  Diffusion probabilistic models (DPMs) represent a class of powerful
generative models. Despite their success, the inference of DPMs is expensive
since it generally needs to iterate over thousands of timesteps. A key problem
in the inference is to estimate the variance in each timestep of the reverse
process. In this work, we present a surprising result that both the optimal
reverse variance and the corresponding optimal KL divergence of a DPM have
analytic forms w.r.t. its score function. Building upon it, we propose
Analytic-DPM, a training-free inference framework that estimates the analytic
forms of the variance and KL divergence using the Monte Carlo method and a
pretrained score-based model. Further, to correct the potential bias caused by
the score-based model, we derive both lower and upper bounds of the optimal
variance and clip the estimate for a better result. Empirically, our
analytic-DPM improves the log-likelihood of various DPMs, produces high-quality
samples, and meanwhile enjoys a 20x to 80x speed up.
