Markov chain

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A Markov chain is a Markov process with a discrete time parameter [1]. The Markov chain is a useful way to model systems with no long-term memory of previous states. That is, the state of the system at time is solely a function of the state , and not of any previous states [2].

A Formal Model

The influence of the values of on the distribution of can be formally modelled as:

Eq. 1

In this model, is any desired subset of the series . These indexes commonly represent the time component, and the range of is the Markov chain's state space [1].

Probability Density

The Markov chain can also be specified using a series of probabilities. If the initial probability of the state is , then the transition probability for state occurring at time can be expressed as:

Eq. 2

In words, this states that the probability of the system entering state at time is a function of the summed products of the initial probability density and the probability of state given state [2].

Invariant Distributions

In many cases, the density will approach a limit that is uniquely determined by (and not ). This limiting distribution is referred to as the invariant (or stationary) distribution over the states of the Markov chain. When such a distribution is reached, it persists forever[2].


  1. 1.0 1.1 Neal, R.M. (1993) Probabilistic Inference using Markov Chain Monte Carlo Methods. Technical Report TR-931. Department of Computer Science, University of Toronto
  2. 2.0 2.1 2.2 Peter M. Lee (2004) Bayesian Statistics: An Introduction. New York: Hodder Arnold. 368 p.