Markov chain Monte Carlo sampling of a normal distribution from an exponential distribution

The following is a simple example to show two important properties of a Markov chain Monte Carlo (MCMC) sampler and to illustrate the basic functionality of the method and issues relating to it’s usage. A Markov chain is a series where the realisation of the next element in the series, Y , is dependent only on the current state, X, and occurs with probability, P (Y |X). So the even number series forms a Markov chain but the Fibonacci sequence does not. A particle trajectory is a Markov chain but the formation of a polymer is not. MCMC is a technique for sampling from space that might be difficult to sample from because it does not have an analytical form or it may be difficult to integrate, in which case a normalising constant and a probability density function (pdf) cannot be found. The elements might be a numerical result from Monte Carlo sampling within a multivariate system, which can not be sampled directly, but are generated from other sampled variables. MCMC enables independent sampling of a target distribution, f , by using samples from a integrable function, f ∗ in a similar fashion to importance sampling. The first important property of MCMC samplers is that the weighting function need not be know in advance but is effectively generated by a stochastic process. Samples are taken from the distribution f ∗ and accepted or rejected depending on the relative probabilities of the new state compared to the old state in the target distribution, f and the relative probabilities of selection in the integrable distribution f. Consequently, the second important property of MCMC samplers is that the probability of a state need never be known to better than a normalising constant. Hastings [1] introduces a general form for an MCMC sampler that is stationary which requires that given a sample from the target distribution, f , subsequent samples will remain in f . The probability of an arbitrary transition is given by P (Y |X) = Q(Y |X)α(X,Y ) where,