Consider the data generation process for observation pair (a, b) as follows:

- a is the outcome of an independent six-faced (possibly loaded) dice-roll. That is, chance of rolling face '1' is p₁, rolling face '2' is p₂, etc., with a total of six distinct possibilities.

- Given the outcome a, b is drawn independently from a density distributed as (where q_a > 0).

(i) List all the parameters of this process. We shall denote the collection of all the parameters as the variable (the parameter vector).

(ii) Suppose we run this process n times independently, and get the sequence: (a₁, b₁),(a₂, b₂), . . . ,(a_n, b_n). What is the likelihood that this sequence was generated by a specific setting of the parameter vector ?

(iii) What is the most likely setting of the parameter vector given the complete observation sequence? that is, find the Maximum Likelihood Estimate of given the observations.

(iv) What is the probability of the partial (incomplete) observation b₁, b₂, . . . , b_n given a specific setting of the parameter vector ?

(v) Derive the Expectation Maximization (EM) algirthm to estimate of the parameters given the incomplete observation sequence