Problem $3\mathrm{.}2$ In this problem, we numerically calculate a posterior distribution. Suppose the

stimulus distribution $p_s(s)$ is Gaussian with mean $20$ and standard deviation $4.$ The measurement

distribution $p_{x|s}(x|s)$ is Gaussian with standard deviation $=5.$ A Bayesian observer infers $s$ from

an observed measurement $x_{\text{o}\text{b}\text{s}}=30.$ We are now going to calculate the posterior probability density

using numerical methods.

a$)$ Define a vector of hypothesized stimulus values $s$:$(0,0\mathrm{.}2,0\mathrm{.}4,$dots,$40).$

b$)$ Compute the likelihood function and the prior on this vector of $s$ values.

c$)$ Multiply the likelihood and the prior pointwise.

d$)$ Divide this product by its sum over all $s($normalization step$).$

e$)$ Convert this posterior probability mass function into a probability density function by dividing

by the step size you used in your vector of $s$ values $($e$.$g$.,0\mathrm{.}2).$

f$)$ Plot the likelihood, prior, and posterior in the same plot.

g$)$ Is the posterior wider or narrower than likelihood and prior? Do you expect this based on the

equations we discussed?

h$)$ Change the standard deviation of the measurement distribution to a large value. What happens

to the posterior? Can you explain this?

i$)$ Change the standard deviation of the measurement distribution to a small value. What

happens to the posterior? Can you explain this?