The weighted geometric average of the target distribution $ p_1 $ is a popular choice for intermediate distributions of AIS.  This sequence is indeed a design choice based on problem domain.

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 partition function for undirected probabilistic models, such as a trained model of 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) 

Deep Learning

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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