Consider a model with likelihood function

L (,,)=P(d,,)=i=1nP(dii,)P(i)

P(dii,)2ie(di)2

P(i,)=e(1+i)

and prior distributions

P()1

P()=e

with fixed . The goal is now to sample from the posterior distribution P(,,d) using a Metropolis-Hastings algorithm in which each parameter gets updated in-turn using symmetric proposal kernels. Note that ,>0 .

Indicate for each of the following statements if it is true or false.

a) The hasting ratio for an update ii is

hi=min(1,iie(ii)[+(di)2])

b) The normal distribution N(0,2) is a possible choice for a symmetric proposal kernel .

c) The R function

q <- function(x,d){return(abs(x + runif(1)*d - d/2));} 

constitutes a valid proposal kernel q(i)

d) The log-Hastings ratio for an update is

log(h)=min(0,i[(di)2(di)2]

e) If the acceptance rate of the parameter is around 45%, the variance of the proposal kernel q() should be decreased.