Question: Joe a casino data scientist wants to use foreign dice for his casino. He is uncertain about the fairness of the dice but does not want to roll the dice indefinitely to find out. To incorporate his belief that all sides have equal probability of landing face up, he assigns all the hyperparameters equally. What distribution is mathematically convenient for his belief?
The correct answer is Dirichlet. Repeated rolling of dice is modeled by a multinomial distribution because there are a discrete number of outcomes (11 for a pair of dice), each event has an independently and identically distributed probability of occurring. Because Joe wants to incorporate his belief that the dice are fair, he wants to use a Dirichlet distribution hyperparameter because the Dirichlet distribution is conjugate to the multinomial distribution. This means that the posterior probability distribution is calculable in closed form. Larger values for the Dirichlet hyperparameters impose a stronger belief and is a way of expressing the certainty Joe has in the fairness of the dice and incorporating his a priori belief into his analysis.
A Gaussian (or normal) distribution is conjugate to itself. A Bernoulli distribution is the distribution for one trial of a system with two outcomes. A multinomial distribution is the distribution of repeated trials of a system with finite discrete outcomes.
Dirichlet-multinomial conjugacy | Wikipedia
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