Abstract：

Post-data statistical inference concerns making probability statements about model parameters conditional on observed data. When a priori knowledge about parameters is available, post-data inference can be conveniently made from Bayesian posteriors. In the absence of prior information, we may still rely on objective Bayes or generalized fiducial inference (GFI). Inspired by approximate Bayesian computation, we propose a novel characterization of post-data inference with the aid of differential geometry. Under suitable smoothness conditions, we establish that Bayesian posteriors and generalized fiducial distributions (GFDs) can be respectively characterized by absolutely continuous distributions supported on the same differentiable manifold: The manifold is uniquely determined by the observed data and the data generating equation of the fitted model. Our geometric analysis not only sheds light on the connection and distinction between Bayesian inference and GFI, but also allows us to sample from posteriors and GFDs using manifold Markov chain Monte Carlo algorithms. A repeated-measures analysis of variance example is presented to illustrate the sampling procedure.

 

Short Bio:

Yang Liu is currently an Associate Professor in the Quantitative Methodology: Measurement and Statistics (QMMS) program of the Department of Human Development and Quantitative Methodology (HDQM) at University of Maryland, College Park. His research focuses on the development of statistical methods for analyzing item response data, as well as applications of measurement models to psychological, educational, and health-related research. Yang Liu received his M.S. in Statistics in 2014 and Ph.D. in Quantitative Psychology in 2015 from the University of North Carolina at Chapel Hill. He served as an Associate Editor for Psychometrika between 2020 and 2023, and is currently an Associate Editor for the Journal of Educational Measurement and a Consulting Editor for the British Journal of Mathematical and Statistical Psychology.