This attachment supports materials presented in the classroom for the fall course curriculum.

1. Problem Set 1
Due: September 25
Problem 1. One of the big scientific mysteries of the late 19th and early 20th centuries
was an anomaly in the orbit of Mercury relative to the predictions of Newtonian physics.
The data at the time could be summarized by saying that the observation of the key orbital
quantity was X ∼ N(µ, 22
); Newtonian mechanics would have predicted µ = 0, but the data
was x = 41.6, so there is clearly something else going on.
a) Formally do an objective Bayesian test of H0 : µ = 0 versus H1 : µ = 0, using the Bayesian
t-test on p. 26 of Lecture 2. Assume that t = 20.8 (the stated observation over the standard
error) and was based on n = 10 observations.
One of the proposed explanations of the anomaly was that gravity followed an inverse
“square + ” law. Under this hypothesis, µ could be any value in the interval (−100, 100).
(Larger values of |µ| would mean other anomalies would have been detected in the orbits of
other planets.) The gravity hypothesis is thus HG : X ∼ N(µ, 22
), where µ ∈ (−100, 100).
In 1915, Einstein proposed general relativity, which predicted that µ = 42.9. The general
relativity hypothesis is thus HE : X ∼ N(42.9, 22
).
b) Find the Bayes factor of HE to HG, utilizing a N(0, 502
) prior for µ under HG (com-
patible with µ being in the interval (−100, 100)). Explain why general relativity is strongly
supported, even though the gravity model would provide a better ‘fit’ to the data.
Problem 2. Decide (and justify) which of the following should be tested as a precise
hypothesis and which should not. Note that this is asking your subjective opinion about
each. Also, you may assume that data sample sizes are moderate.
• H0 : Eating broccoli does not cause lung cancer.
• H0 : Access to health care is equal in cities and rural areas.
• H0 : Learning calculus on-line is the same as learning calculus with an instructor.
• H0 : The effectiveness of a new vaccine will be the same for men and women.
Problem 3. The data X1, . . . , Xn (Xi > 0) arise from one of the following two scale
distributions:
f1(x | σ) =
2
σ
√
π
e−(x/σ)2
or f2(x | σ) =
1
σ
e−x/σ
,
with σ > 0.
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2. a) What is the standard objective prior to use for σ?
b) What is the subsequent Bayes factor to use for objective Bayesian testing between these
two distributions? (Provide a closed form answer.)
Problem 4. In an industrial setting, when a production machine is operating properly, it
produces parts of diameter X, where X ∼ N(x | 5, 0.01) (i.e., mean 5 and variance .01).
Frequently the machine fails in a certain way, and then produces parts with X ∼ N(x |
5, 0.02). The quality control policy is to periodically sample 5 parts.
After obtaining a sample x1, x2, x3, x4, x5, what is the Bayes factor for the machine op-
erating properly to having failed?
Problem 5. In the five location-scale models example, the goal is to predict the next
observation xn+1. Suppose you can compute the predictive density p(xn+1 | x1, . . . , xn, M)
for each of the five models M. (You do not need to do these computations; just use the
generic expressions for the predictive densities.) What is the best overall predictive density
for each of the four data sets?
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