标题： 后验分布在线性反问题中的相容性 显示英文标题

标题： Consistency of the posterior distribution in generalised linear inverse problems

摘要： 对于不适定的反问题，正则化解可以在贝叶斯框架下被解释为后验分布的一个众数。 这个框架丰富了解的集合，因为可以使用其他的后验估计作为反问题的解，例如后验均值，在实践中可能更容易计算。 本文证明了一类有限维设定下的不适定线性反问题贝叶斯解在Ky Fan度量下的相容性，该结果适用于一类广义似然函数和先验分布。 这一结果可以通过让未知参数的维度趋于无穷来应用于无限维问题的研究，这可以看作是在网格上的离散化或者无限维问题的谱逼近。 假设似然函数和先验分布具有指数形式（包括指数族分布），并且可微。 观测值可以是相关的。 不需要假定观测值有有限矩，例如期望或方差，因此允许非正则的似然函数，并允许非共轭和非适当先验。 如果方差存在，它可能是异方差的，即可能依赖于未知函数。 当我们将这一结果应用于谱逼近框架时，我们观察到一个相当令人惊讶的现象：可以达到参数收敛率，也就是说问题变得自正则化了。 我们还考虑了未知参数位于参数集边界上的特殊情况，并证明在这种情况下收敛速度比内部点参数更快。 显示英文摘要

摘要： For ill-posed inverse problems, a regularised solution can be interpreted as a mode of the posterior distribution in a Bayesian framework. This framework enriches the set the solutions, as other posterior estimates can be used as a solution to the inverse problem, such as the posterior mean that can be easier to compute in practice. In this paper we prove consistency of Bayesian solutions of an ill-posed linear inverse problem in the Ky Fan metric for a general class of likelihoods and prior distributions in a finite dimensional setting. This result can be applied to study infinite dimensional problems by letting the dimension of the unknown parameter grow to infinity which can be viewed as discretisation on a grid or spectral approximation of an infinite dimensional problem. Likelihood and the prior distribution are assumed to be in an exponential form that includes distributions from the exponential family, and to be differentiable. The observations can be dependent. No assumption of finite moments of observations, such as expected value or the variance, is necessary thus allowing for possibly non-regular likelihoods, and allowing for non-conjugate and improper priors. If the variance exists, it may be heteroscedastic, namely, it may depend on the unknown function. We observe quite a surprising phenomenon when applying our result to the spectral approximation framework where it is possible to achieve the parametric rate of convergence, i.e the problem becomes self-regularised. We also consider a particular case of the unknown parameter being on the boundary of the parameter set, and show that the rate of convergence in this case is faster than for an interior point parameter.