We want to estimate a single unknown parameter $$ in a certain model. Assume that in R we

have defined a function log$\_$post to calculate the log of the unnormalized posterior density as

a function of $.$ This function and the data $y$ being analysed are not shown in the code extract

below. The posterior density is $p(|y).$ Consider the following R code:nm $=10000$

theta $=$ vector$($length$=$nm$)$

s $=0\mathrm{.}4$

theta$0=2$

log$\_$postQ $=$ log$\_$post$($theta$\backslash$theta$)$

for$($i in $1$:$($nb$+$nm$))\{$log$\_$post$1=$ log$\_$post$($theta$1)$ theta$0=$ theta$1\}\}$

stheta $=$ sort$($theta$)$

stheta$[$nm$/2]$

stheta$[$nm$*0\mathrm{.}025]$

stheta$[$nm$*0\mathrm{.}975]$Except where stated, an explanation in words is all that is needed for this question.

$($a$)$ What is the name of the algorithm that the code is carrying out?

$($b$)$ Explain what the command theta$1,$ mean$=$thetad, $sd=s$ is doing in

the context of the algorithm.

$($c$)$ Explain what the command if$($log$($runif$(1))$ log$\_$post$1-$log$\_$post $)$ is doing

in the context of the algorithm. In your answer, include a formula involving $p(|y)$ that

the code is implementing.

$($d$)$ What are the effects on the behaviour of the algorithm of making the variable called $s$

smaller? What are the effects of making it larger?

$($e$)$ What is the purpose of the variable called nb$?$

$($f$)$ When the code has run, what will the vector theta contain?

$($g$)$ In statistical terms, what will the command stheta$[$nm$/2]$ output?

$($h$)$ In statistical terms, what will the last two lines of code output?