Analysis of Financial Time Series, Third Edition

12.4 Alternative Algorithms

In many applications, there are no closed-form solutions for the conditional posterior distributions. But many clever alternative algorithms have been devised in the statistical literature to overcome this difficulty. In this section, we discuss some of these algorithms.

12.4.1 Metropolis Algorithm

This algorithm is applicable when the conditional posterior distribution is known except for a normalization constant; see Metropolis and Ulam (1949) and Metropolis et al. (1953). Suppose that we want to draw a random sample from the distribution , which contains a complicated normalization constant so that a direct draw is either too time-consuming or infeasible. But there exists an approximate distribution for which random draws are easily available. The Metropolis algorithm generates a sequence of random draws from the approximate distribution whose distributions converge to . The algorithm proceeds as follows:

1. Draw a random starting value θ0 such that .

2. For t = 1, 2, … ,

a. Draw a candidate sample θ* from a known distribution at iteration t given the previous draw θt−1. Denote the known distribution by , which is called a jumping distribution in Gelman ...