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.

翻译：Van der Vaart(1991年)从关于存在非零渔业信息的半参数统计中得出的关于非零渔业信息的半参数统计结果,在无限的、非线性高斯回归设置中加以审查,只有在回归图的线性化符合一定范围条件的情况下,才有可能就未知参数的各个方面作出信息理论上的最佳推断,这表明这一范围条件在通常研究的椭圆反问题中可能失败,而分裂形式方程式则存在这种问题,在这种情况下无法有效地估计导电参数的一大批光线性功能。特别是,Gausian `Bernstein von Mises'-Bayesian 远地点分布的近似值在这一环境中并不存在。