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后验分布的偏斜对称近似AI 解读

Abstract Popular deterministic approximations of posterior distributions from, e.g. the Laplace method, variational Bayes and expectation-propagation, generally rely on symmetric families, often taken to be Gaussian. This choice facilitates optimization and inference, but typically affects the quality of the approximation. In fact, even in basic parametric models, posterior distributions often display asymmetries that yield bias and reduced accuracy when considering…

Vecchia 高斯过程的概率与统计性质AI 解读

Gaussian Processes (GPs) are widely used to model dependencies in spatial statistics and machine learning. However, exact inference is computationally intractable for GP regression, with a time complexity of O(n3). The Vecchia approximation scales up computation by introducing sparsity into the spatial dependency structure, represented by a directed acyclic graph (DAG). Despite its practical popularity, this approach lacks rigorous theoretical foundations, and the choice of…

利用空间分布式高斯过程实现自适应AI 解读

We consider the accuracy of an approximate posterior distribution in nonparametric regression problems by combining posterior distributions computed on subsets of the data defined by the locations of the independent variables. We show that this approximate posterior retains the rate of recovery of the full data posterior distribution, where the rate of recovery adapts to the smoothness of the true regression function. As particular examples we consider Gaussian process…

偏斜 Bernstein–von Mises 定理与偏斜模态近似AI 解读

Gaussian deterministic approximations are routinely employed in Bayesian statistics to ease inference when the target posterior is intractable. Although these approximations are justified, in asymptotic regimes, by Bernstein-von Mises type results, in practice the expected Gaussian behavior might poorly represent the actual shape of the target posterior, thus affecting approximation accuracy. Motivated by these considerations, we derive an improved class of closed-form and…

筛先验下贝叶斯可信集的渐近频率学覆盖性质AI 解读

We investigate the frequentist coverage properties of (certain) Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem (in context of sieve type of priors), we introduce an extra assumption on the functional parameters, the so-called “general polished tail” condition. We then show that under standard assumptions, both the…

最大边际似然经验贝叶斯后验的渐近行为AI 解读

We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First, we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood…