One author has referred to Bayesian Inference as "probability in action." Probability is all about looking
forward in time - computing the likelihood of an event occurring given that you know everything about the model. In
practice, we are often working backward; we have data generated from some process, but we are not fully aware of the
model that governs this process. The Bayesian framework is one approach to statistical inference that quantifies
the prior knowledge available about a process through probability distributions.
This course offers an introduction to inference under the Bayesian framework. Building from Bayes' Rule for
probability computations, we develop a framework of estimation and hypothesis testing. We examine inference in
several scenarios, including regression analysis. Modern applications will be covered, time permitting. We discuss
the construction of prior distributions given prior information about a parameter and give an introduction to
computational tools for Bayesian inference, including Markov Chain Monte Carlo (MCMC) methods.
This course is particularly useful for students considering an analytical career, those interested in
computational psychology, and those interested in traffic engineering at Rose-Hulman. This course is unique at
Rose-Hulman in that our other statistics courses consider inference from a Frequentist perspective. This course,
therefore, assumes no prior knowledge of statistics but does assume knowledge of probability.
Learning Objectives
After completing this course, students will be able to accomplish the following tasks:
Given a research goal, identify the parameter(s) of interest and, if applicable,
formulate the goal as measurable statements about the parameter(s) of interest.
Describe the importance of considering data collection when interpreting the results
of a study. Specifically, determine whether it is reasonable that a sample is
representative of the underlying population and justify your rationale.
Construct and interpret graphical and numerical summaries of
data to address a research goal.
Given a description of the data generating process, construct a probability model
that represents that process as a function of unknown parameters.
Given a question of interest, construct a probability distribution that captures the
information on the parameter prior to conducting the study, and use the data to update
the information on the parameter.
Comment on the adequacy of a statistical method for addressing a given question of
interest by assessing the assumptions underlying the method.
Given a posterior distribution, summarize the uncertainty in the parameter of interest
and interpret the resulting output in context of the research question.
Identify the value of statistical methodology in the advancement of science as well as
recognize its limitations.
Clearly communicate an analysis and its implications in written format.
Describe the role of probability theory in the Bayesian framework for making inference
on a population and appreciate the use of probability for capturing uncertainty about a
parameter and the use of data to inform that uncertainty.
Course Structure
This course is given as a traditional lecture class. The course is divided into the following modules:
Statistical Process
Bayesian Fundamentals
Bayesian Computations
Comparing Groups
Regression
Instead of exams, each module is accompanied by a lengthy homework assignment and applied analysis task.
These tasks pose a problem in a specific context and ask you to use the available data to address the problem.
These focus on both the implementation as well as communication of the solution.
Beyond a handful of classic examples, the Bayesian framework requires specialized computing tools. The
class makes use of Stan through the rstan package in R. While R is supported, students have had
success implenting Stan in Python.
