Bayesian Inference in Nonparametric Logistic Regression
Raghavan, Nandini
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https://hdl.handle.net/2142/72584
Description
Title
Bayesian Inference in Nonparametric Logistic Regression
Author(s)
Raghavan, Nandini
Issue Date
1993
Doctoral Committee Chair(s)
Cox, Dennis D.
Department of Study
Statistics
Discipline
Statistics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Statistics
Abstract
We consider the problem of regressing a dichotomous response variable on a predictor variable. Our interest is in modelling the probability of occurrence of the response as a function of the predictor variable, and in inferences about the estimated function. The log-odds (logit) of the probability is estimated nonparametrically, using generalized smoothing splines.
For purposes of inference, a partially improper stochastic process prior, proposed by Wahba (1978), is specified on the logit. The posterior is rather complicated and a number of questions need to be addressed.
We study the properties of the posterior distribution and give necessary and sufficient conditions under which it is a proper probability measure. These conditions are shown to be equivalent to those for the existence of the posterior mode (which is the smoothing spline estimator) and of the m.l.e. defined on the subspace of prior impropriety or the null space of the prior precision. A simple test for the case of polynomial regression in 1-dimension is also derived.
A Gaussian approximation to the posterior has been proposed by other authors. Our results on the tail behavior of the posterior suggest there are problems with this, but a more complete answer requires finite sample calculations, a computationally imposing task. We use Monte Carlo importance sampling for analysis of the posterior. Some of the posterior quantities that are estimated using this approach are posterior moments and pointwise and simultaneous posterior credibility bands. A comparison of these quantities with those based on the Gaussian approximation provides an assessment of how well the Gaussian approximation works for finite-sample sizes. The frequentist properties of the inferences based on the Bayesian model can also be investigated using this approach.
This Bayesian model with a partially improper prior is mathematically equivalent to the more classical generalized random effects models that have been used. The implications of these results for the logistic regression model with random effects are discussed.
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