Statistics I Assignment 2 SOLVED

Math 525 – Statistics I
Assignment 2
Problem 1: With this problem, you will set up a Bayesian model for continuous measurements.
Consider data wn, where n = 1, . . . , N indexes individual measurements. Each wn is obtained
through a Normal(µ, v) distribution. Assume the variance v has a known value. On µ use a
Normal(M, V ) prior, where M and V have also known values.
(i) Write down the model using statistical notation.
(ii) Derive the posterior of µ.
(iii) Use the provided data w1:N and compare graphically the prior and posterior of µ. For
concreteness, use the following values v = 1, M = 0, V = 10.
(iv) As in step (iii), use the provided data w1:N and v = 1, M = 0, V = 10 to compute the prior
and posterior probability that µ takes a negative value. To carry out the involved integrations,
you might use integral or any other integration strategy you prefer.
(v) As in step (iii), use the provided data w1:N and v = 1, M = 0, V = 10 to compute analytically
the maximum a posteriori estimate of µ.
(vi) Provided that each measurement is reported in adu units, deduce the units of µ, v, M, V .
Associated data: For steps (iii), (iv), and (v) use the data in normal normal.mat.
Problem 2: With this problem, you will set up a Bayesian model for discrete measurements.
Consider data wn, where n = 1, . . . , N indexes individual measurements. Each wn is obtained
through a Geometric(π) distribution. This distribution is supported on the non-negative integers
and its probability mass is given by:
p(w) = (1 − π)
wπ
On π use a Beta(α, β) prior, where α and β have known values. This distribution is supported
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on the real numbers that lie between 0 and 1 and its probability density is given by
p(π) = π
α−1
(1 − π)
β−1
B(α, β)
where B(α, β) is the Beta function.
(i) Write down the model using statistical notation.
(ii) Derive the posterior of π.
(iii) Use the provided data w1:N and compare graphically the prior and posterior of π. For
concreteness, use the following values α = 1 and β = 1.
(iv) As in step (iii), use the provided data w1:N and α = 1, β = 1 to compute the prior and
posterior probability that π takes a value above 0.15. To carry out the involved integrations, you
might use integral or any other integration strategy you prefer.
(v) As in step (iii), use the provided data w1:N and α = 1, β = 1 to compute analytically the
maximum a posteriori estimate of π.
Associated data: For steps (iii), (iv), and (v) use the data in geo beta.mat.
Problem 3: With this problem, you will investigate a Bayesian model with a non-standard form.
Consider the statistical model
(φ, ψ) ∼ H(A, B, r, s)
wn|φ, ψ ∼ Gamma (φ, ψ), n = 1, . . . , N
In this model, the distribution H(A, B, r, s) has a probability density of the form
H(φ, ψ; A, B, r, s) ∝
Aφ−1
e
−B/ψ
(Γ(φ))r
ψφs , φ > 0,ψ > 0
(i) Derive the posterior of φ, ψ.
(ii) Indicate whether the model is conjugate or not.
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