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High-dimensional Bayesian inference via the unadjusted Langevin algorithm
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Journal Article
High-dimensional Bayesian inference via the unadjusted Langevin algorithm
2019
Overview
We consider in this paper the problem of sampling a high-dimensional probability distribution π having a density w.r.t the Lebesgue measure on ℝd, known up to a normalization constant
x
↦
π
x
=
e
−
U
x
/
∫
ℝ
d
e
−
U
y
d
y
. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that U is continuously differentiable, ▽U is globally Lipschitz and U is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order 2 and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.
Publisher
International Statistical Institute (ISI),Bernoulli Society for Mathematical Statistics and Probability
Subject
