The beta-binomial is a discrete distribution generally used to describe the number of success  for  Bernoulli trials when the probability of success  at each trial is unknown and assumed to follow a beta distribution with parameters and :

That is, to find the probability of observing the outcome y, we average over all the possible (and continuous) values of . And hence the beta-binomial can be considered as a continuous mixture model. If the beta-binomial model sounds familiar to you, it is because you have being paying attention to the first two chapters of the book! This is the model we use for the coin-flipping problem, although we explicitly use a beta and a binomial distribution, instead of using the already mixed beta-binomal distribution.

In a similar fashion, we have the negative-binomial distribution, which can be understood as a gamma-Poisson mixture. For this model we have a mixture of Poisson distributions were the rate parameter is gamma distributed. This distribution is often used to circumvent a common problem encountered when dealing with count data. This problem is known as over-dispersion. Suppose you are using a Poisson distribution to model count data, and then you realize that the variance in your data exceeds that of the model; the problem with using a Poisson distribution is that mean and variance are linked (in fact they are described by the same parameter). So one way to solve this problem is to model the data as a (continuous) mixture of Poisson distributions with rates coming from a gamma distribution, which gives us the rationale to use the negative-binomial distribution. Since we are now considering a mixture of distributions, our model has more flexibility and can better accommodate the observed mean and variance of the data. Both the beta-binomial and the negative-binomial can be used as part of linear models and both also have zero-inflated versions of them. And also, both are implemented on PyMC3 as ready-to-use distributions.