Assignment 1

These problems are for credit and to be handed in on September 23rd at 23:59 the latest.

Problem 1.1

Consider the geometric distribution \(\mathsf{geom}(p)\) with mass function \[\begin{align*} f(y; p) = p(1-p)^{y}, \qquad y = 0, 1, 2, \ldots. \end{align*}\] This distribution is used to model the number of failures \(Y\) from independent trials until a first success, which occurs with probability \(p\).

  1. If \(Y \mid P=p \sim \mathsf{geom}(p)\) and \(P \sim \mathsf{beta}(\alpha_1, \alpha_2)\), show that \(P \mid Y=y\) is beta distributed and give explicit expressions for its shape parameters.
  2. Obtain the marginal distribution of \(Y\) and show that it is a special case of the beta-negative binomial distribution.
  3. Using the tower property, compute the unconditional mean and variance of \(Y\). Hint: the formula will depend on the reciprocal moments of a beta distribution, \(\mathsf{E}_P(P^{-1})\) and \(\mathsf{E}_P(P^{-2})\). Complete the kernel to obtain these using the property \(\Gamma(\alpha+1) = \alpha \Gamma(\alpha)\).
  4. Generate 10K samples from the marginal distribution of \(Y\) if \(\alpha_1=4.5\) and \(\alpha_2=1.5\). Plot the resulting marginal distribution using a bar plot and use the samples to verify the formulas for the expected value and variance derived previously using Monte Carlo integration.

Forward sampling: generate data from the marginal of \(Y\) as follows:

  • Simulating first from \(P_i\sim \mathsf{beta}(4.5, 1.5)\),
  • Next, generate \(Y_i \mid P_i\) for \(i=1, \ldots, 10 000\).
  • Discard the values of \(P_i\) and keep only those for \(Y_i\)’s.

Problem 1.2

The waiting dataset contains waiting times (in seconds) from 17:59 until the departure of the next metro at the Universite de Montreal station during week-days over three consecutive months.

Assume first that the waiting time are independent and identically distributed as exponential.

  1. Use a conjugate gamma prior such that the average waiting time \(1/\lambda\) has mean 30 seconds and std. deviation 30 seconds.1
    • Give the values of the corresponding shape and rate parameters of the prior.
    • Plot an histogram of \(1000\) prior predictive draws.
  2. Derive the posterior distribution and report its parameter values.
  3. Calculate the posterior probability of waiting more than 30 seconds analytically and verify the result via Monte Carlo integration.
  4. Perform a prior sensitivity analysis.

The post_waiting_weibull contains 10K random samples from the posterior of a Weibull model \(\mathsf{Weibull}(\lambda, \alpha)\) with a penalized-complexity prior for the shape parameter \(\alpha \sim \mathsf{PC}(\theta=0.5)\) (Niekerk et al., 2021) and \(\lambda \sim \mathsf{inv. gamma}(\gamma, \omega)\) with scale \(\gamma=90\) and shape \(\omega=4\).

  1. Draw \(B=1000\) posterior predictive samples of size \(n=62\) from the Weibull and exponential models. For each posterior draw, generate a sample of size \(n=62\).
  2. For each, compute the sample mean, ( the sample std. deviation and the empirical proportion of samples exceeding 30 seconds. Plot an histogram for each of the three summary and each model (Weibull and exponential). Superimpose a vertical line indicating the corresponding function for the original waiting sample. Hence comment on the adequacy (or lack thereof) of the two models.

References

Niekerk, J. van, Bakka, H., & Rue, H. (2021). A principled distance-based prior for the shape of the Weibull model. Statistics & Probability Letters, 174, 109098. https://doi.org/10.1016/j.spl.2021.109098

Footnotes

  1. Hint: if \(\Lambda \sim \mathsf{gamma}(\alpha, \beta)\), then the reciprocal rate follows \(1/\Lambda \sim \mathsf{inv. gamma}(\alpha, \beta)\) with \(\mathsf{E}(\Lambda^{-1}) = \beta/(\alpha-1)\) for \(\alpha>1\) and \(\mathsf{Va}(\Lambda^{-1}) = \beta^2/\{(\alpha-1)^2(\alpha-2)\}\). Solve to find the values of the parameters and check numerically by generating data from the inverse gamma distribution.↩︎