APPLIED MATHEMATICS & STATISTICS (AMS)
Bayesian Modeling, Inference, Prediction and Decision-Making
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The Bayesian statistical approach to uncertainty quantification, which involves combining information, both internal and external to your available data sources, into an overall information summary, is both logically internally consistent and simple to describe: there's one equation for inference (drawing valid conclusions about the underlying data-generating process), one for prediction of observables, and one for optimal decision-making.
However, specifying the ingredients that, when combined, formulate a good model for your uncertainty is a process -- combining elements of both art and science, intuition and rigor -- that can take a lifetime to master.
In this course, for one full day per week over a ten-week span, I'll provide a 60-hour introduction to the Bayesian paradigm, based on a series of real-world case studies and featuring a wealth of computational detail in the statistical freeware environments R and WinBUGS.
Topics will include prior, likelihood, posterior and predictive distributions, maximization of expected utility, conjugate and non-conjugate analysis, Markov-chain Monte Carlo computational methods, one-parameter and multi-parameter problems, and hierarchical and mixture modeling; in addition to introductory topics, in 10 one-day sessions we'll be able to explore a variety of intermediate- and advanced-level ideas, including optimal Bayesian model specification, Bayesian nonparametric methods, and approximate Bayesian computation with large data sets.
(2) Statistics relies on probability for uncertainty quantification, but there's more than one way to think about probability.
Over the past 350 years, two main approaches to the meaning of probability have been developed:
The Bayesian approach is more flexible (it includes the frequentist approach as a special case), and indeed most statistical work around the world was Bayesian from about 1650 (when probabilistic ideas were first considered systematically) through about 1920, a period of time during which the problems whose solutions were attempted by statistical means gained in complexity.
But calculations in the Bayesian approach are more difficult, and for this reason the frequentist approach enjoyed greater popularity from about 1920 (when scientific problems became sufficiently complicated that Bayesian closed-form mathematical solutions became impossible to find) to around 1990, when computers became fast enough for so-called MCMC algorithms (first proposed in the 1950s to solve problems like the Bayesian computation problem) to become practically feasible.
Since the early 1990s there's been an explosion of Bayesian methods and applications taking advantage of this breakthrough in computing, and it's fair to say that the subject of statistics has undergone a contemporary revolution.
Many people have predicted that the 21st century will once again be a Bayesian century, but this time with almost no limit to the complexity of the problems that can be solved, and this appears to be a solid prediction worth investing time and money in.
Notes: (1) You may need to watch a short ad before getting to one or more of these videos; the revenue for these ads goes to support the International Society for Bayesian Analysis (ISBA), so I ask your indulgence in wading through them. Some of these ads will offer you a chance to exit from them before they're finished; if you want to help generate the full revenue for ISBA, please just let them finish. (2) In one of the videos (in a discussion about calibration), I describe a conversation I had with Jay Kadane, a prominent subjective Bayesian. I now realize that this conversation was actually with another prominent subjectivist, who shall remain nameless to avoid further controversy. I apologize unreservedly to Jay, whose views on the subject of calibration I have not yet fully elicited.
(last modified: 3 Sep 2013)