Bayesian modelling of a variation of dice problem

Question

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.

Answer 1

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

Using a Dirichlet Prior with parameters: α_A, α_B, α_C.

Say, the probability of A, B, and C are θ_A, θ_B and θ_C.Then,

• The likelihood of success when asked for face A: θ_A
• The likelihood of failure when asked for face A: (1 - θ_A)
• The likelihood of success when asked for face B: θ_B
• The likelihood of failure when asked for face B: (1 - θ_B)
• The likelihood of success when asked for face C: θ_C
• The likelihood of failure when asked for face C: (1 - θ_C)

Let's indicate the number of successes when asked for face A as N_AS, and the number of failures when asked for face A as N_AF. Similar definitions for N_BS, N_BF, N_CS, N_CF.

Then the net likelihood function is θ_A^N_AS * (1 - θ_A)^N_AF * θ_B^N_BS * (1 - θ_B)^N_BF * θ_C^N_CS * (1 - θ_C)^N_CF

From Bayes theorem: Posterior ∝ Prior x Likelihood

Therefore, Posterior distribution P(θ_A, θ_B, θ_C) ∝ (θ_A^{α_A - 1} * θ_B^{α_B - 1} * θ_C^{α_C - 1}) * (θ_A^N_AS * (1 - θ_A)^N_AF * θ_B^N_BS * (1 - θ_B)^N_BF * θ_C^N_CS * (1 - θ_C)^N_CF)