Problem 6 (Inference via Bayes’ Rule)
Suppose we are given a coin with an unknown head probability ? ?{0.3,0.5,0.7}. In order to infer the value ?, we experiment withthe coin and consider Bayesian inference as follows: De?ne eventsA1 = {? = 0.3}, A2 = {? = 0.5}, A3 = {? = 0.7}. Since initially wehave no further information about ?, we simply consider the priorprobability assignment to be P(A1) = P(A2) = P(A3) = 1/3.
(a) Suppose we toss the coin once and observe a head (for ease ofnotation, we de?ne the event B = {the ?rst toss is a head}). Whatis the posterior probability P(A1|B)? How about P(A2|B) andP(A3|B)? (Hint: use the Bayes’ rule)
(b) Suppose we toss the coin for 10 times and observe HHTHHHTHHH(for ease of notation, we de?ne the event C = {HHTHHHTHHH}).Moreover, all the tosses are known to be independent. What is theposterior probability P(A1|C), P(A2|C), and P(A3|C)? Given theexperimental results, what is the most probable value for ??
(c) Given the same setting as (b), suppose we instead choose to usea di?erent 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 themost probable value for ??
