|Title: Classical Linear Mixed Model Estimation with Dirichlet Process Random Effects.
Chen Li (Feb. 14)
The Dirichlet process has been used extensively in Bayesian nonparametric
modeling, and has proven to be very useful. Here we discuss the linear
mixed model with Dirichlet process random
e ects from a classical view, and derive the best linear unbiased
estimator (BLUE) of the fixed e ffects. We also characterize the
relationship between the BLUE and the OLS, and show how confidence
intervals can be approximated. At Last, show some simulations studies and
|Title: Honest Exploration of Intractable Probability Distributions Via Markov
Chain Monte Carlo.
Professor James Hobert (Feb. 7)
Two important questions that must be answered whenever a Markov chain
Monte Carlo algorithm is used are (Q1) What is an appropriate burn-in?
and (Q2) How long should the sampling continue after burn-in? One
method of developing rigorous answers to these questions involves
establishing drift and minorization conditions, which together imply
that the underlying Markov chain is geometrically ergodic. In this
talk, I will explain what drift and minorization are as well as how and
why these conditions can be used to form rigorous answers to (Q1) and
|On the Strong Law of Large Numbers for Weighted Sums of Random Elements in Banach Spaces
Yuan Liao (Mar. 13)
Strong Law of Large Numbers for Weighted Sums of Random Elements in Banach
Spaces are obtained for the following two broad cases; the results are new
even when the underlying Banach space is the real line.
(i) The random elements are independent. The underlying Banach space is
assumed to satisfy the geometric condition that it is of Rademacher type p
(p \in[1,2]). Special cases include results of Woyczy\'nski (1980),
Teicher (1985), Adler, Rosalsky, and Taylor (1989), and Sung (1997).
(2) Conditions are provided under which a general SLLN is obtained
irrespective of the joint distributions of the random elements. No
geometric conditions are imposed on the underlying Banach space. The
results are general enough to include as special cases results of Petrov
(1973), Teicher (1985), Sung (1997), and Rosalsky and Stoica (2010).
|Good Confidence Intervals for Categorical Data Analyses
Professor Alan Agresti, University of Florida (Emeritus) (Mar. 20)
This talk surveys confidence intervals that perform well for
estimating parameters used in categorical data analysis. Considerable
research has now shown that intervals resulting from inverting score
tests perform much better than inverting Wald tests and usually better
than inverting likelihood-ratio tests. For small samples, `exact'
methods are conservative inferentially, but inverting a score test
using the mid-P value provides a sensible compromise. Finally, we
briefly review an approach for proportions and their differences that
approximates the score intervals and is much better than the ordinary
Wald intervals by adding pseudo data before forming the Wald intervals.
|Problems in High-Dimensional Bayesian Regression: Posterior Inconsistency of g-Priors
Doug Sparks (Mar. 27)
Like most classical methods, linear regression lends itself easily to a Bayesian formulation. A variety of useful Bayesian techniques have evolved to meet specific needs, and many of these techniques provide advantages over their frequentist counterparts. However, analysis of the frequentist properties of Bayesian regression models sometimes reveals subtle but serious problems, especially in settings where the number of covariates p grows with the sample size n. In particular, Zellner's g-prior and its empirical and hierarchical extensions are a commonly used family of models for which the posterior can become inconsistent in surprising circumstances. These results will be provided and discussed, but broader issues with Bayesian regression models will also be addressed throughout the talk.