# Multinomial parameter estimation

In Segment 5. Bernoulli Trials, we did Bayesian parameter estimation of the rate parameter of a binomial distribution.

The setup was: we saw the outcomes of a series of independent trials. There were two possible outcomes to each trial: the jailer says B, or the jailer says C. There was one parameter of interest: x, the probability with which the jailer says B. (What about the probability that the jailer says C?) The goal was to compute the posterior distribution, given data in the form of counts of outcomes observed, of x.

In this exercise, we will generalize this to a multinomial setting.

Each trial is now a chess game, to which there are three possible outcomes: white wins, black wins, or the players draw.

We want to use data to learn how likely each outcome is. In other words, we assume that due to the structure of the game of chess, there is some inherent probability of each outcome occurring, and we want to figure out what these probabilities are. The parameters of interest are w, the probability that white wins, and b, the probability that black wins. (What about the probability that they draw?) We will take data in the form of counts of outcomes observed and compute the joint posterior distribution of w and b.

Notational conventions:

- w = probability that white wins
- b = probability that black wins
- d = probability that the players draw
- N = total number of games observed
- W = number of white wins in these games
- B = number of black wins in these games
- D = number of draws in these games

1. What does a joint uniform prior on w and b look like?

2. Suppose we know that w=0.4, b = 0.3, and d = 0.3. If we watch N = 10 games, what is the probability that W = 3, B = 5, and D = 2?

3. For general w, b, d, W, B, D, what is P(W, B, D | w, b, d)?

4. Applying Bayes, what is P(w, b, d | W, B, D)? What is the Bayes denominator?

5. Using the data from last Friday, count the outcomes of the first N games and produce a visualization of the joint posterior of the win rates for N = 0, 3, 10, 100, 1000, and 10000.

If you choose to do this in Python, some snippets demonstrating library functions that you may find useful can be found here.