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Specific information for this piece of coursework
• Answer ALL questions.
• Students must work alone.
• AI tools cannot be used.
• You may submit only one answer to each question.
• The total number of marks is 50.
• The numbers in square brackets indicate the relative weights attached to each part question.
• Marks are awarded not only for a final answer but also for the clarity and coherence of your solution. Do not include large amounts of R output.
• Your answers should be submitted as one pdf document (no larger than 100MB) through the submission link on the STAT0031 moodle page, within the ‘In-course Assessment’ section. Your answers may include hand-written and/or type sections.
• You must complete an ICA cover sheet to include as the first page of your submitted document. The cover sheet can be found within the ‘In-course Assessment’ section of the STAT0031 moodle page.
• Your answers will be marked anonymously. Please DO NOT include your name in any submitted material.
• You may use your course materials to answer questions.
• You may contact the course lecturer to ask questions concerning the ICA using the ‘In-course assessment discussion forum’ on the course moodle page. Please note, the lecturer will limit the amount of help ofered to students and cannot check answers (or part-answers) or give detailed guidance.
You may use the following notation and results:
The Poisson distribution, Poisson(λ), has probability mass function
where λ > 0. The mean is λ .
The Gamma distribution, Gamma(Q, β), has probability density function
where Q > 0 and β > 0. The mean is Q/β and the variance is Q/β2.
A multinational electrical components manufacturer wants to limit the number of faulty components that it produces. The company chooses C factories for the experiment. At each factory, B large batches of components are tested. The number of faulty components in the i-th batch at the j-th factory is Yi,j . Let Y = (Y1,1, . . . , YB,1, Y1,2, . . . , YB,2, . . . , Y1,C, . . . , YB,C ). The data can be downloaded in the file faulty_comp . txt from the ICA section of the STAT0031 Moodle page and contains results for 10 batches taken at 60 factories.
The company’s Bayesian statistician proposes the model
Yi,jjθj ~ Poisson(θj ), i = 1, 2, . . . , B, j = 1, 2, . . . , C
where Yi,j are independent given θj and
θj i.
.d. Gamma(Q, β), j = 1, 2, . . . , C.
1. Choose appropriate hyperpriors for α and β and briefly justify your choice. [3]
2. Derive the full conditional distribution of θj . [6]
3. Use NIMBLE to implement a Gibbs sampler to sample from the posterior distribution of this model with the data in the file faulty_comp . txt. In your answer include all R code needed to run the sampler with two chains and to monitor the chains for Q and β , i.e. all steps needed to use the function nimbleMCMC. [13]
4. Draw trace plots and densities of Q and β using the first 1000 iterations of your Gibbs sampler and comment on the convergence and mixing. [4]
5. Decide on an appropriate burn-in (using graphs of the Gelman-Rubin diagnostic) and a suitable run length to answer the next question (question 6) (briefly justify your choices). [7]
6. Use your code to provide estimates (by reporting the posterior mean and a 95% central credible interval for each parameter) of the following:
7. The company’s Bayesian statistician is worried that some batches have been incor- rectly tested. She proposes extending the model by introducing the parameter Zi,j which indicates whether the i-th batch at the j-th factory was accurately tested (Zi,j = 0) or inaccurately tested (Zi,j = 1). The new model is
Yi,jjθj , Zi,j ~ Poisson((1 + 2Zi,j )θj ), i = 1, 2, . . . , B, j = 1, 2, . . . , C
Zi,j ~ Bernoulli(φ), i = 1, 2, . . . , B, j = 1, 2, . . . , C
θj ~ Gamma(2, β), j = 1, 2, . . . , C
β ~ Gamma(1, 0.001)
(a) Choose a prior for φ and extend your NIMBLE code in Question 3 to sample from the posterior of this model with the data in file faulty_comp . txt. You should only include the model definition (i.e. the nimbleCode part). [4]
(b) Consider the first 3 factories (j = 1; 2; 3) and use your model and MCMC output to decide which batches are faulty. Briefly justify your answer. [7]