Let's now return to the example of modeling count data. We will see two examples; one with a time varying rate and one with a 2D-spatially varying rate. In order to do this, we will use a Poisson likelihood and the rate will be modeled using a Gaussian process. Because the rate of the Poisson distribution is limited to positive values, we will use an exponential as the inverse link function, as we did for the zero-inflated Poisson regression from Chapter 4.

In the literature, the variable rate also appears with the name intensity; thus, this type of problem is known as intensity estimation. Also, this type of model is often referred to as a Cox model. A Cox model is a type of Poisson process, where the rate is itself a stochastic process. Just as a Gaussian process is a collection of random variables, where every finite collection of those random variables has a multivariate normal distribution, a Poisson process is a collection of random variables, where every finite collection of those random variables has a Poisson distribution. We can think of a Poisson process as a distribution over collections of points in a given space. When the rate of the Poisson process is itself a stochastic process, such as, for example, a Gaussian process, then we have what is known as a Cox process.