Homework Set 15, Due Wednesday November 18, 2016.
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Assigned 11/9/2016, modified 11/12, due Friday 11/18
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Again use binomial model for data X_i as in HW 14, in a Bayesian setting, 

with X_i ~ Binomial(n_i, p_i) , i=1,...,10,  but now with the prior

p_i ~ f(p) = (9/49)*dbeta(p,.4,.4) + (15/49)*dbeta(p,1,1) + 
          (25/49)*dbeta(p,2,2)

and the data:

> nvec
 [1] 105 150 140 81 52 95 74 112 140 75

> Xvec
 [1]  48 111  74  15  46  44  24  81  85  33

The coefficients and parameters are chosen in this mixture density 
so that the mean and mean-square are very close to the respective values 
.5 and 1/3 of the Unif[0,1] that we used before, although the prior 
density is VERY different. (Graph it and you will see !)

(a) Find the exact posterior densities of the numbers p_i based on the 
data (n_i, X_i, i=1,...,10) arising from the model, and form 99% credible 
intervals for the random-effect parameters p_1,...,p_10,

### Hint: this prior is a mixture of beta's, and so will the posterior be.
If you first imagine that you know which of the three mixture-components each
X_i comes from, and then un-condition, you can prove this easily and use the
standard conjugate-prior property of the beta and binomial to read off the
mixture components in the posterior.


(b) Obtain the posterior density values by (a single) numerical 
integration for each p_i, as in the first segment of the 
Nov7F16.RLog  log, by a Laplace-method change of variable.


(c) How close would each of your numerically integrated 
posterior-density integrals be to the truth if you replaced 
the respective posterior densities by the exponentials of the 
SECOND-order Taylor series approximations to those densities 
at the respective points

p.st[i] = maximizer of  log f(p|X_i)  on (0,1)    

as we discussed under the heading of "Laplace Method" in class 
on November 9 ?