Problem Statement: Give an inference on the posterior distribution of (yTx) using the common Bayesian approximated inference: p(yTx) ~ p(xyT)p(yTyH)
. Now, let’s define each of these variables:

x
is a data vector with sizencol x 1
, where the density function ofxyT
is:1/[(2pi)^0.5*sigma_x)e^(x  A*yT)^2/(2*sigma_x^2)
(this really just meansxyT ~ N(A*yT, sigma_x)),
sigma_x = vector of size ncol x 1 with all entries equal to a positive constant,
A = matrix with size nrow x ncol
defined as:
A = matrix(c(1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1),
nrow, ncol, byrow= TRUE);

yT = a parameter vector with size ncol x 1
where the density function ofyT
is:1/[(2pi)^0.5*sigma_y)e^(yT  yH)^2/(2*sigma_y^2)', sigma_y = vector of size ncol x 1 with all entries equal to a positive constant,
sigma_y # sigma_x

yH = data vector with size ncol x 1, and yH is constructed as follows:
lambda = matrix(sample.int(100, size = ncol*nrow, replace = T),nrow,ncol);
lambda = lambda  diag(lambda)*diag(x=1, nrow, ncol);
yH = rpois(ncol,lambda) + rtruncnorm(ncol,0,1,mean = 0, sd = 1);
 Key constraint:
x = A*(yh+epsilon)
I hope this is sufficient for you to code. Please let me know if anything does not make sense to you.