15. Monte Carlo Methods

The Monte Carlo method is a general algorithm for estimating a definite integral based on outcomes of a simulation experiment. One can apply this algorithm to estimate the expectation of a function of a random variable where a random sample is drawn from the associated probability distribution. There is an associated standard error of a Monte Carlo estimate which provides insight into the accuracy of this simulation-based calculation. One can use Monte Carlo to simulate the sampling distribution of a statistical estimate which is helpful in computing the bias and standard error of one estimate, or in comparing the mean absolute error of two estimators. The Monte Carlo method is useful in determining the probability of coverage of an interval procedure. A Markov chain Monte Carlo algorithm is a general method for simulating from an arbitrary probability distribution. Metropolis-Hastings and Gibbs sampling are introduced as general methods for simulating from distributions. A variety of plots in R graphics and ggplot graphics are used to illustrate the concepts.