Assignment 4: Bayesian Statistical Inference

1. Consider the Beta-Binomial model for \pi with Y|\pi \sim \text{Bin}(n, \pi) and \pi \sim \text{Beta}(3, 8). Suppose that in n=10 independent trials, you observe Y=2 successes.

a. Simulate the posterior model of \pi with RStan using 4 chains and 10000 iterations per chain.

b. Produce the trace plot for the four chains.

c. Create a density plot of the values for each of the four chains.

d. Use what we know about conjugate families to find the posterior distribution.

e. Overlay the true posterior distribution on the density plot of the simulated posterior distribution. Do they match?

f. What do you think the most likely value of \pi is? Why?

2. Consider the Gamma-Poisson model for \lambda with Y|\lambda \sim \text{Poi}(\lambda) and \lambda \sim \text{Gamma}(20, 5). Suppose you observe n=3 independent data points, \left(Y_1, Y_2, Y_3\right) = (0, 1, 0).

a. Simulate the posterior model of \lambda with RStan using 4 chains and 10000 iterations per chain.

b. Produce the trace plot for the four chains.

c. Create a density plot of the values for the four chains.

d. Use what we know about conjugate families to find the posterior distribution.

e. Overlay the true posterior distribution on the density plot of the simulated posterior distribution. Do they match?

f. What do you think the most likely value of \lambda is? Why?

3. Consider the Normal-Normal model for \mu with Y|\mu \sim \text{N}(\mu, 1.3^2) and \mu \sim \text{N}(10, 1.2^2). Suppose you observe n=4 independent data points, \left(Y_1, Y_2, Y_3, Y_4\right) = (7.1, 8.9, 8.4, 8.6).

a. Simulate the posterior model of \mu with RStan using 4 chains and 10000 iterations per chain.

b. Produce the trace plot for the four chains.

c. Create a density plot of the values for the four chains.

d. Use what we know about conjugate families to find the posterior distribution.

e. Overlay the true posterior distribution on the density plot of the simulated posterior distribution. Do they match?

f. What do you think the most likely value of \mu is? Why?