SQB7009 Bayesian Statistics Assignment 1
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SQB7009 Bayesian Statistics
Assignment 1
2022
1. Suppose we have 1, … , with
| ~ Bern().
(a) Using the improper prior distribution () ∝ −1(1 − )−1 , find the posterior
distribution (|) up to proportionality, and find a Normal approximation about the mode to the posterior (|).
(b) Show that the prior () is equivalent to a uniform prior on
= log ()
and find the posterior (|). Find a Normal approximation about the mode to the posterior (|).
(c) For which parameterization does it make more sense to use a Normal approximation? Explain.
(d) Based on your answer in part (c), write a program in R to use the Normal approximation. Use the data = ∑1 generated below, where = 30:
set.seed(id) p = round(runif(1, 0.2, 0.8), 2) S = rbinom(1, 30, p) list (p=p, S=S) |
(i) Plot the density of the Normal approximation about the mode to the posterior of the corresponding parameter.
(ii) Using 1,000 samples from this Normal approximation (use set.seed(id)),
calculate the mean and variance of these samples. Compare them to the theoretical values from the Normal approximation.
(iii) Using the samples from part (ii), find the (approximate) 99% credible
interval for the posterior mean of the corresponding parameter.
2. Capture-recapture models are used to estimate the size of a closed animal population. Suppose is unknown, and each animal in the population has the probability of being caught at any given time. At the first time point we capture 1 animals at random, tag them and release them back into the wild. At time 2 we capture 2 animals at random; of these 11 are already tagged. Let = 1 + 2 be the total number of captures made, and = − 11 be the total number of unique animals captured.
(a) Each animal can be in one of four situations: caught on both occasions, caught on
the first but not the second, caught on the second but not the first, and never caught. Let be the number of animals in each situation, where = 0, 1 denotes not captured and captured respectively on the first occasion, and = 0, 1 denotes not captured and captured respectively on the second occasion. Find and the corresponding probabilities .
(b) Assume that the data have multinomial distribution with parameters and
distribution given by
(00 , 01, 10, 11 |, ) = 0(00) 1(01) 0(10)
Find the likelihood of (, |, ).
(c) Take independent priors for and with | ~ Po(), where is known, and ~ (0, 1). Find the joint posterior distribution (, |, ).
(d) Find an expression for the marginal distribution, (|, ), up to proportionality.
(e) Consider the reparameterisation = − . Describe how we may use the
Gibbs sample to produce samples from the joint posterior (, |, ), and hence how we might estimate the marginal mean [|, ].
(f) Suppose we wish to estimate the size of the dolphin population in a marine area
of New Zealand. Experts believe that there are somewhere between 75 and 125 dolphins in the area; an appropriate choice of prior is as above with = 100. On the first occasion we capture 36 dolphins on camera. A week later, we capture 48 on camera, of whom 27 are identified as having been observed on the first occasion also.
(i) Write a Gibbs sampler program in R to estimate the size of the population . Note to include burn-in as part of your program.
(ii) Using 1,000 samples and burn-in of 100 (use set.seed(id)), estimate the
value of . Use the time series plots of and and other plots/methods to check for convergence.
3. Let , = 1, … , , be exchangeable where | ~ Inv − Gamma(, ), where is
known. Consider a prior distribution ~ Gamma(, ).
(a) Find the posterior distribution | .
(b) We wish to use the Metropolis-Hastings algorithm to sample from the posterior
distribution | using a normal distribution with mean and chosen constant variance 2 as the symmetric proposal distribution.
(i) Describe how the Metropolis-Hastings algorithm works for this example, giving the acceptance probability in its simplest form.
(ii) Suppose that, at time , the proposed value ∗ ≤ 0. Briefly explain why the
corresponding acceptance probability is zero for such a ∗ and thus that the sequence of values generated by the algorithm are never less than zero.
(iii) Does this normal proposal distribution result in an efficient algorithm? How
can this algorithm be improved? Explain.
(c) Suppose instead for the Metropolis-Hastings algorithm in part (b), the proposal ∗ is obtained by first sampling ̃ from a normal distribution with mean and chosen variance 2 and then setting
∗ = { ̃ if ̃ ≥ 0
Show that the associated proposal distribution is symmetric.
(d) (i) Write a program in R to estimate the posterior mean of using the the Metropolis-Hastings algorithm in part (c). Note to include burn-in as part of your program.
(ii) Let the data , = 1, … , be generated below, where = 20.
set.seed(id) Y = round(1/rgamma(20, 1, 0.5),2) list (Y=Y) |
Here = 1. Use the prior values = 0.5, = 0.5. Using 10,000 samples and burn-in of 1000 (use set.seed(id)), and the proposal variance 2 = 252 , estimate the posterior mean of . Compare it to the theoritical mean obtained using the distribution in part (a).
Also, use the time series plot of and other plots/methods to check for convergence.
2022-06-27