  We introduce a theoretical framework for sampling from unnormalized densities
based on a smoothing scheme that uses an isotropic Gaussian kernel with a
single fixed noise scale. We prove one can decompose sampling from a density
(minimal assumptions made on the density) into a sequence of sampling from
log-concave conditional densities via accumulation of noisy measurements with
equal noise levels. Our construction is unique in that it keeps track of a
history of samples, making it non-Markovian as a whole, but it is lightweight
algorithmically as the history only shows up in the form of a running empirical
mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi
& Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of
(log-concave) Markov chains. The "jump" from the accumulated measurements is
obtained by empirical Bayes. We study our sampling algorithm quantitatively
using the 2-Wasserstein metric and compare it with various Langevin MCMC
algorithms. We also report a remarkable capacity of our algorithm to "tunnel"
between modes of a distribution.
