Homework #6 Due April 25.


1. Suppose the data we can observe are

X1, X2, ... Xn   which are independent and having a Poisson
distribution with parameter lambda.

We shall take a Bayes approach, so assume the parameter lambda
is random and having a (prior) distribution of gamma(alpha=1.5, rate=2)

a. Find the posterior distribution.

b. find the mean of this posterior distribution. Compare to MLE.

c. If n gets larger and larger, but  sum(Xi)/n = 3 for all n
plot the posterior density for n=10, 100, 1000.

(notice the rate parameter in gamma usually were called lambda, but
here I already used lambda for the Poisson, so I have to use another
Greek letter....I just call it the rate.)

2. What if we take the (improper) prior that the lambda is "uniform"
on the (0, infinity) ?

a. find the posterior

b. find the mean of the posterior, Compare to MLE.

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More problems will be posted later here, as take home exam.

I.  Suppose X1, X2, ... Xn are independent random variables
with distribution N( \mu , 4 ). (where 4 is the variance, \mu is 
the mean and is an unknown parameter).

Taking a Bayesian point of view, the parameter \mu in the above 
is also random and assumed to have a prior distribution  N(10, 9 ).

(a). Compute the posterior distribution. 

(b). assume n=16, and the average of X1, X2, ... X16 = \bar X = 12, 
plot the posterior density. And identify the 90\% highest
posterior density (HPD) interval for \mu from the posterior. 

(c). If we take the prior distribution
to be  N(0, 0.25) but everything else remain the
same, what will be the HPD interval now?

II. Write (find, copy, steal, Google, cut and paste ...) a paragraph arguing 
for the Bayesian method; and another paragraph against the Bayesian
method.

You may include one example where you think Bayesian is more appropriate;
and another one that you think the maximum likelihood estimation
is more appropriate.


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That's it. Good luck.