Explores the Metropolis Hastings (MH) Algorithm - Statistics Assignment Help

Assignment Task:

Task:

Task 1. This task explores the Metropolis-Hastings (MH) algorithm used in the second computer lab a bit further. In that lab, we used an MH algorithm with a proposal distribution that was independent of the current location of the chain. While such MH algorithm can be used, they are typically quite inefficient if the proposal distribution is not “close” to the distribution one wants to sample from.

Another popular way of implementing an MH algorithm is to use a random walk MH algorithm. In such an implementation the proposed new state is obtained by adding some random noise (typically using a normal distribution with mean zero) to the current state. In other words, the proposal distribution is a normal distribution with mean equal to the current state and some variance (which serves can serve as a tuning parameter for the MH algorithm). The difficulty with using a random walk MH algorithm in the example used in the computer lab is that we are modelling a probability, which is always between 0 and 1. Adding random noise to the current state could lead to a proposal that is outside the interval [0, 1]! We can circumvent this difficulty by first transforming the current state (a value between 0 and 1) via the logit transformation to the corresponding log-odds (a value between −∞ and ∞). We can then add some noise to the transformed current state to obtain a proposal on the log-odds scale and then transform this proposal back to the interval [0, 1].

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