Brief Summary
The Monte Carlo approach, combined with the theory for Markov chains can be summarized as follows:
A Markov chain Monte Carlo method for the simulation of a distribution w is any method producing an
ergodic Markov chain of events x whose stationary distribution is w . The Metropolis algorithm can be phrased as
- Generate an initial value x^{(i)} .
- Generate a trial value y_t with probability T(y_t|x^{(i)}) . The latter quantity represents the probability of generating y_t given x^{(i)} .
- Take a new value
\begin{equation*}
x^{(i+1)}= \left\{\begin{array}{cc} y_t & \mathrm{with\hspace{0.1cm}probability} = A(x^{(i)}\rightarrow y_t) \\ x^{(i)} & \mathrm{with \hspace{0.1cm}probability} = 1-A(x^{(i)}\rightarrow y_t)\end{array}\right .
\end{equation*}
- We have defined the transition (acceptance) probability as
\begin{equation*} A(x\rightarrow y)= \mathrm{min}\left\{\frac{w(y)T(x|y)}{w(x)T(y|x)},1\right\}.
\end{equation*}