Think Bayes, 2nd Edition

Chapter 8. Poisson Processes

This chapter introduces the Poisson process, which is a model used to describe events that occur at random intervals. As an example of a Poisson process, we’ll model goal-scoring in soccer, which is American English for the game everyone else calls “football”. We’ll use goals scored in a game to estimate the parameter of a Poisson process; then we’ll use the posterior distribution to make predictions.

And we’ll solve the World Cup Problem.

The World Cup Problem

In the 2018 FIFA World Cup final, France defeated Croatia 4 goals to 2. Based on this outcome:

1. How confident should we be that France is the better team?

2. If the same teams played again, what is the chance France would win again?

To answer these questions, we have to make some modeling decisions.

• First, I’ll assume that for any team against another team there is some unknown goal-scoring rate, measured in goals per game, which I’ll denote with the Python variable lam or the Greek letter $\lambda$, pronounced “lambda”.

• Second, I’ll assume that a goal is equally likely during any minute of a game. So, in a 90-minute game, the probability of scoring during any minute is $\lambda /90$.

• Third, I’ll assume that a team never scores twice during the same minute.

Of course, none of these assumptions is completely true in the real world, but I think they are reasonable simplifications. As George Box said, “All models are wrong; some are useful” (https://oreil.ly/oeTQU).

In this case, the model is ...