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Bayesian Statistics
Introduction
Many analysts still find Bayesian Statistics (also known as Bayesian Probability) to be
unintelligible
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We are now only interested in learning about machine learning
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Even if there is data involved in these problems, it does not always assist us
in solving business problems
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A British mathematician named Thomas Bayes developed what was known as the
"Bayes Theorem" in the 1770s
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In fact, several of the greatest colleges in the world currently offer in-depth
courses on this subject
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I've made an effort to provide examples to help simplify the ideas
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If you want to learn everything there is to know about
statistics and probability, you should take this course
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Theory
Bayesian statistics uses probabilities to solve statistical issues
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You understand that? Here's an illustration to help me explain:
Let's say that of the four F1 championship races that Niki Lauda and James Hunt
competed in, Niki won three of them while James only managed one
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The twist is in this
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But the key query is: by how much? We need to identify ourselves with a few
theories in order to comprehend the issue at hand, the first of which is conditional probability
(explained below)
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You can study the fundamentals of linear algebra by visiting Khan Academy
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Probability and Basic Statistics: Check out this course from Khan Academy
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The possibility of a result happening
based on the likelihood that a comparable outcome has already happened in the past is known
as conditional probability
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Bayes' Theorem can be used in finance to assess the risk associated with loaning money
to potential borrowers
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Thus, using new information that is or could be related to an event, Bayes' Theorem
predicts the likelihood of that happening
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Think about drawing just one card from a whole deck of 52 cards
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69%
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Let's say it is discovered that the chosen card is a face
card
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3%
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So, let's find out how it
operates!
tremendous effort to innovate, notably by relating to real-world data problems and using
Bayesian approaches to overcome these challenges
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Bayesian Data Analysis, A
First Course in Bayesian Statistical Methods, Bayesian Essentials with R, Statistical Rethinking,
and Bayesian Statistical Methods, among others, are just a few of the textbooks that are
currently available
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Additionally, statisticians have enabled nonstatisticians to learn about Bayesian theory
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Most
recently, Eadie et al
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created a web simulator to teach the Bayes theorem with a focus on the 1968
search for the nuclear submarine USS Scorpion
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In addition to being a topic in introductory or statistical inference/mathematical
statistics courses, we are among the many Bayesian statistics instructors who believe in the
substantial benefits of incorporating Bayesian concepts within the undergraduate statistics
curriculum
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The primary

learning goals are for students to: (1) comprehend fundamental concepts in Bayesian statistics,
such as the Bayes rule, prior, posterior, and posterior predictive distributions; and (2) apply
Bayesian inference techniques to problems in science and daily life
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Three crucial elements of our
proposed course are case studies, discussing and evaluating journal articles, and course
projects
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As Bayesian computing is
a crucial and integral component of this course, Section 2
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After that, in Section 3, we go into
more depth about the proposed course, including its three main components: case studies,
reading and discussing journal articles, and course assignments where we offer advice based on
our experience
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The course schedule,
sample homework, sample case studies, sample computing labs in R Markdown, and a reading
guide for a journal article include the additional materials
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Overview
For students majoring in statistics or minoring in statistics, our proposed course is a
popular elective
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75
minutes are allotted for each class meeting, and some lectures double as computer labs
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We place particular
emphasis on joint densities of conditionally independently and identically distributed random
variables and transformation of random variables among the calculus and probability topics
because these are essential abilities in expressing joint posterior distributions
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While it isn't necessary, a typical
student in this course may have prior experience to statistics
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We assign three DataCamp1 courses
within the first several weeks of the course: Introduction to R, Intermediate R, and Introduction
to the Tidyverse, all of which are available through DataCamp for the classroom, to make sure
students are prepared for statistical computing in the course
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We have two primary learning goals: students must be able to: (1) comprehend the
basics of Bayesian statistics, such as the Bayes rule and the prior, posterior, and posterior
predictive distributions; and (2) apply Bayesian inference techniques to real-world and scientific
topics
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The Markov chain Monte Carlo (MCMC) diagnostics, coding one's own Gibbs sampler,
and using Just Another Gibbs Sampler (JAGS) for MCMC estimates are all topics covered in the
section on the Gibbs sampler and MCMC
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In this course, students are introduced to a considerably
larger range of Bayesian approaches: some are creative expansions of methodologies discussed
in class through case studies; others are far more sophisticated methodologies students
encounter in their course projects
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In
Section 3, we will go into further detail and explain why we selected these features
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The estimation of a regression model, various prior selections, MCMC estimation, and
predictions are covered in Bayesian linear regression
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We begin by outlining
the course's homework, computer labs, and tests
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These three crucial elements are presented, discussed, and our
suggestions for improving students' learning experiences are included
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The primary assessment tool for the first 1/3 of the semester is homework, which is
given four times every two weeks and has a written and a R component
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While the R
section gives students plenty of practice using R for Bayesian inference, the textual portion
helps them better comprehend fundamental Bayesian ideas
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Each computing lab's theme is closely related to the lecture material
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The labs are set up
and required to be completed using R Markdown; they significantly aid students in developing
their familiarity and proficiency with this crucial tool
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Both have
a take-home component that emphasizes Bayesian computing to apply appropriate techniques
to address applied issues and an in-class component that emphasizes Bayesian thinking and

