We analyze the posterior contraction rates of parameters in Bayesian models
via the Langevin diffusion process, in particular by controlling moments of the
stochastic process and taking limits. Analogous to the non-asymptotic analysis
of statistical M-estimators and stochastic optimization algorithms, our
contraction rates depend on the structure of the population log-likelihood
function, and stochastic perturbation bounds between the population and sample
log-likelihood functions. Convergence rates are determined by a non-linear
equation that relates the population-level structure to stochastic perturbation
terms, along with a term characterizing the diffusive behavior. Based on this
technique, we also prove non-asymptotic versions of a Bernstein-von-Mises
guarantee for the posterior. We illustrate this general theory by deriving
posterior convergence rates for various concrete examples, as well as
approximate posterior distributions computed using Langevin sampling
procedures.

通过控制随机过程的矩和取极限，我们通过 Langevin 扩散过程分析贝叶斯模型中参数的后验收缩速率，并特别关注了结构和随机摄动边界对于结果的影响，此外，基于该技术，我们也证明了后验的 Bernstein-von-Mises 保证的非渐近版本。