Hey guys, I'm really stuck on this problem:
Consider the following two groups:
x* = 216 y* = 210
σ_x^2 = 2210 σ_^2 = 2618
n = 130 m = 119
where given µ_x, x_1,... ,x_n are independent and identically distributed normal
random variables with mean µ_x and variance σ_x^2
(which is known). Same for the y’s.
(d) Let us now assume that µ_x ∼ µ_y ∼ N(β,τ^2) as we did in class. Find the
posterior density f(µ_x −µ_y|x* − y*). Note this is not the same as f(µ_x −µ_y|x*, y*).
The advantage of this alternate conditioning is that it removes the dependence
on the parameter β (why?) and simplifies the posterior. Repeat part (b) for
two cases τ^2 = σ^2 and τ^2 = 2σ^2, where σ^2 = σ_x^2/n + σ_y^2/m. Hint on how to
approach this: let w = µ_x − µ_y and d = x* − y* and observe that given w, d is
normal with mean w.
Now I know the posterior density is proportional to f(d|w) * f(w) (by bayes rule), however when I do the math I get an enormous variance (something like 4σ^4) and a mean of 0. I think the problem is my assumption that f(µ_x −µ_y) ~ N(0, 2τ^2) (I'm basically assuming they have the same variance and mean, but I don't think I can and don't really know what else I can do.)
EDIT: Is there anyway to do latex on here (or something similar) the formatting is quite ugly.
Any help would really be appreciated; this hw is due tomorrow, but there is a test coming up this week and I really need to understand these concepts.