Bayesian Statistics
UCSC's Applied Math & Statistics (AMS) department is developing statistical methods for dealing with some of the thorniest problems facing modern society, such as how to evaluate the quality of hospitals and schools, and how to assess the risks of nuclear waste disposal. This includes the study of problems ranging from rainfall prediction to the interpretation of electrocardiogram readings from heart patients.
Bayesian ideas has stemmed from philosophical resistance to letting subjectivity play a role in the scientific process."
David Draper
Department Chair,
Applied Math & Statistics
As the Department of Applied Mathematics and Statistics grows, the emphasis in the statistics group is clearly Bayesian. The AMS department applies Bayesian statistical methods to approach these complexities. Within the past decade or so, the Bayesian approach has gone from being a controversial theory on the margins of mainstream statistics to being widely accepted as a valuable alternative to more common frequentist approach.
The Bayesian approach grew out of mid-18th century theorem by the Reverend Thomas Bayes. Bayes developed a mathematical formula for revising subjective probability in the light of new evidence. Bayes used this approach to draw inferences about future events based on the results of previous trials. Important contributions from other mathematicians allowed Bayes's ideas to blossom into a whole new approach to statistics.
“The two main areas in which the department aims to achieve excellence are Bayesian statistical methods and mathematical modeling of complex natural phenomena,” AMS Dept. Chair David Draper says. “The focus in both cases is on solving real-world problems in engineering and the sciences.”
“There are simple things that we can all agree on the probability of, but when you get into more complicated situations you discover there are always elements of judgment involved,” he says. “People have gradually and grudgingly come to understand that the objectivity we hoped to get from the frequentist approach is a myth, and what we should be doing instead is to be as clear as possible about what we assume and to show whether different assumptions all lead to the same outcome or not.”
The biggest stumbling block for Bayesian statistics, however, isn't subjectivity but the complexity of the math. At the heart of the Bayesian approach is a tremendously difficult mathematical task involving a type of calculus called multiple integration. Recently, however, discoveries of mathematical techniques for handling high-dimensional integration problems, and the advent of computers fast enough to actually do the calculations in a reasonable time, have changed the use of Bayesian methods.
One of the desirable features of the Bayesian approach to statistical inference and decision making is that it provides a straightforward way to combine new information with existing knowledge. The AMS department works to fuse the Bayesian and frequentist paradigms, using a Bayesian approach to formulate inferences and predictions, and then evaluate how good they are, using frequentist methods.
Draper has been working on health policy issues since the mid-1980s, when he was part of a large project at the RAND Corporation, a southern California think tank, studying the cost-effectiveness of the Medicare system on behalf of the federal government. The problem of how to measure the quality of care that hospitals offer their patients is still a major focus of his research. In a nutshell, there are good ways of assessing quality that are too expensive to be practical on a large scale, and there are cheaper ways that yield less reliable information. Draper's work is aimed at finding a combination of assessment strategies that can yield good information at a reasonable cost.
Bayesian methods have proven useful in a wide range of disciplines, in part because they are more flexible and general than other approaches. Bayesians are not stymied by incomplete data sets or multiple sources of uncertainty. Raquel Prado, an assistant professor of applied mathematics and statistics, uses Bayesian methods to analyze the signals from biomedical devices, such as electroencephalograms and electrocardiograms. Her work may enable physicians to extract more information about a patient's health or prospects for recovery from these kinds of tests. Bruno Sansó, a visiting associate professor in UCSC's statistics group, uses Bayesian methods to predict rainfall and gain insights into climate patterns. He and his colleagues in the department also hope to develop collaborations in new areas of research. “Statistics is a tool, and you can apply the same methods to many different issues - in engineering, psychology, economics, the environment - that's one of the beauties of this job,” Sansó says.
The People
Laying the foundation for an outstanding Statistics program within the UC Santa Cruz School of Engineering are an excellent faculty that includes:
David Draper - Bayesian hierarchical modeling, nonparametric methods, Markov chain Monte Carlo, quality assessment in health and education, stochastic optimization, applications in the natural and social sciences.
Herbie Lee - Bayesian statistics, spatial inverse problems, model selection and model averaging, statistical computing, neural networks, nonparametric regression and classification.
Raquel Prado - Bayesian analysis of nonstationary time series, bioinformatics, medical and other applications.
Bruno Sansó - Bayesian predictive modeling of rainfall at macro and micro levels of aggregation in space and time, Bayesian spatial modeling, environmental and geostatistical applications.
Athanasios Kottas - Bayesian nonparametrics, survival analysis, quantile regression modeling, categorical data analysis, spatial statistics.