  In statistical applications, it is common to encounter parameters supported
on a varying or unknown dimensional space. Examples include the fused lasso
regression, the matrix recovery under an unknown low rank, etc. Despite the
ease of obtaining a point estimate via the optimization, it is much more
challenging to quantify their uncertainty -- in the Bayesian framework, a major
difficulty is that if assigning the prior associated with a $p$-dimensional
measure, then there is zero posterior probability on any lower-dimensional
subset with dimension $d<p$; to avoid this caveat, one needs to choose another
dimension-selection prior on $d$, which often involves a highly combinatorial
problem. To significantly reduce the modeling burden, we propose a new
generative process for the prior: starting from a continuous random variable
such as multivariate Gaussian, we transform it into a varying-dimensional space
using the proximal mapping.
  This leads to a large class of new Bayesian models that can directly exploit
the popular frequentist regularizations and their algorithms, such as the
nuclear norm penalty and the alternating direction method of multipliers, while
providing a principled and probabilistic uncertainty estimation.
  We show that this framework is well justified in the geometric measure
theory, and enjoys a convenient posterior computation via the standard
Hamiltonian Monte Carlo. We demonstrate its use in the analysis of the dynamic
flow network data.
