In recent years, the shortcomings of Bayes posteriors as inferential devices has received increased attention. A popular strategy for fixing them has been to instead target a Gibbs measure based on losses that connect a parameter of interest to observed data. While existing theory for such inference procedures relies on these losses to be analytically available, in many situations these losses must be stochastically estimated using pseudo-observations. The current paper fills this research gap, and derives the first asymptotic theory for Gibbs measures based on estimated losses. Our findings reveal that the number of pseudo-observations required to accurately approximate the exact Gibbs measure depends on the rates at which the bias and variance of the estimated loss converge to zero. These results are particularly consequential for the emerging field of generalised Bayesian inference, for estimated intractable likelihoods, and for biased pseudo-marginal approaches. We apply our results to three Gibbs measures that have been proposed to deal with intractable likelihoods and model misspecification.

翻译：近年来，贝叶斯后验作为推断工具存在的缺陷受到越来越多关注。一种流行的改进策略是转向基于连接目标参数与观测数据的损失函数的吉布斯测度。尽管现有关于此类推断过程的理论要求这些损失函数具有解析形式，但在许多实际场景中必须使用伪观测值进行随机估计。本文填补了这一研究空白，首次推导了基于估计损失函数的吉布斯测度的渐近理论。我们的发现表明，精确逼近真实吉布斯测度所需的伪观测值数量取决于估计损失偏差和方差的收敛速率。这些结果对广义贝叶斯推断、不可解析似然函数估计及有偏伪边际方法等新兴领域具有重要影响。我们将所得结论应用于三个针对不可解析似然函数和模型误设问题提出的吉布斯测度。