Definitions

A Markov chain is a series where the realisation of the next element, Y, in the series, (X,Y,...), is dependent only on the current state, X, and occurs with probability, P(Y|X).

• Markov chains: even number series, particle trajectories in diffuse media
• Not Markov chains: Fibonacci sequences, the formation of a polymer

Markov chain Monte Carlo (McMC) computes a series of estimates of a quantity, x, from a known posterior distribution $\pi$.

Samples are taken instead from a more convenient distribution q with similarities to $\pi$.

The transition probability, Q(Y|X), is the prior estimate of P(Y|X) irrespective of $\pi$.

As the chain progresses, the contribution from the most recent, $n^{\mbox{th}}$, iteration, is added to previously estimates which form a running average of increasing precision which improves at a rate 1/$\sqrt n$.

The use of a Markov chain as a means to compute parameters associated with an average is the focus of this work.