Bayes estimators are well known to provide a means to incorporate prior knowledge that can be expressed in terms of a single prior distribution. However, when this knowledge is too vague to express with a single prior, an alternative approach is needed. Gamma-minimax estimators provide such an approach. These estimators minimize the worst-case Bayes risk over a set $\Gamma$ of prior distributions that are compatible with the available knowledge. Traditionally, Gamma-minimaxity is defined for parametric models. In this work, we define Gamma-minimax estimators for general models and propose adversarial meta-learning algorithms to compute them when the set of prior distributions is constrained by generalized moments. Accompanying convergence guarantees are also provided. We also introduce a neural network class that provides a rich, but finite-dimensional, class of estimators from which a Gamma-minimax estimator can be selected. We illustrate our method in two settings, namely entropy estimation and a prediction problem that arises in biodiversity studies.

翻译：贝叶斯估计器以能够整合可用单一先验分布表示的先验知识而著称。然而，当这种知识过于模糊而无法用单一先验表达时，需要采用替代方法。Gamma-最小最大估计器提供了这样一种途径。这类估计器在相容于可用知识的先验分布集合Γ上最小化最坏情况下的贝叶斯风险。传统上，Gamma-最小最大性针对参数模型定义。本研究将Gamma-最小最大估计器推广至一般模型，并在先验分布集合受广义矩约束的情形下，提出对抗元学习算法以计算此类估计器。同时提供了相应的收敛性保证。我们还引入了一类神经网络结构，该结构提供了丰富但有限维度的估计器类，可从中选取Gamma-最小最大估计器。我们通过熵估计和生物多样性研究中的预测问题两个实例展示了所提方法。