This course is designed to introduce students to statistical
inference from the Bayesian point of view. The emphasis will be on
the creation of models for data, the incorporation of varying sources
of information, and the computational aspects of assessing posterior
distributions.
| Prerequisites: |
Probability and statistics (MATH 3800 or 4320)
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| Computing skills - can use Matlab or Mathematica
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This is generally taught each fall semester.
| Topics |
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- Fundamentals of the Bayesian Paradigm
The biggest benefit of Bayesian statistics over classical
statistics is its ability to utilize information available
outside of the data itself. This prior information and its
related uncertainty must be encoded into probabilities. Then
it can be combined with data to assess the total value of the
combined information. The basic structure of combining
a prior distribution with a likelihood function for data to
produce a posterior distribution for a quantity of interest
follows the simple probability rule known as Bayes' Theorem.
But while simple in theory, the computation of the posterior
distribution often requires more effort.
- Computational Techniques
The primary reason that Bayesian statistics has become so
popular in recent years is that the advance in computing
technology has made its use feasible. Numerical optimization,
approximation techniques, and Monte Carlo simulations are valuable
tools for determining the properties of a posterior distribution.
And Markov Chain Monte Carlo (MCMC) has recently become the tool
of choice for simulating sequences of points from a posterior
distribution, for MCMC can often be employed in models whose
complexity would otherwise make them intractable.
- Conditional Independence Models
A big benefit of the Bayesian paradigm is that additional
parameters can easily be added to a model without seriously
adding to the complexity of the statistical analysis, provided
that those parameters fit into a conditional independence structure.
That is, provided the dependence of the new parameters to the
existing data and parameters can be made explicit, assessing the
new parameters is often a simple matter of additional computing
time. Some of the most common models employed in the sciences
-- hierarchies and networks -- typically fall into the category
of conditional independence models.
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