  Stein variational gradient descent (SVGD) refers to a class of methods for
Bayesian inference based on interacting particle systems. In this paper, we
consider the originally proposed deterministic dynamics as well as a stochastic
variant, each of which represent one of the two main paradigms in Bayesian
computational statistics: variational inference and Markov chain Monte Carlo.
As it turns out, these are tightly linked through a correspondence between
gradient flow structures and large-deviation principles rooted in statistical
physics. To expose this relationship, we develop the cotangent space
construction for the Stein geometry, prove its basic properties, and determine
the large-deviation functional governing the many-particle limit for the
empirical measure. Moreover, we identify the Stein-Fisher information (or
kernelised Stein discrepancy) as its leading order contribution in the
long-time and many-particle regime in the sense of $\Gamma$-convergence,
shedding some light on the finite-particle properties of SVGD. Finally, we
establish a comparison principle between the Stein-Fisher information and
RKHS-norms that might be of independent interest.
