STAT 578 – Advanced Bayesian Modeling Assignment 1 solution

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1. The film review aggregator website rottentomatoes.com publishes ranked lists of movies
based on the number of positive critical reviews out of a total number counted for each
movie. See, for example, https://www.rottentomatoes.com/top/bestofrt/?year=2019.
Because the site uses an “Adjusted Score,” a movie with a higher approval percentage
sometimes ranks lower on the list.
Consider the following hypothetical scenario:
Movie 1: 425 positive reviews out of 500 (85%)
Movie 2: 9 positive reviews out of 10 (90%)
Assume that reviews of Movie i are independent with a common probability pi of being
positive (depending on the movie). Assume a U(0, 1) prior on each pi
.
(a) [4 pts] Determine the posterior distribution of p1 and of p2 (separately). (Name the
type of distribution and give the values of its defining constants.)
(b) [3 pts] Which movie ranks higher according to posterior mean? According to posterior
median? According to posterior mode? Show your computations. (For median, use R
function qbeta. For mean and mode, use formulas in BDA3, Table A.1. Do not use
simulation, as it may not be sufficiently accurate.)
2. File randomwikipedia.txt contains the ID number and number of bytes in length for 20
randomly selected English Wikipedia articles.
(a) (i) [2 pts] Draw a histogram of article length, and describe the distribution.
(ii) [2 pts] Transform article length to the (natural) log scale. Then re-draw the
histogram and describe the distribution.
(iii) [1 pt] Based on your histograms, explain why the log scale would be better to use
for the remainder of the analysis. (Read below.)
(b) [2 pts] Let yi be length of article i on the log scale (i.e., the natural logarithm of the
number of bytes). Compute the sample mean and sample variance of y1, . . . , y20.
In the remaining parts, assume the yis have a normal sampling distribution with
mean µ and variance σ
2
.
(c) Assume σ
2
is known to equal the sample variance. Consider a flat prior for µ. Use it to:
(i) [3 pts] Compute the posterior mean, posterior variance, and posterior precision
of µ.
(ii) [2 pts] Plot the prior density and the posterior density of µ together in a single
plot. Label which is which.
(iii) [2 pts] Compute a 95% central posterior interval for µ.
(d) Now let µ and σ
2 have prior
p(µ, σ2
) ∝

σ
2
−1
σ
2 > 0

Use it to:
(i) [3 pts] Compute the posterior mean, posterior variance, and posterior precision
of µ. (If you cannot compute explicitly, use a good computational approximation.)
(ii) [2 pts] Approximate a 95% central posterior interval for µ.
(iii) [2 pts] Approximate a 95% central posterior interval for σ
2
.
(e) Assume the prior of the previous part. Use simulation in R to answer the following,
based on 1,000,000 draws from the posterior.
(i) [2 pts] Approximate a 95% central posterior predictive interval for the length (in
bytes) of a single (new) randomly selected article. (Note that this is on the
original scale, not the log scale.)
(ii) [2 pts] Approximate the posterior predictive probability that the length of a single
(new) randomly selected article will exceed the maximum article length in the
data.
(iii) [2 pts] Approximate the posterior predictive probability that the maximum length
of 20 (new) randomly selected articles will exceed the maximum article length in
the data. (Be careful! All 20 randomly selected articles have the same value for µ
and for σ
2
.)
Reminder: Show the R code you used and also a summary of the approximate inference results
that you used to answer the preceding parts.
Total: 34 pts
2