ASYMPTOTIC NORMALITY AND VALID INFERENCE FOR GAUSSIAN VARIATIONAL APPROXIMATION

By Peter Hall, Tung H. Pham, Matt P. Wand, S. Wang. 2011

65 citations recorded by the source corpus.

We derive the precise asymptotic distributional behavior of Gaussian variational approximate estimators of the parameters in a single-predictor Poisson mixed model. These results are the deepest yet obtained concerning the statistical properties of a variational approximation method. Moreover, they give rise to asymptotically valid statistical inference. A simulation study demonstrates that Gaussian variational approximate confidence intervals possess good to excellent coverage properties, and have a similar precision to their exact likelihood counterparts. 1. Introduction. Variational approximation methods are enjoying an increasing amount of development and use in statistical problems. This raises questions regarding their statistical properties, such as consistency of point estimators and validity of statistical inference. We make significant inroads into answering such questions via thorough theoretical treatment of one of the simplest nontrivial settings for which variational approximation is beneficial: the Poisson mixed model with a single predictor variable and random intercept. We call this the simple Poisson mixed model. The model treated here is also treated in [7], but there attention is confined to bounds and rates of convergence. We improve upon their results by obtaining the asymptotic distributions of the estimators. The results reveal that the estimators are asymptotically normal, have negligible bias and that their variances decay at least as fast as m −1 ,w herem is the number of groups. For the slope parameter, the faster (mn) −1 rate is obtained, where n is the number of repeated measures. An important practical ramification of our theory is asymptotically valid statistical inference for the model parameters. In particular, a form of studentization leads to theoretically justifiable confidence intervals for all model parameters. Unlike those based on the exact likelihood, all Gaussian variational approximate point estimates and confidence intervals can be computed without the need for numer

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Variational Inference: A Review for Statisticians (2016, background): “Hall, Ormerod, and Wand (2011a) and Hall et al. (2011b) examined a simple Poisson mixed-effects model, one with a single predictor and a random intercept.”
Variational Approximations for Generalized Linear Latent Variable Models (2017, unknown): “…consistency of the VA estimators for GLLVMs, and substantial research remains to be done to broaden the results of Ormerod and Wand (2012) and Hall et al. (2011) to show that variational approximations in general produces estimators that are consistent and asymptotically normal, and what…”