Suppose 202 animals are classified into 4 categories as follows: Y = (31, 32. 13, B4) = (125, 18, 25, 34) The corresponding cell probabilities are known to have the structure 1 02 2(1 - 01) 1 - 02 41 3 3 3 3 where of and 62 are unknown and to be estimated. Assume the uniform distribution on (0, 1) for 6 as well as for 02- 1. Write down the posterior distribution of (61, #2) up to a constant, i.e., specify the distribution except the normalizing constant. 2. We want to simulate from the above posterior distribution, say, to get 95% Bayesian intervals for 61 and 62. This may be done using the idea of data imputation, e.g., by suitably splitting the observed count in the first cell, so that the resulting posterior is a familiar one. Describe with sufficient details how you would do this and how you would then simulate from the posterior distribution of (61, 02). HINT: Data augmentation is done by augmenting the parameter vector (61, 62) to (61, 02, 2) with a probability model p(2 | 01, 02, y), changing the original posterior p( | y) = cxp(@) xp(y | 0) to p(6, = | y) = cxp(0)xp(y | 8) x p(= [ y, #). In this example, using p(z [ ..) = Binly1, 1/(61 + 62)] neatly simplifies things