1,518 research outputs found
Stochastic Approximation and Modern Model-Based Designs for Dose-Finding Clinical Trials
In 1951 Robbins and Monro published the seminal article on stochastic
approximation and made a specific reference to its application to the
"estimation of a quantal using response, nonresponse data." Since the 1990s,
statistical methodology for dose-finding studies has grown into an active area
of research. The dose-finding problem is at its core a percentile estimation
problem and is in line with what the Robbins--Monro method sets out to solve.
In this light, it is quite surprising that the dose-finding literature has
developed rather independently of the older stochastic approximation
literature. The fact that stochastic approximation has seldom been used in
actual clinical studies stands in stark contrast with its constant application
in engineering and finance. In this article, I explore similarities and
differences between the dose-finding and the stochastic approximation
literatures. This review also sheds light on the present and future relevance
of stochastic approximation to dose-finding clinical trials. Such connections
will in turn steer dose-finding methodology on a rigorous course and extend its
ability to handle increasingly complex clinical situations.Comment: Published in at http://dx.doi.org/10.1214/10-STS334 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Recursive Parameter Estimation: Convergence
We consider estimation procedures which are recursive in the sense that each
successive estimator is obtained from the previous one by a simple adjustment.
We propose a wide class of recursive estimation procedures for the general
statistical model and study convergence.Comment: 25 pages with 1 postscript figur
Truncated Stochastic Approximation with Moving Bounds: Convergence
In this paper we propose a wide class of truncated stochastic approximation
procedures with moving random bounds. While we believe that the proposed class
of procedures will find its way to a wider range of applications, the main
motivation is to accommodate applications to parametric statistical estimation
theory. Our class of stochastic approximation procedures has three main
characteristics: truncations with random moving bounds, a matrix valued random
step-size sequence, and dynamically changing random regression function. We
establish convergence and consider several examples to illustrate the results
Design Issues for Generalized Linear Models: A Review
Generalized linear models (GLMs) have been used quite effectively in the
modeling of a mean response under nonstandard conditions, where discrete as
well as continuous data distributions can be accommodated. The choice of design
for a GLM is a very important task in the development and building of an
adequate model. However, one major problem that handicaps the construction of a
GLM design is its dependence on the unknown parameters of the fitted model.
Several approaches have been proposed in the past 25 years to solve this
problem. These approaches, however, have provided only partial solutions that
apply in only some special cases, and the problem, in general, remains largely
unresolved. The purpose of this article is to focus attention on the
aforementioned dependence problem. We provide a survey of various existing
techniques dealing with the dependence problem. This survey includes
discussions concerning locally optimal designs, sequential designs, Bayesian
designs and the quantile dispersion graph approach for comparing designs for
GLMs.Comment: Published at http://dx.doi.org/10.1214/088342306000000105 in the
Statistical Science (http://www.imstat.org/sts/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Conditionally conjugate mean-field variational Bayes for logistic models
Variational Bayes (VB) is a common strategy for approximate Bayesian
inference, but simple methods are only available for specific classes of models
including, in particular, representations having conditionally conjugate
constructions within an exponential family. Models with logit components are an
apparently notable exception to this class, due to the absence of conjugacy
between the logistic likelihood and the Gaussian priors for the coefficients in
the linear predictor. To facilitate approximate inference within this widely
used class of models, Jaakkola and Jordan (2000) proposed a simple variational
approach which relies on a family of tangent quadratic lower bounds of logistic
log-likelihoods, thus restoring conjugacy between these approximate bounds and
the Gaussian priors. This strategy is still implemented successfully, but less
attempts have been made to formally understand the reasons underlying its
excellent performance. To cover this key gap, we provide a formal connection
between the above bound and a recent P\'olya-gamma data augmentation for
logistic regression. Such a result places the computational methods associated
with the aforementioned bounds within the framework of variational inference
for conditionally conjugate exponential family models, thereby allowing recent
advances for this class to be inherited also by the methods relying on Jaakkola
and Jordan (2000)
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