The basic of the Metropolis Algorithm

• Sample a possible new state $j$ with some probability $T_{i\rightarrow j}$.
• Accept the new state $j$ with probability $A_{i \rightarrow j}$ and use it as the next sample.
• With probability $1-A_{i\rightarrow j}$ the move is rejected and the original state $i$ is used again as a sample.

We wish to derive the required properties of $T$ and $A$ such that ${\cal w}_i^{(n\rightarrow \infty)} \rightarrow p_i$ so that starting from any distribution, the method converges to the correct distribution. Note that the description here is for a discrete probability distribution. Replacing probabilities $p_i$ with expressions like $p(x_i)dx_i$ will take all of these over to the corresponding continuum expressions.

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