Research School of Finance, Actuarial Studies and Statistics

Test 1 - Practice Paper A

STAT3016/6016 - Introduction to Bayesian Data Analysis

Problem 1 [12 marks]

The Weibull distribution is commonly used in product failure analysis to model the lifetime of a product, for example, a light bulb.

Let T be the lifetime of a lightbulb. Let’s assume T|k, λ ∼ W eibull(k, λ) where k is the shape parameter and λ is the scale parameter. The probability density function of T is

(a) [1 mark] What is the support of each of the parameters k and λ? In other words, are there any restrictions on the values of k and λ?

(b) [2 marks] Assume we observe the lifetime of n lightbulbs. So our lifetime data is t1, t2, ...., tn. Write down the joint likelihood function of the data given k and λ. That is, write down the form. of the likelihood function f(t1, ..., tn|λ, k).

(c) [2 marks] Let’s reparametise the model and define θ = λ k . Rewrite the likelihood function from part (b) given the new parameter θ and k. That is, write down the form. of the likelihood function f(t1, ..., tn|θ, k).

(d) [2 marks] Let’s assume k is fixed and known. Derive a conjugate prior distribution for the parameter θ.

(e) [2 marks] What is the posterior distribution of θ|t1, ..., tn, k?

(f) [1 mark] From part (e) what are the sufficient statistics of the Weibull distribution?

(g) [2 marks] What is the posterior mean of θ? Express the posterior mean as some linear combination of the prior mean and a sample mean.

Problem 2 [8 marks]

It is a common problem for measurements to be observed in rounded form. For a simple example, suppose we weigh an object five times and measure weights, rounded to the nearest kilogram, of 20, 21, 20, 20, 22. Assume the unrounded measurements are normally distributed with a noninformative prior distribution on the mean µ and variance σ 2 .

(a) [2 marks] Give the posterior distribution for (µ, σ2 ) obtained by pretending that the observations are exact unrounded measurements.

(b) [3 marks] Give the correct posterior distribution (up to a proportionality constant) for (µ, σ2 ) treating the measurements as rounded.

(c) [3 marks] How do the incorrect and correct posterior distributions differ? Compare means, vari-ances and contour plots using the following plots and summary estimates of the posterior distri-butions.