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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! To better grasp the concept behind Bayesian Inference, let's use a coin tossing
example
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The mathematical representation of the observed occurrences is called a model
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For instance, the
parameter of coin expressed by may be used to define coin fairness while tossing a coin
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Now respond to this
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e
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Is that question even the proper one? No
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5) given a result (D) should be of greater
relevance to us
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It is acceptable to think that a coin can be fairly biased in any way
between 0 and 1
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This gives
the likelihood of observing the number of heads in a specific number of flips if we knew the
coin was fair
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This is the probability of the data calculated by adding up (or
integrating) all conceivable values of, weighted by how strongly we think those specific values
of are true
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P(θ|D) is the parameter's posterior belief as determined by the evidence, or the number
of heads
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Not to worry
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We need two mathematical models beforehand in order to define our model effectively
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The posterior belief P(θ|D) distribution is the product of these two
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So have that in mind
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As a result, the Bayes theorem is supported by a variety of functions
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Conclusion
The purpose of this essay was to encourage you to consider the various statistical philos
ophies that are available and how each of them cannot be used in every circumstance
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The Section on Statistical Education
organized an invited discussion on the advantages, disadvantages, justifications, and techniques
for teaching an introductory statistics course from a Bayesian perspective at the Joint Statistical
Meetings in 1996
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In example, Moore claimed that it was untimely to introduce the
concepts and procedures of Bayesian inference in an introductory statistics course
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In reality, prior to the discovery and development of the Gibbs sampler and other
MCMC algorithms in the late 1980s and early 1990s, there was very little classroom instruction
and very little practical use of Bayesian methods
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The teaching and learning of Bayesian techniques has flourished, albeit primarily at the
graduate level, as a result of the revolutionary advances in computation and the rapid
expansion of Bayesian techniques utilized in applied applications
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Some of these textbooks, such Doing Bayesian Data Analysis, are suitable
for undergraduate statistics students as well
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How about undergraduate-level Bayesian education? The majority of academic
innovation is in introductory statistics courses, one of which covers Bayesian inference
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created an active-learning Bayesian inference exercise using M&Ms, while
Barcena et al
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Kuindersma and Blais developed a teaching tool
of Bayesian model comparison used in a physics application of a three-sided coin, and Rouder
and Morey proposed teaching Bayes' theorem by looking at strength of evidence as predictive
accuracy for advanced-level undergraduate statistics courses, where Bayesian inference is
typically covered as a topic in a statistical inference/mathematical statistics course
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We suggest in this article that undergraduates with a background in multivariable
mathematics and probability take a semester-long Bayesian statistics course
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We place a strong emphasis
on the development of Bayesian reasoning, the importance of statistical computation, and the
utilization of actual data to achieve these learning objectives
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We provide a course overview in Section 2 and discuss the topics we select for the
course after introducing the prerequisites and students' backgrounds
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1 outlines our decisions, strategies,
and computing philosophies in the context of the course
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In this section, we also talk about the course exams
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These can be found in the online
supplementary sources
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For 13 weeks, we meet twice a week for a total of 26 class meetings
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Probability and multivariable calculus are prerequisites for the course
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At the start of
the course, we give a thorough overview of these subjects
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We do not assume that students
have prior R experience, given the likelihood that they do
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5 Although we've had students
with mod backgrounds choose to use tidyverse, we have chosen to primarily use base R instead
of tidyverse in this course because we do not assume prior R experience
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Our selection of subjects and their substance is guided by these training objectives:

Bayesian conclusions for a percentage and a mean: covering the Bayes theorem,
conjugate prior, posterior distribution, and predictive distribution, with a comparison of the
exact answers to the Monte Carlo simulation solutions in one-parameter Bayesian models as
the main point of emphasis
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The aforementioned are the course's main topics, which are intended to give a sufficient
overview of fundamental Bayesian concepts, inference procedures, and computing techniques
in a few specific practical scenarios
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These elements are intended to provide students plenty of
time and space to conduct research using what they have learned to complete their goals
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How to specify a multistage (hierarchical) prior distribution, MCMC estimation,
prediction, and study of pooling/shrinkage effects caused in hierarchical models are all covered
within the topic of Bayesian hierarchical modeling
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Features and Recommendations
Further information about the course is provided in this section
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In order to meet our learning goals, our
course includes three crucial elements: case studies, journal article analysis and discussion and
course assignments
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We provide all course
evaluation elements along with their percentages of the overall grade and a discussion of how
