期刊
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
卷 115, 期 530, 页码 822-835出版社
TAYLOR & FRANCIS INC
DOI: 10.1080/01621459.2019.1585253
关键词
Bayesian density estimation; Bayesian logistic regression; Gradient ascent algorithm; Infinite-dimensional Riemannian optimization; Square-root density
资金
- NSF [DMS 1613054, CCF 1740761]
- NIH [R37 CA214955]
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 alpha-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 Frechet 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. for this article are available online.
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