.
The website for this book contains 50 trivariate data points drawn from the
N3(
μ
,_
) distribution. Some data points have missing values in one or more coordinates. Only 27 of the 50 observations are complete.
a.
Derive the EM algorithm for joint maximum likelihood estimation of
μ
and
_
. It is easiest to recall that the multivariate normal density is in the exponential family.
b.
Determine the MLEs from a suitable starting point. Investigate the performance of the algorithm, and comment on your results.
c.
Consider Bayesian inference for
μ
when is known. Assume independent priors for the three elements of
μ
. Specifically, let
the
jth prior be
f
(μj) = exp{−(μj
−
αj)/βj} where (α1, α2, α3) = (2,
4,
6) and
βj
= 2 for
j
= 1,
2,
3. Comment on difficulties that would be faced in implementing a standard EM algorithm for estimating the posterior mode for
μ
. Implement a gradient EM algorithm, and evaluate its performance.
d.
Suppose that
_
is unknown in part (c) and that an improper uniform prior is adopted, that is,
f
(_) ∝ 1 for all positive definite
_
. Discuss ideas for how to estimate the posterior mode for
μ
and
_
.