theoretical derivations
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We will now go through the three distinctive and significant aspects of our course
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In the final 2/5 of the semester, we offer three case studies in place of homework
assignments
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Each case study requires students to work in pairs and has a one-week
deadline
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Students are encouraged to expand on what they have learned and think critically
because case studies are open-ended
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Are there a knowledgeable group and a random guessing group present
in the dataset that consists of two clusters of students' multiple-choice test results, for
instance? If so, can we tell them apart? Can we also draw conclusions about other group
characteristics, such how accurately a question is answered? Students typically use the
hierarchical modeling approach with a predetermined set of two categories based on their prior
experience
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Students can retake this case study after receiving a sufficient degree of model introduction
and some sample R/JAGS syntax and learn the model's specifics and estimating method on their
own
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Additionally, they highlight and spark
conversations about the shortcomings of well-known approaches and present chances for
better, albeit inherently more complex, modeling approaches
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We advise the following methods to improve students' learning
experiences in addition to creating appropriate case studies
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Permit students to collaborate with a different partner on
each case study, if at all practicable
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Third, allow students to discuss their techniques and conclusions in small groups before
allowing the class as a whole to discuss and evaluate various approaches
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"
If the papers have the proper substance at the right level, we think undergraduate students
may and should read scholarly journal articles as part of their education
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George and Casella have been our favorite
professors for our Bayesian course
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Additionally, because it was a
preliminary study, several features of the Gibbs sampler, such as how to achieve independent
parameter draws, might not be applicable to modern methods
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We recommend the following methods to improve students' learning experiences in
addition to choosing articles with the appropriate content and level
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A good mix of
questions includes a variety of different types, some of which ask you to check mathematical
expressions and others of which you must explain your process
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After class discussion, if at all possible, pose further

questions
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The supporting materials include our
reading list and computer lab
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Students have developed their confidence and skills through
this process, which has inspired them to accept the challenge of reading, comprehending,
implementing, and occasionally extending methods from journal articles in their course
projects
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Course Projects
A course project is the next step in our implementation of Cobb's "teach through
research" demand after exposing students to journal articles
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Our first lecture includes a few video clips of student projects from prior semesters in
addition to some instructor-selected examples of Bayesian approaches handling intriguing
practical problems (a 2-min introduction video of the course project is required as part of a
students' project submission)
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The variety of project subjects and interests inspires kids to choose what they want to do by
showcasing what they can do in their projects
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In the case of Vassar College, there are a number of double
majors in mathematics/statistics and economics, which results in student groups working on
projects in economics and finance; the cognitive science program has a faculty member who is
a Bayesian cognitive scientist, which results in student groups analyzing experimental data to
explore learning theories
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Examples of these topics include neural
networks and natural language processing
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