in (neelon) to sample the ZINB model with a logistic mode:

Use the Gibbs sampler in (polson2013) to sample the parameter of the logistic regression model;
Use the Gibbs sampler in to sample the parameter of the negative binomial;
Update $r$ using a MCMC step;
Update the $w_{j}$ from their discrete full conditional distributions.