Problem $3\mathrm{.}7$ In this chapter, we assumed that $$ is independent of the stimulus $s.$ This assumption is

often violated in real problems, leading to heteroskedasticity. Assume the measurement distribution

$p_{x|s}(x|s)=\frac{1}{\sqrt[\phantom{2}]{2(s{)}^2}}{e}^{-\frac{(x-s{)}^2}{2(s{)}^2}},$

where $(s)$ is the following function of $s$:$(s)=1+s^2.$

a$)$ For $s=0,1,2,$ plot the measurement distribution $($three curves in one plot, color$-$coded$).$ All

three should look Gaussian.

b$)$ For $x_{\text{o}\text{b}\text{s}}=0,1,2,$ plot the likelihood function over hypothesized $s($three curves in one plot,

color$-$coded$).$ None of them should look Gaussian.

c$)$ Explain how it is possible that the measurement distributions are all Gaussian but the

likelihoods are not.

d$)$ If the prior were Gaussian, would the posterior be Gaussian as well? Explain your answer

using math.