Question

Suppose we are given two separate groups of data, y1 and y2, which are given by the following: y1: (y1,...yn)^T and y2: (yn+1,yn+2....yn+m)^T such that yi ~ N (µ, σ^2), i = 1,2,...,n and yi ~ N (µ*, σ^2*), i = n+1, n+2...,n+m. Assume all data are mutually independent and IID, and T = transpose.

a) if σ^2 = σ^2* = (σ^2), where (σ^2) is known, find the posterior dist p(µ,µ*|y1,y2) and find bayes estimator for µ and µ* (assume normal prior)

b) assume µ, µ*, σ^2, σ^2* are all unknown. assume normal prior for both µ, µ*, and Inverse gamma for σ^2, σ^2*. Find posterior and marginal density.

Answer 1