Moment conditions are widely used to identify parameters in models where the full likelihood is either unknown or intentionally left unspecified. Empirical likelihood methods address this problem by assigning probability weights to the observed data so that the sample moment conditions hold exactly. Building on this idea, we propose a nonparametric Bayesian framework based on exponentially tilted empirical likelihood. This Bayesian formulation is particularly appealing in settings where prior information is more naturally specified on the observables rather than on the underlying parameters. Such settings arise in the presence of auxiliary data sources or synthetic data generated by modern generative AI models.Inference proceeds by projecting posterior draws from a Dirichlet process onto the moment-restricted model, yielding a computationally efficient procedure that is naturally amenable to parallelization. We establish new Bernstein--von Mises and consistency theorems for the resulting projection posterior under both vanishing-prior and persistent-prior regimes. In an application to return prediction using overnight news headlines, we show that AI-generated auxiliary data can provide a useful source of indirect regularization when informative priors on the parameter itself are unavailable.

翻译：矩条件广泛应用于识别模型中的参数，当完整似然函数未知或有意不予设定时尤为如此。经验似然方法通过为观测数据分配概率权重，使得样本矩条件精确成立，从而解决该问题。基于这一思想，我们提出一个以指数倾斜经验似然为基础的非参数贝叶斯框架。当先验信息更适合设定在可观测变量而非潜在参数上时，该贝叶斯公式尤为具有吸引力。此类情形常见于存在辅助数据源或由现代生成式AI模型产生的合成数据时。推论过程通过将狄利克雷过程的后验样本投影到矩约束模型上实现，从而得到一种计算高效且天然适合并行化的方法。我们在先验消失和先验持续两种机制下，为由此产生的投影后验建立了新的伯恩斯坦-冯米塞斯定理与一致性定理。在一项利用隔夜新闻标题进行收益预测的应用中，我们展示了当参数本身缺乏有效先验信息时，AI生成的辅助数据可提供一种有益的间接正则化来源。