A series of Markov chain transition functions are applied to define the conditional probability distribution. By chaining the importance weights for the jumps between the distributions throughout the sampling process, we can derive the importance weight, and estimate the ratio of partition functions:

$w^{(k)}=\frac{\tilde{p}_{\eta_{1}}\left(\boldsymbol{x}_{\eta_{1}}^{(k)}\right)}{\tilde{p}_{0}\left(\boldsymbol{x}_{\eta_{1}}^{(k)}\right)} \frac{\tilde{p}_{\eta_{2}}\left(\boldsymbol{x}_{\eta_{2}}^{(k)}\right)}{\tilde{p}_{\eta_{1}}\left(\boldsymbol{x}_{\eta_{2}}^{(k)}\right)} \cdots \frac{\tilde{p}_{1}\left(\boldsymbol{x}_{1}^{(k)}\right)}{\tilde{p}_{\eta_{n-1}}\left(\boldsymbol{x}_{\eta_{n}}^{(k)}\right)}$

$\frac{Z_{1}}{Z_{0}} \approx \frac{1}{K} \sum_{k=1}^{K} w^{(k)}$