ON BAYESIAN CONSISTENCY FOR FLOWS OBSERVED THROUGH A PASSIVE SCALAR

• 成果类型: Article
• 署名作者: Borggaard, Jeff; Glatt-Holtz, Nathan; Krometis, Justin
• 署名单位: Virginia Polytechnic Institute & State University; Tulane University; Virginia Polytechnic Institute & State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1542
• 发表日期: 2020
• 页码: 1762-1783
• 关键词: posterior contraction rates; DIFFUSIONS; inference

摘要

We consider the statistical inverse problem of estimating a background fluid flow field v from the partial, noisy observations of the concentration theta of a substance passively advected by the fluid, so that theta is governed by the partial differential equation partial derivative/partial derivative(t )theta (t, x) = - v(x) . del theta(t, x) + kappa Delta theta(t, x), theta(0, x) =theta(0)(x) for t is an element of [0, T], T > 0 and x is an element of T-2 = [0, 1](2). The initial condition theta(0) and diffusion coefficient kappa are assumed to be known and the data consist of point observations of the scalar field theta corrupted by additive, i.i.d. Gaussian noise. We adopt a Bayesian approach to this estimation problem and establish that the inference is consistent, that is, that the posterior measure identifies the true background flow as the number of scalar observations grows large. Since the inverse map is ill-defined for some classes of problems even for perfect, infinite measurements of theta, multiple experiments (initial conditions) are required to resolve the true fluid flow. Under this assumption, suitable conditions on the observation points, and given support and tail conditions on the prior measure, we show that the posterior measure converges to a Dirac measure centered on the true flow as the number of observations goes to infinity.