论文标题
最佳运输框架中概率内核的Lipschitz连续性
Lipschitz continuity of probability kernels in the optimal transport framework
论文作者
论文摘要
在贝叶斯统计中，相对于可观察变量的后验分布的连续性特性至关重要，因为它表达了适当的性能，即相对于数据测量的误差的稳定性。本质上，这需要分析相对于条件变量的条件概率分布的概率内核的连续性。 在这里，我们为在最佳运输框架（例如Wasserstein Metric）中产生的度量结构的Lipschitz连续性提供了一般条件。对于有限维空间上的主导概率内核，我们显示了Lipschitz的连续性结果，Lipschitz常数在Fisher-Information功能和加权PoincaréStonstants方面享有明确的界限。我们还为具有移动支持，无限维空间和非主导内核的内核提供结果。我们显示了贝叶斯统计中几个问题的应用，例如通过混合物和后验一致性近似后验分布。
In Bayesian statistics, a continuity property of the posterior distribution with respect to the observable variable is crucial as it expresses well-posedness, i.e., stability with respect to errors in the measurement of data. Essentially, this requires to analyze the continuity of a probability kernel or, equivalently, of a conditional probability distribution with respect to the conditioning variable. Here, we give general conditions for the Lipschitz continuity of probability kernels with respect to metric structures arising within the optimal transport framework, such as the Wasserstein metric. For dominated probability kernels over finite-dimensional spaces, we show Lipschitz continuity results with a Lipschitz constant enjoying explicit bounds in terms of Fisher-information functionals and weighted Poincaré constants. We also provide results for kernels with moving support, for infinite-dimensional spaces and for non dominated kernels. We show applications to several problems in Bayesian statistics, such as approximation of posterior distributions by mixtures and posterior consistency.
