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Suppose you have a six-sided die that you suspect is not fair and toss it N times. What would be a Bayesian approach to estimating the probability of the six outcomes given that you suspect the die may be unfair?

I know that I have to come up with a prior density and a likelihood in order to define what the posterior would be.

I was thinking of using the multinomial as the likelihood, but I am not sure what to use for a prior since the die is suspected to be not fair.

Any help is appreciated... Thank you!

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Based on the information you have provided, I would suggest the conjugate Dirichlet prior with the unit vector as the parameter with the multinomial likelihood. This doesn't account for the fact that you suspect the die to be unfair, but you have also not specified how you expect the die to be unfair. As you collect data, the estimated probabilities for each outcome will converge to their true probabilities which seems sufficient. If you would rather encode how you believe the die to be unfair, you can easily modify the vector parameter of the Dirichlet to put more (or less) weight on some outcomes.

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