Penalized Least Squares and Frequentist and Bayesian Mixed-Effects Models Yolanda Muñoz Maldonado Division of Biostatistics, School of Public Health University of Texas Health Science Center at Houston Abstract
Penalized least squares and mixed effects models
using frequentist and Bayesian approaches are commonly used tools in a
wide range of settings like, for example, clustered data, smoothing
problems, ridge regression, and functional data analysis, and they have
a broad range of applications in areas like Biology, Environmental
Sciences, Survey Sampling, etc. However, this broad range of
applications has also made that some interesting relationships between
these three different methodologies rest unnoticed. In this talk, I
will use as platform, the well known connections between smoothing
splines estimators, a particular Gaussian mixed effects model, and a
Bayesian Gaussian stochastic process, to build a general framework that
encompasses Penalized Least Squares Techniques and Frequentist and
Bayesian Mixed Effects Models. This proposed framework has the
implication that, in many cases of practical interest, an efficient,
O(n), Kalman Filter algorithm can be used to obtain the desired
predictors and corresponding Bayesian confidence intervals. This
algorithm also permits the evaluation of the exact likelihood function
with the same level of computational efficiency. To illustrate the
range of applicability of our main results we use examples from three
different settings: varying coefficient models, ridge regression and
randomized block designs.
Keywords: Smoothing splines, ridge regression, Bayesian prediction, varying coefficients, confidence bands, Kalman filtering. |