STA360 at Duke University
Suppose the target distribution we wish to sample from is given by probability mass function
\[ \pi(\theta) = \theta / w \text{ for } \theta \in \{1, 2, \ldots 6\} \]
in words, we wish to roll a die with probability \(1/w\) of landing on face 1, \(2/w\) of landing on face 2, etc.
Write a Metropolis algorithm to approximate the target distribution using a proposal \(J(\theta = j | \theta^{(s)} = i) = 1/6\) for all \(j\), i.e. propose a new state \(j\) uniformly. Run your Markov chain for \(S=10000\) states.
The Metropolis algorithm requires a symmetric proposal \(J\). Explain why this proposal is symmetric.
Plot a histogram of the output. Does the plot match your intuition?
Compare the estimated probabilities of each outcome to the truth (compute \(w\)).
Metropolis-Hastings lets us work with non-symmetric proposals. Re-write the algorithm of the previous exercise using the non-symmetric proposal \(J(\theta = j | \theta^{(s)} = i)\) such that
\[ \theta = \begin{cases} 1 & \text{ with prob } & 0.05\\ 2 & \text{ with prob } & 0.15\\ 3 & \text{ with prob } & 0.2\\ 4 & \text{ with prob } & 0.15\\ 5 & \text{ with prob } & 0.15\\ 6 & \text{ with prob } & 0.3\\ \end{cases} \]