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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.

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1 Answer 1

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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)

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