核心概念
The core message of this paper is to develop a Bayesian approach for testing hypotheses about specific features of the unknown quantity of interest in statistical inverse problems, using the maximum a posteriori (MAP) test.
摘要
The paper focuses on a Bayesian approach to testing hypotheses in statistical inverse problems. The authors consider a linear inverse problem with Gaussian noise and a Gaussian prior distribution on the unknown quantity of interest.
The key highlights and insights are:
The authors introduce the maximum a posteriori (MAP) test, which naturally arises from the posterior distribution and allows for inference about specific features of the unknown quantity.
They provide a frequentistic analysis of the MAP test's properties, such as its level and power. They show that without further a priori assumptions, it is impossible to derive a non-trivial bound for the size of the MAP test.
The authors characterize the MAP test as a regularized test in the sense of Kretschmann et al. (2024) and show that optimal detection properties are almost reachable by choosing the prior covariance appropriately.
Under classical spectral source conditions and Gaussian priors, the authors provide lower bounds for the power of the MAP test.
Numerical simulations illustrate the superior performance of the MAP test in both moderately and severely ill-posed situations.