  In this work we tackle the problem of estimating the density $ f_X $ of a
random variable $ X $ by successive smoothing, such that the smoothed random
variable $ Y $ fulfills the diffusion partial differential equation $
(\partial_t - \Delta_1)f_Y(\,\cdot\,, t) = 0 $ with initial condition $
f_Y(\,\cdot\,, 0) = f_X $. We propose a product-of-experts-type model utilizing
Gaussian mixture experts and study configurations that admit an analytic
expression for $ f_Y (\,\cdot\,, t) $. In particular, with a focus on image
processing, we derive conditions for models acting on filter-, wavelet-, and
shearlet responses. Our construction naturally allows the model to be trained
simultaneously over the entire diffusion horizon using empirical Bayes. We show
numerical results for image denoising where our models are competitive while
being tractable, interpretable, and having only a small number of learnable
parameters. As a byproduct, our models can be used for reliable noise level
estimation, allowing blind denoising of images corrupted by heteroscedastic
noise.
