Description
1This problem highlights the difference between the maximum a posteriori (MAP) and conditional mean (CM) Bayesian point estimators. Consider a random variable X : S →R, and assume the posterior density is given by
where 0 < α < 1, σ0,σ1 > 0, and φ(x) is the Gaussian distribution
as in Kaipio and Somersalo (2005) Ch. 3, Example 1. (a) Using their respective definitions, show that
 ,    if

xMAP =
 ,     if
and
xCM = 1 − α .
Hint: If you come across some complicated integration in computing xCM, stop and think about which useful definitions (or densities, actually) could help!
(b) Make figures in MATLAB that show where xMAP and xCM lie with respect to the posterior πpost(x) for the following cases:
- α = 0.5, σ0 = 0.08, σ1 = 0.04
- α = 0.01, σ0 = 0.001, σ1 = 0.1
Does the MAP estimate or CM estimate give a better representation of the underlying posterior density? Explain your findings for each case.
If the conditional covariance
is given by
how does this spread estimator change in each of the above cases? Comment on your results.
2 (10 points) Given the random variables X : S →Rn and Y,E : S →Rm and the forward operator G : Rn →Rm, consider the multiplicative noise model
Y = E. ∗ G(X)
where .∗ denotes component-wise multiplication. One way to construct a likelihood function corresponding to this noise model is to transform it into an additive noise model in the natural log space, assuming that E is log-normally distributed, i.e., that the natural log of the noise follows a normal distribution N(0,σ2Im) with variance σ2.
Write the likelihood function for logY conditioned on X = x when σ2 is both known and unknown.
Note: For any of the above problems for which you use MATLAB to help you solve, you must submit your code/.m-files as part of your work. Your code must run in order to receive full credit. If you include any plots, make sure that each has a title, axis labels, and readable font size, and include the final version of your plots as well as the code used to generate them.
ALSO: Please note that this is your last official homework assignment for MA 590!  Your homework for the remainder of the course is to focus on your final project report and presentation – more details on this to follow next week!
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