Markov chain Monte Carlo for computing averages

In particle transport problems expectations of useful quantities, E(), are computed by forming averages, in which case the stochastic process is a random walk.

The law of large numbers states that the variation in the estimate approaches the normal distribution so a model for the probability of an MC generated result is the chi-square difference between the computed result and the measured result i.e. $\chi^2=(E[x]-\overline{\pi})^2$.

Likewise the variance of the MC calculation is given as $\sigma^2=E[x^2]-E[x]^2$.

Subsequently, $\pi(Y)\propto e^{-\chi^2/\sigma^2}/\sigma.$

In the limit where the normal distribution becomes valid the transition probabilities approaches a symmetric relation Q(X|Y)=Q(|Y-X|) and cancel in equation 1.

Further, in the limit of small differences from the mean, a normal distribution can be approximated by a square, Dirichlet, function, i.e. $\Delta(t)=1$ where |t|<1, otherwise 0.

The acceptance probability is,

$$\alpha \min\left[1,\frac{\Delta(\chi(Y)/\sigma(Y))}{\Delta(\chi(X)/\sigma(X)}\right]$$ =

$$\Delta(\chi(Y)/\sigma(Y)),$$ = (3)

hence a step is accepted if $\chi^2<\sigma^2$.

This is a Markov chain that converges to $\overline\pi$ and is stationary as previously shown.