A popular choice for the intermediate distributions in Annealed Importance Sampling (AIS) is the weighted geometric average of the initial distribution $$p_0$$ and the target distribution $$p_1$$. The specific sequence of these intermediate distributions is a design choice tailored to the problem domain.

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University of Michigan - Ann Arbor

Annealed Importance Sampling (AIS) is a strategy of importance sampling applied to an extended state with a sequence of intermediate distributions ($$p_0$$ to $$p_1$$). AIS is the most common way of estimating the partition function for undirected probabilistic models, such as a trained Restricted Boltzmann Machine.

Annealed Importance Sampling

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

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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:

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

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

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