STAT 431 — Applied Bayesian Analysis Homework 3

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1. The following are times (days) since last modification for 10 randomly-chosen English
Wikipedia articles:
481 144 93 446 69 170 383 63 79 181

On the log scale, they are roughly normally distributed. Take natural logarithms of these
values, then model the resulting (log-transformed) values as a two-parameter normal
sample, with both mean µ and variance σ
2 unknown.

For the following parts, use a normal-inverse gamma prior: For the conditional normal part,
specify a mean of zero and an equivalent prior sample size of 1. For the inverse gamma part,
let σ

2 have a (marginal) prior mean of 1 and a (marginal) prior variance of 1.
(a) [3 pts] Determine the numerical values of all of the parameters of the normal-inverse
gamma prior.

(b) [2 pts] Determine the marginal posterior distribution of µ: Name it, and give the
value(s) of its parameter(s).

(c) [1 pt] Compute a posterior 95% equal-tailed credible interval for µ.
(d) [2 pts] Determine the marginal posterior distribution of σ
2
: Name it, and give the
value(s) of its parameter(s).

(e) [1 pt] Compute a posterior 95% equal-tailed credible interval for σ
2
.

2. Consider the same data and data model as in the previous problem, but now use the
standard noninformative prior
p(µ, σ2
) ∝
1
σ
2

2 > 0)
in the following parts:

(a) [2 pts] Determine the marginal posterior distribution of µ: Name it, and give the
value(s) of its parameter(s).

(b) [1 pt] Compute a posterior 95% equal-tailed credible interval for µ.

(c) [2 pts] Determine the marginal posterior distribution of σ
2
: Name it, and give the
value(s) of its parameter(s).

(d) [1 pt] Compute a posterior 95% equal-tailed credible interval for σ