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Poisson regression
Haberman (1978) considers an experiment involving subjects reporting one stressful event. The
collected data are
y1, y2 ,..., y18 , where yi
interview. Suppose
yi
is the number of events recalled i months before the
is Poisson distributed with mean
i
, where the i  satisfy the regression
model
log  i   0  1i .
The data file stress.csv contains the observed values.
a. (2 points) If
 0 , 1  is assigned a joint uniform prior 0,1 , show that the logarithm of the
posterior density is given up to an additive constant, by
log  f  0 , 1  | data   i1  yi  0  1i   exp  0  1i  .
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b. (1 point) Write an R function to compute the logarithm of the posterior density of
 0 , 1  .
c. (1 point) Suppose we are interested in estimating the posterior mean and standard deviation for
the intercept and the slope. Through using optim() (method = "BFGS") with the data, find
the maximum likelihood estimate for
 0 , 1  and the corresponding estimated matrix of
variance covariance, using the starting value for
0  1 and 1  1 .
d. (3 points) Using the output of optim(), and the multivariate t distribution as the proposal
distribution, set the seed to be 1234, and using the data, construct a Metropolis-Hastings
random walk Algorithm for simulating from the posterior density. Compute the posterior mean,
median, mode, and standard deviation for
0
and
1 , using 10000 iterations. The multivariate t
distribution can be compute in R using the following function rmvt(n, sigma=cov*(df-2)/df,
df=df) in package rmvt, where df = 16, cov is the covariance matrix estimated from part c).
e. (1 point) Compare the result in part d) with what we get in part c) using optim().
f. (1 point) Estimate the highest density interval (HDI).
g. (1 point) Construct the histogram of
0
and
1 with the density overlay on the top, with Mean,
Mode, Median and standard deviation display on the graph.