Problem $4\mathrm{.}6$ An observer infers a stimulus $s$ from a measurement $x_{\text{o}\text{b}\text{s}\text{.}}.$ As in the chapter, the

measurement distribution $p(x|s)$ is Gaussian with mean $s$ and variance ${}^2.$ Unlike in the chapter,

we use the prior

$p(s)=e^{-s},$

where $$ is a positive constant. This is an improper prior $($see Section $3\mathrm{.}5\mathrm{.}2)$ but that does not stop

us$.$

a$)$ Derive an equation for the posterior mean estimate.

b$)$ Derive an equation for the distribution of the posterior mean estimate for given $s.$