In the Bayes paradigm and for a given loss function, we propose the construction of a new type of posterior distributions, that extends the classical Bayes one, for estimating the law of an $n$-sample. The loss functions we have in mind are based on the total variation and Hellinger distances as well as some $\mathbb{L}_{j}$-ones. We prove that, with a probability close to one, this new posterior distribution concentrates its mass in a neighbourhood of the law of the data, for the chosen loss function, provided that this law belongs to the support of the prior or, at least, lies close enough to it. We therefore establish that the new posterior distribution enjoys some robustness properties with respect to a possible misspecification of the prior, or more precisely, its support. For the total variation and squared Hellinger losses, we also show that the posterior distribution keeps its concentration properties when the data are only independent, hence not necessarily i.i.d., provided that most of their marginals or the average of these are close enough to some probability distribution around which the prior puts enough mass. The posterior distribution is therefore also stable with respect to the equidistribution assumption. We illustrate these results by several applications. We consider the problems of estimating a location parameter or both the location and the scale of a density in a nonparametric framework. Finally, we also tackle the problem of estimating a density, with the squared Hellinger loss, in a high-dimensional parametric model under some sparsity conditions. The results established in this paper are non-asymptotic and provide, as much as possible, explicit constants.

翻译：在贝叶斯范式下，针对给定的损失函数，我们提出了一种新型后验分布的构建方法，该方法扩展了经典贝叶斯后验分布，用于估计$n$个样本的分布律。我们所考虑的损失函数基于全变差距离、Hellinger距离以及某些$\mathbb{L}_{j}$距离。我们证明，在接近1的概率下，当所选损失函数对应的数据分布律属于先验支撑集或其足够接近时，这种新的后验分布会将其质量集中在数据分布律的邻域内。因此，我们确立了该新型后验分布对先验（更确切地说是其支撑集）可能存在的误设具有稳健性。对于全变差损失和平方Hellinger损失，我们还证明，当数据仅独立（不必独立同分布）时，只要其大部分边缘分布或这些边缘分布的平均值足够接近先验赋予足够质量的某个概率分布，后验分布仍能保持其集中性。因此，该后验分布对等分布假设也具有稳定性。我们通过多个应用实例说明了这些结果：研究非参数框架下位置参数或位置与尺度联合估计问题；最后，在稀疏条件下处理高维参数模型中基于平方Hellinger损失的密度估计问题。本文所建立的结论均为非渐近的，并尽可能给出了显式常数。