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Especially we appreciate your suggestion of underdamped Langevin for further consideration.

For MCMC methods like Metropolis-Hastings, tuning the proposal distribution is important in practice for effective sampling from the target distribution \pi. In this paper, we present Sample Adaptive MCMC (SA-MCMC), a MCMC method based on a reversible Markov chain for \pi^{\otimes N} that uses an adaptive proposal distribution based on the current state of N points and a sequential substitution procedure with one new likelihood evaluation per iteration and at most one updated point each iteration. The SA-MCMC proposal distribution automatically adapts within its parametric family to best approximate the target distribution, so in contrast to many existing MCMC methods, SA-MCMC does not require any tuning of the proposal distribution. Instead, SA-MCMC only requires specifying the initial state of N points, which can often be chosen a priori, thereby automating the entire sampling procedure with no tuning required. Experimental results demonstrate the fast adaptation and effective sampling of SA-MCMC.

We will incorporate the answers/other feedback into the revised manuscript. The general quality of samples seems to be negatively impacted. We agree other wavelets could be potentially interesting. SR is not claimed as our primary goal/contribution. Rather, it is a fortuitous byproduct of the conditional structure that WF enables. A more thorough exploration of WF for SR is a promising direction for future work.

Generative artificial intelligence (AI) has made unprecedented advances in vision language models over the past two years. During the generative process, new samples (images) are generated from an unknown high-dimensional distribution. Markov Chain Monte Carlo (MCMC) methods are particularly effective in drawing samples from such complex, high-dimensional distributions. This makes MCMC methods an integral component for models like EBMs, ensuring accurate sample generation. Gradient-based optimization is at the core of modern generative models. The update step during the optimization forms a Markov chain where the new update depends only on the current state. This allows exploration of the parameter space in a memoryless manner, thus combining the benefits of gradient-based optimization and MCMC sampling. MCMC methods have shown an equally important role in physically based rendering where complex light paths are otherwise quite challenging to sample from simple importance sampling techniques. A lot of research is dedicated towards bringing physical realism to samples (images) generated from diffusion-based generative models in a data-driven manner, however, a unified framework connecting these techniques is still missing. In this course, we take the first steps toward understanding each of these components and exploring how MCMC could potentially serve as a bridge, linking these closely related areas of research. Our course aims to provide necessary theoretical and practical tools to guide students, researchers and practitioners towards the common goal of generative physically based rendering. All Jupyter notebooks with demonstrations associated to this tutorial can be found on the project webpage: https://sinbag.github.io/mcmc/

We are very grateful to all the reviewers for their thoughtful feedback. All typos and minor points will also be fixed. Prop. 3 implies that any inference problem can be decomposed into a sequence of Another consideration, as highlighted by the example of 4.3, is that reducing the Bayesian computation, as the two methods have different computational cost patterns. This is required for each optimization step as well. Currently, however, we haven't found problems where the basis derived from H In the discussion after Prop. 1, we should have The phrase "lack of precision" in 4.4 refers to the finite number of samples drawn from

We will incorporate the answers/other feedback into the revised manuscript. The general quality of samples seems to be negatively impacted. We agree other wavelets could be potentially interesting. SR is not claimed as our primary goal/contribution. Rather, it is a fortuitous byproduct of the conditional structure that WF enables. A more thorough exploration of WF for SR is a promising direction for future work.

Early works include the Adaptive Direction Sampler by Gilks et al.