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.