Intro to Bayesian Learning Exercise Solutions. Ata Kaban The University of Birmingham 2005. 1) In a casino, two differently loaded but identically looking dice are thrown in repeated runs. The frequencies of numbers observed in 40 rounds of play are as follows:
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The University of Birmingham
Die 1, [Nr, P_1(Nr)]: [1, 0.125], [2,0.075], [3,0.250], [4,0.025], [5,0.250], [6,0.275]
Die 2, [Nr, P_2(Nr)]: [1,0.250], [2,0.275], [3,0.100], [4,0.250], [5,0.075], [6,0.050]
Since we have a random sequence model (i.i.d. data) D, the probability of D under the two models is
Since there is no prior knowledge about either dice, we use a flat prior, i.e. the same 0.5 for both hypotheses.
Because P_1(D) < P_2(D), and the prior is the same for both hypothesies, we conclude that the die in question is the die no. 2.
(s1): A B B A B A A A B A A B B B
(s2): B B B B B A A A A A B B B B
(M1): a random sequence model with parameters P(A)=0.4, P(B)=0.6
(M2): a first order Markov model with initial probabilities 0.5 for both symbols and the following transition matrix: P(A|A)=0.6, P(B|A)=0.4, P(A|B)=0.1, P(B|B)=0.9.
Which of s1 and s2 is more likely to have been generated from which of the models M1 and M2? Justify your answer both using intuitive arguments and also by using Bayes’ rule. (As there is no prior knowledge given here, then consider equal prior probabilities.)
Now the required quantities are known from the problem. These are the following:
Replacing, we have:
So clearly it is more probable that the disease is not present.