# Bayesian modelling of a variation of dice problem

I'll start with the normal case, finding the probability for each face of a dice:

1. Start with uniform prior - Dirichlet distribution with all alphas 1
2. Roll the dice, and depending on the result, compute the posterior distribution

The variation of the problem that I'm dealing with is:

1. Choose a number randomly, say 2
2. Roll the dice
3. You will only know if the result was 2 or not: true/false. If the dice threw 3, you will get false. If it was 2, you will get true as feedback

An example: I ask for one throw of a dice, and ask if the face was '3'. The dice thrower will throw the dice once, and will only tell me if the face was '3' or not.

How should I model this? I can start with the same uniform prior using Dirichlet distribution (with all alphas 1). I can update the prior if the dice result was true. But I'm unsure as to how to update the prior when the dice result is false.

For illustration purposes, let's assume the dice has 3 faces named A, B, and C.

Using a Dirichlet Prior with parameters: αA, αB, αC.

Say, the probability of A, B, and C are θA, θB and θC.Then,

• The likelihood of success when asked for face A: θA
• The likelihood of failure when asked for face A: (1 - θA)
• The likelihood of success when asked for face B: θB
• The likelihood of failure when asked for face B: (1 - θB)
• The likelihood of success when asked for face C: θC
• The likelihood of failure when asked for face C: (1 - θC)

Let's indicate the number of successes when asked for face A as NAS, and the number of failures when asked for face A as NAF. Similar definitions for NBS, NBF, NCS, NCF.

Then the net likelihood function is θANAS * (1 - θA)NAF * θBNBS * (1 - θB)NBF * θCNCS * (1 - θC)NCF

From Bayes theorem: Posterior ∝ Prior x Likelihood

Therefore, Posterior distribution P(θA, θB, θC) ∝ (θAαA - 1 * θBαB - 1 * θCαC - 1) * (θANAS * (1 - θA)NAF * θBNBS * (1 - θB)NBF * θCNCS * (1 - θC)NCF)