Problem 2: EM algorithm (5 + 5 = 10 points) Consider a scenario with K chess players. On day t, one of the K players plays mt games and wins wt of those mt games. We have access to the data of how many games were played and how many games were won each day, but we don't know which of the K players played on which day. We use a probabilistic mixture model to model this data. For each player If, we model the prob- ability that they win on any given game with some unknown parameter pic 6 [0,1]. We use a mixture of K binomials with parameters p1, . . . ,pK to model the observed data for n days given 01 by (m1, 101), . . . , (m.m run). The generative story is that on day t, we rst pick one player out of the K at random according to the distribution 7r as ct ~ 77. Then, given player ct plays mt games, the number of wins wt out of the mt games is given by the binomial distribution. Derive the EM algorithm for this problem: 1. E-step: Write down the Estep update. 2. Mstep: Derive the M-step update for pg), . . . ,p? the K model parameters on iteration 'i, in terms of data and the output of the Estep