In this exercise covariates x$1$ and x$2$ are simulated as$1$

x$1=$ rand$(1,40)$; x$2=$ floor$(10*$ rand$(1,40))+1$;

and the response variable y is obtained as

y $=2+6*$ x$1-0\mathrm{.}5*$ x$2+0\mathrm{.}8*$randn$($size$($x$1))$;

Write a program using a Probabilistic Programming Language $($PPL$),$ like BUGS or

PyMC$,$ that takes $20$ triples $($x$1,$ x$2,$ y$)$ to train the linear regression model $$y $=$ b$0+$b$1$x$1+$b$2$x$2$

and then uses the remaining $20$ triples to evaluate the model by comparing the original

responses yi$,$ i $=21,...,40,$ with regression$-$predicted values $$yi$,$ i $=21,...,40.$

The comparison would involve calculating the MSE, the mean of $($yi $$yi$)2,$ i $=21,...,40.$

This is an example of how to implement a basic train$/$test split to test a Bayesian inference

model$$s predictive accuracy. How do the Bayesian estimators of $\backslash$beta $0,\backslash$beta $1,\backslash$beta $2,$ and $\backslash$sigma compare

to the $$true$$ values $2,6,0\mathrm{.}5,$ and $0\mathrm{.}8?$