BIS 567A- Assignment 4- Due 10/29/2020
1. Suppose we assume Yi|λi ind∼ Poisson (λi), i = 1, . . . , n with ln (λi) = β0 + β1xi + φi and
φi|σ2 iid∼ N
(
0, σ2
)
. We specify the following prior distributions:
• β0 ∼ N
(
0, 1002
)
;
• β1 ∼ N
(
0, 1002
)
;
• σ2 ∼ Inverse Gamma (0.01, 0.01).
This represents a Poisson regression model that accounts for potential overdispersion that is
often observed in actual count data (greater variability in the data than expected). We will
fit this model in two different ways (both Bayesian) and compare the performance of each
algorithm. Our interest is in estimating β0, β1, and σ
2. The R file “HW4.RData” contains
n = 1, 000 observed data points (y) along with the covariate of interest (x). Please do not
use a Bayesian software package for this assignment (e.g., JAGS, Proc MCMC).
(a) Create an MCMC sampling algorithm to sample from the joint posterior distribution
f
(
ln (λ1) , . . . , ln (λn) , β0, β1, σ
2|Y ). Provide full details on the full conditional distribu-
tions, commented code for fitting your MCMC algorithm, and a table that summarizes
your posterior inference for each of the parameters (posterior mean, posterior standard
deviation, 95% credible interval). Do not include inference for ln (λi) in the table.
(b) Create an MCMC sampling algorithm to sample from the joint posterior distribution
f
(
φ1, . . . , φn, β0, β1, σ
2|Y ). Provide full details on the full conditional distributions,
commented code for fitting your MCMC algorithm, and a table that summarizes your
posterior inference for each of the parameters (posterior mean, posterior standard devi-
ation, 95% credible interval). Do not include inference for φi in the table.
(c) For each algorithm, provide some evidence to suggest that the algorithm has converged
and that you have collected enough posterior samples to make accurate inference.
(d) How do the results in parts (a) and (b) compare? Based on your experience, which
algorithm do you prefer and why?
1

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