Chain Statistics

Abstract

Chains as introduced in the previous chapter are stochastic processes, because selection and acceptance functions assign probabilities to state pairs. Probability theory is therefore another basic discipline needed for deriving and controlling the properties of chains. In many of these manipulations a key role is played by the equilibrium density function. We want to derive properties of this function from the properties of selection and acceptance function, and we want to translate requirements on the equilibrium density function into constraints on the selection and acceptance function. These constraints do not determine the equilibrium density function completely. The score function also has a major influence, and is derived from the instance of the optimization problem at hand. In typical applications the number of states is too large to determine the equilibrium density function exactly. We therefore have to estimate the properties of this function by observing state and score frequencies. This falls in the domain of statistics.