Question

Problem 6 (Inference via Bayes’ Rule)
Suppose we are given a coin with an unknown head probability θ ∈ {0.3,0.5,0.7}. In order to infer the value θ, we experiment with the coin and consider Bayesian inference as follows: Define events A1 = {θ = 0.3}, A2 = {θ = 0.5}, A3 = {θ = 0.7}. Since initially we have no further information about θ, we simply consider the prior probability assignment to be P(A1) = P(A2) = P(A3) = 1/3.
(a) Suppose we toss the coin once and observe a head (for ease of notation, we define the event B = {the first toss is a head}). What is the posterior probability P(A1|B)? How about P(A2|B) and P(A3|B)? (Hint: use the Bayes’ rule)
(b) Suppose we toss the coin for 10 times and observe HHTHHHTHHH (for ease of notation, we define the event C = {HHTHHHTHHH}). Moreover, all the tosses are known to be independent. What is the posterior probability P(A1|C), P(A2|C), and P(A3|C)? Given the experimental results, what is the most probable value for θ?
(c) Given the same setting as (b), suppose we instead choose to use a different prior probability assignment P(A1) = 2/5,P(A2) = 2/5,P(A3) = 1/5. What is the posterior probabilities P(A1|C), P(A2|C), and P(A3|C)? Given the experimental results, what is the most probable value for θ?

Answer 1