How to apply the multivariate Poisson distribution?

From my recent question here: Calculating the probability of a car accident

I wish to learn how to apply a multivariate poisson distribution in the example that car-accidents vary for each year, therefore this model will require in effect to take into account this variation.

A simple multivariate Poisson model looks like this:

The joint probability function is given by \begin{align} P(X)&=P(X_1=x_,X_2=x_2,\ldots,X_m=x_m) \\&=\exp\left(-\sum_{i=1}^m \theta_i\right)\prod_{i=1}^m \frac{\theta_i^{x_i}}{x_i!}\sum_{i=0}^s \prod_{j=1}^m \binom{x_j}{i}i!\left(\frac{\theta_0}{\prod_{i=1}^m \theta_i}\right)^i\,, \end{align}

where $$s=\min(x_1,x_2,\ldots,x_m).$$

Though I'm unfamiliar with the equation such as why is the formula on the right of the poisson distribution introduced the way that it is? And what's an example of it's use (preferably an R example)

• It seems to have been invented to give some correlation by using a hidden $X_0$ Sep 8 '21 at 11:01
• @Henry Pardon my questions: What do you mean by 'hidden'? As in it's not currently visible in this model, but if broken down then it's visible? If so, what correlation are you referring to? Sep 8 '21 at 11:08
• Your formula comes from slide 17. Slides 16 and 5 suggest a $Y_0$ or $X_0$ term which is shared but unknown and the right hand $\sum\prod$ term in the formula deals with this Sep 8 '21 at 11:15
• This distribution is derived at stats.stackexchange.com/questions/108705. The derivation gives strong indications of potential uses, as @Henry has intimated. There are simpler ways to analyze Poisson processes (variation in counts over time), though. This might be a dead end for you as far as that's concerned.
– whuber
Sep 8 '21 at 18:29