Saha Abhijoy, Bharath Karthik, Kurtek Sebastian
Department of Statistics, The Ohio State University.
School of Mathematical Sciences, University of Nottingham.
J Am Stat Assoc. 2020;115(530):822-835. doi: 10.1080/01621459.2019.1585253. Epub 2019 Apr 30.
We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold of probability density functions. Under the square-root density representation, the manifold can be identified with the positive orthant of the unit hypersphere in , and the Fisher-Rao metric reduces to the standard metric. Exploiting such a Riemannian structure, we formulate the task of approximating the posterior distribution as a variational problem on the hypersphere based on the -divergence. This provides a tighter lower bound on the marginal distribution when compared to, and a corresponding upper bound unavailable with, approaches based on the Kullback-Leibler divergence. We propose a novel gradient-based algorithm for the variational problem based on Fréchet derivative operators motivated by the geometry of , and examine its properties. Through simulations and real data applications, we demonstrate the utility of the proposed geometric framework and algorithm on several Bayesian models.
我们基于概率密度函数流形上的非参数Fisher-Rao度量，为贝叶斯模型中的变分推断提出了一种新颖的黎曼几何框架。在平方根密度表示下，该流形可与 中单位超球面的正卦限等同，且Fisher-Rao度量简化为标准 度量。利用这种黎曼结构，我们将逼近后验分布的任务表述为基于 -散度在超球面上的变分问题。与基于Kullback-Leibler散度的方法相比，这为边际分布提供了更紧的下界，以及基于Kullback-Leibler散度的方法所没有的相应上界。我们基于由 的几何结构激发的Fréchet导数算子，为变分问题提出了一种新颖的基于梯度的算法，并研究了其性质。通过模拟和实际数据应用，我们展示了所提出的几何框架和算法在几个贝叶斯模型上的效用。
