Instructions

Please make sure your final output file is a pdf document. You can submit handwritten solutions for non-programming exercises or type them using R Markdown, LaTeX or any other word processor. All programming exercises must be done in R, typed up clearly and with all code attached. Submissions should be made on gradescope: go to Assignments \(\rightarrow\) Homework 4.

Questions

  1. Hoff problem 7.4.
    • Part (a): 1 point for all students
    • Part (b): 1 point for all students
    • Part (c): 2 points for all students
    • Part (d)(i): 2 points for all students
    • Part (e): 1 points for all students

    You can find the data mentioned in the question here: http://www2.stat.duke.edu/~pdh10/FCBS/Exercises/.

    For 7.4(d), parts (ii) and (iii) are NOT required but feel free to attempt for practice; you are only required to submit answers to part (i). For 7.4(d)(i), there is no need to read or complete the entire Exercise 7.1, simply rely on the form of the Jeffrey’s prior (from class slides) and the corresponding full conditionals (from the discussion session).

  2. Simulate data assuming \(\boldsymbol{y_i} = (y_{i1},y_{i2})^T \sim \mathcal{N}_2(\boldsymbol{\theta}, \Sigma)\), \(i = 1, \ldots, 100\), with \(\boldsymbol{\theta} = (0,0)^T\) and \(\Sigma\) chosen so that the marginal variances are \(1\) and correlation \(\rho = 0.8\). Assuming independent normal & inverse-Wishart priors for \(\boldsymbol{\theta}\) and \(\Sigma\), that is, \(\pi(\boldsymbol{\theta}, \Sigma) = \pi(\boldsymbol{\theta}) \pi(\Sigma)\), run Gibbs sampler (hyperparameters up to you but you must justify your choices) to generate posterior samples for \((\boldsymbol{\theta}, \Sigma)\).
    Now, generate 50 new “test” data from the same sampling distribution, that is, \(\boldsymbol{y_i}^\star = (y_{i,1}^\star,y_{i,2}^\star)^T \sim \mathcal{N}_2(\boldsymbol{\theta}, \Sigma)\), \(i = 1, \ldots, 50\). Keep the \(y_{i,2}^\star\) values but set the \(y_{i,1}^\star\) values to NA (make sure to save the “true” values somewhere!).
    Using the posterior samples for \((\boldsymbol{\theta}, \Sigma)\), based on the 100 “train” data, answer the following questions:

    • Part (a): Generate predictive samples of \(y_{i,1}^\star\) given each \(y_{i,2}^\star\) value you kept, for the 50 test subjects. Show your sampler.
      (2 points for all students)
      You should view this as a “train –> test” prediction problem rather than a missing data problem on an original data. That is, given the posterior samples of your parameters, and the test values for \(y_{i2}^\star\), draw from the posterior predictive distribution of \((y_{i,1}^\star | y_{i,2}^\star, \{(y_{1,1},y_{1,2}), \ldots, (y_{100,1},y_{100,2})\})\),. You may find it useful to think of this in terms of compositional sampling, that is, for each posterior sample of \((\boldsymbol{\theta}, \Sigma)\), sample from \((y_{i,1} | y_{i,2}, \boldsymbol{\theta}, \Sigma)\), which is just from the form of the sampling distribution. Do not incorporate the prediction problem into your original Gibbs sampler!

    • Part (b): Using the samples from the predictive density obtained above, obtain \(\mathbb{E}[y_{i,1}^\star | y_{i,2}^\star]\) for each of the test subjects, as well as a 95% posterior predictive interval. Make a plot containing all the intervals for each of the 50 subjects. In the plot, indicate where each \(\mathbb{E}[y_{i,1}^\star | y_{i,2}^\star]\) falls within each interval.
      (2 points for all students)

    • Part (c): What is the coverage of the 95% predictive intervals out of sample? That is, how many of the 95% predictive intervals contain the true \(y_{i,1}^\star\) values?
      (2 points for all students)

  3. Suppose data consist of reaction times \(y_{ij}\) for subjects \(i = 1, \ldots, n_j\) in experimental conditions \(j = 1, \ldots, J\). Researchers inform you that it is reasonable to assume that reaction times follow an exponential distribution.
    • Part (a): Describe a Bayesian hierarchical model for borrowing information across experimental conditions. Specify priors that will allow you to borrow information across the \(J\) conditions.
      (2 point for all students)
      When setting the priors, make sure you select priors that respect the support/parameter space of each parameter. Also, think about specifying conjugate or semi-conjugate priors will make your life much easier.
    • Part (b): Derive the Gibbs sampling algorithm for fitting your hierarchical model. What are the full conditionals?
      (2 point for all students)
    • Part (c): Simulate data from the assumed model with \(J = 5\) and the \(n_j\)’s set to your preferred values, but with each set to at most 25. Also, set all parameter values as you like, but make sure they are reasonable (that is, avoid very extreme values). Implement the Gibbs sampler, present point and interval estimates of the group-specific mean reaction times.
      (2 point for all students)
    • Part (d): Compare results from hierarchical specification to the true parameter values that you set. How well does your Gibbs sampler perform?
      (1 point for all students)

Grading

20 points.