非线性统计反问题的信息论问题
On some information-theoretic aspects of non-linear statistical inverse problems
摘要：
在无限维非线性高斯回归中，回顾了van der Vaart（1991）关于非零Fisher信息存在性的半参数统计结果。当且仅当回归映射的线性化的伴随满足一定的范围条件时，关于未知参数的信息理论上的最优推断是可能的。结果表明，这种范围条件可能会失败，在一个共同研究的椭圆型反问题的散度型方程，并在这种情况下，一个大类的光滑线性泛函的电导率参数不能有效地估计。特别是，高斯'伯恩斯坦冯米塞斯'型近似贝叶斯后验分布不持有在此设置。
英文摘要：
Results by van der Vaart (1991) from semi-parametric statistics about the existence of a non-zero Fisher information are reviewed in an infinite-dimensional non-linear Gaussian regression setting. Information-theoretically optimal inference on aspects of the unknown parameter is possible if and only if the adjoint of the linearisation of the regression map satisfies a certain range condition. It is shown that this range condition may fail in a commonly studied elliptic inverse problem with a divergence form equation, and that a large class of smooth linear functionals of the conductivity parameter cannot be estimated efficiently in this case. In particular, Gaussian `Bernstein von Mises'-type approximations for Bayesian posterior distributions do not hold in this setting.
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