(In this problem we will be covering the material from Bishop section 2.1.1. This material was also covered in class as the Bayesian parameter estimation problem from March 31, 2021) We will be following the use of Bayes' theorem as expressed in Bishopo equation (1.44): posterior o likelihood x prior Assume that we observe a series of / i.i.d. coin flips D = (x(), ..., x("}, where each coin flip is modelled as a Bernoulli random variable: p(x ") [) = (1 - p)(1-z) for i = 1, ... , N The above assumptions lead to the form of the likelihood function seen in Bishop equation (2.5). Make sure you understand this process. For the prior on /, we assume a Beta distribution with a = 1, b = 1. p(p; a = 1, b = 1) = r(1+1) 1-1(1-p)1=1 r(1)r(1) a) [2 points] By plugging the above forms for the likelihood function and the prior distribution over / into the Bishop equation (1.44), show that the posterior distribution has the form seen in Bishop (2.17): p(u[D) = p(u; m, l, a, b) oc umta-1(1 - ()'+6-1 where m = _ (the number of heads) and / = _ _ (1 - x(") (the number of tails). Make particular note of the interpretation of the hyperparameters a and b as "effective observations" mentioned on Bishop page 72. Suppose we now want to predict the outcome of the next trial (i.e. flip of the coin). The Bayesian way to do this is to evaluate the predictive distribution of the next coin flip a, given the observed data set D. The steps to do this are desbried on Bishop page 73, from equation (2.19) to (2.20)