In Bayesian statistics, posterior probability distribution is typically used to update our prior beliefs after accounting for the likelihood of observing certain data. In complicated models, it may not be feasible to work out the posterior distribution, especially with unknown parameters. Thus, simulation methods are used to approximate the posterior instead. This blog post presents the application of a simulation method termed grid approximation to estimate the win rate of top football teams in the English Premier League.

## Grid Approximation

Bayesian statistics encompass three key components: prior probability, likelihood and posterior probability. The latter is what we wish to solve for using the former two components. Grid approximation is a simple simulation method that estimates the posterior probability distribution by using a grid of possible parameter values and computes the posterior probability at each of the values. For example, let’s assume that we have a biased coin and we like to estimate its probability of flipping heads. We perform some coin flips to observe some data. We then use a grid of possible posterior probability values (i.e., ranging from 0 to 1), and computes the a) prior probability at each of these values, b) the likelihood of observing the data at each of these values. Consequently, this will generate multiple posterior probability values, which form the approximated posterior probability distribution.

Grid approximation works extremely well in simple models with few parameters. In this blog post, I applied this simulation method to estimate the win rate of top football teams in the English Premier League given the results that we have observed so far in this 22/23 season.