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：贝叶斯估计器因其能够融合用单一先验分布表达的先验知识而闻名。然而，当这些知识过于模糊而无法用单一先验描述时，就需要采用替代方法。Gamma-极小极大估计器为此提供了解决方案。这类估计器在与可用知识兼容的先验分布集$\Gamma$上，最小化最坏情况下的贝叶斯风险。传统上，Gamma-极小极大性针对参数模型定义。本文中，我们为一般模型定义了Gamma-极小极大估计器，并提出了对抗元学习算法来计算这些估计器（当先验分布集受广义矩约束时）。同时给出了相应的收敛性保证。我们还引入了一类神经网络，它提供丰富但有限维的估计器类，可从中选择Gamma-极小极大估计器。我们在两个场景中验证了该方法：熵估计以及生物多样性研究中出现的预测问题。