HW 20 Stat 705  Assigned Monday 12/5/16 Due Tuesday 8pm 12/20/2016
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NOTE: UNLIKE PREVIOUS HW'S THE DEADLINE FOR THIS ONE IS NOT EXTENDABLE, 
SINCE GRADES FOR THE COURSE NEED TO BE SUBMITTED BY 12/23.

ALSO: YOU MUST SUBMIT THIS ONE ELECTRONICALLY, EVEN IF YOU SUMBIT EARLIER. 
I WILL NOT BE ON CAMPUS FOR THE BEGINNING OF THAT WEEK TO COLLECT PAPERS.


The topic of this HW is an important Gibbs Sampler Example:

(1)  A pair of random variables (U,V) is assumed to 
have joint density such that  

given U ,      V ~ Gamma(A, B1+B2*(U-b)^2)

given V ,      U ~ N(D*V/(V+C), 1/(V+C))

where A, B1,B2,b,C,D are constants.

Write an R function to simulate random pairs (U_i,V_i) from 
this joint density (not necessarily independent across i), 
and use that function to find (either through a picture, a 
density function, or some other representation) of the density 
function of U.


(2) Show that the conditional density structure given in (1)
holds conditionally given  the data  (X_1,..., X_n)  in a normal
N(mu, 1/tau)  sample, for the two variables

V = n*tau ,   U = mu

where the prior densities of independent  tau  and  mu  are 
respectively assumed to be

    tau ~ Gamma(nu/2, 1/2)     and    mu ~ N(0, 1/eps)

and find the constants  A,B1,B2,b,C,D as functions of  n, nu, 
eps, and X_1,...,X_n (or Xbar  and S^2).

Note that 1/eps and 1/tau denote normal variances in this example.


(3) Use your Gibbs Sampler found in part (1), in the setting of 
part (2), to find the posterior expectations of  mu   and  1/tau  
given the data example of 50 frontal-lobe sizes (in millimiters), 
assumed iid N(mu,1/tau), of the 50 female blue crabs in the 
dataset "crabs" located in the MASS library in R, where  
eps = 0.05   and  nu = 3.