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Derive the equation for the posterior mean by expanding the square in the exponential
for each i, collecting all similar power terms, and making a perfect square again. Note that the product of exponentials can be written as the exponential of a sum of terms.
- For this exercise, we use the dataset corresponding to Smartphone-Based Recognition of Human Activities and Postural Transitions, from the UCI Machine Learning repository (https://archive.ics.uci.edu/ml/datasets/Smartphone-Based+Recognition+of+Human+Activities+and+Postural+Transitions). It contains values of acceleration taken from an accelerometer on a smartphone. The original dataset contains x, y, and z components of the acceleration and the corresponding timestamp values. For this exercise, we have used only the two horizontal components of the acceleration x and y. In this exercise, let's assume that the acceleration follows a normal distribution. Let's also assume a normal prior distribution for the mean values of acceleration with a hyperparameter for a mean that is uniformly distributed in the interval (-0.5, 0.5) and a known variance equal to 1. Find the posterior mean value by using the expression given in the equation.
- Write an R function to compute the Fisher information matrix. Obtain the Fisher information matrix for this problem by using the dataset mentioned in exercise 1 of this section.
- Set up an MCMC simulation for this problem by using the mcmc package in R. Plot a histogram of the simulated data.
- Set up an MCMC simulation using Gibbs sampling. Compare the results with that of the Metropolis algorithm.
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