The question then is how can we model anything under such a severe lack of knowledge? The Metropolis algorithm comes to our rescue here. Since $W(j\rightarrow i)$ is unknown, we model it as the product of two probabilities, a probability for accepting the proposed move from the state $j$ to the state $j$, and a probability for making the transition to the state $i$ being in the state $j$. We label these probabilities $A(j\rightarrow i)$ and $T(j\rightarrow i)$, respectively. Our total transition probability is then $$\begin{equation*} W(j\rightarrow i)=T(j\rightarrow i)A(j\rightarrow i). \end{equation*}$$ The algorithm can then be expressed as

• We make a suggested move to the new state $i$ with some transition or moving probability $T_{j\rightarrow i}$.
• We accept this move to the new state with an acceptance probability $A_{j \rightarrow i}$. The new state $i$ is in turn used as our new starting point for the next move. We reject this proposed moved with a $1-A_{j\rightarrow i}$ and the original state $j$ is used again as a sample.