they are used to evaluate learning objectives and determine grades
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The written portion of
each homework is devoted to exercises on topics like calculating posterior distributions, and
the R component is devoted to using Bayesian inference techniques on actual data
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Students are required to complete
five in-class computing labs throughout the semester in order to improve their programming
abilities
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We'll go over a
computing lab intended to help students better understand a journal article
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There are two midterm exams
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The supplemental resources include a course schedule with evaluation
elements like homework, laboratories, and exams
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Case Studies
We agree with Allenby and Rossi that case studies are essential tools for demonstrating
applied Bayesian analysis: one constructs suitable prior distributions, develops the likelihood,
computes the posterior distributions, and then explains their findings to address the pertinent
problems
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In these case studies, students are urged to investigate the extension of
previously learned methodologies and/or develop new approaches in order to address openended practical issues
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While some case studies require several tries and discussions, others are easier to
understand
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They also offer possibilities to introduce advanced
modeling techniques
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After a first round of case study reports and discussion, we identify a glaring flaw in

the methodology: Are the predetermined groups reasonable, particularly for those scores that
are on the fence (for example, accuracy rates of 60% and 70%, which are higher than the
random guessing of 50% but lower than the knowledgeable score of 70%)? Latent class models,
which don't attribute each observation to a specific category, are a better fit for this scenario
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Case studies like this one reinforce students' learning by simultaneously introducing new
Bayesian techniques and significant applied problems
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The supplemental resources
include this case study
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To promote teamwork, first assign
students to case studies in pairs
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Before the class discussion, instruct pairs to post their case
study summaries on the learning management system (LMS, such as Moodle, Canvas, etc
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Journal Articles
We strongly agree with Cobb's five directives to "rethink our undergraduate curriculum
from the bottom up," particularly the final and most crucial directive: "teach through research
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For undergraduates, journals like The American Statistician are excellent sources of
high-quality and easily accessible journal papers
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The paper provides a brief and well-written description of
how and why the Gibbs sampler operates, illustrates its characteristics with a two-by-two basic
scenario, designs simulation studies, and evaluates the outcomes
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All of them offer fantastic
chances for students to study about the history of the Gibbs sampler, learn about alternative
approaches, and discuss and evaluate them
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First, give students a
reading guide with a number of questions to take them through the material
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Second, to encourage smallgroup and full-class conversations in class, ask students to post their answers to these
questions on the LMS ahead of time
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In our illustration with Casella and George, replicating one of their simulation studies
necessitates students to be very clear about how many Gibbs samplers are run and how many
iterations each run needs to take, which undoubtedly strengthens their comprehension of
Gibbs samplers and improves their computing skills
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Beyond the articles themselves, reading, talking about, reviewing, and copying journal
articles has several advantages
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In these projects, we frequently see students shift their attention from completing
assignments to focusing on conducting research
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Starting on day one of our course, we encourage students to come up with project
ideas
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This quickly switches the attention from topics that we, the
instructors, find intriguing and engaging to those that they, the other students, are working on
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There may be groups of interests and themes depending on the academic backgrounds
and career goals of the pupils
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Students are inevitably interested in trending issues, which are
reflected in their project interests and topic selections
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Here, we provide further details about the procedures we employ to create project
teams, maintain the timelines for each one, and provide project results
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Students are encouraged to publish a self introduction on the LMS during the
first week of the semester to describe their project interests
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At this point, students start to discover common interests, and
project teams start to take shape gradually
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Before submitting the project proposal, each
project team (up to 3 students) must meet with their instructor, and by week 7, or halfway
through the 13-week semester, thorough feedback on the project's viability and suggestions
are provided
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Every team must produce a methodology draft by Week 10 and a project
draft by Week 12 in order to receive credit for their work
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The
75-minute poster session lasts
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There is a 5-minute discussion break in between sessions, and the session ends with a 20minute final debate
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After learning about other students' work, there will be a 5-minute
discussion break where students can express their opinions
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The
75-minute poster session lasts
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There is a 5-minute discussion break in between sessions, and the session ends with a 20minute final debate
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After learning about other students' work, there will be a 5-minute
discussion break where students can express their opinions
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Presenters can use these two-minute videos to help them make a strong argument for
their ideas
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Presenters can use these two-minute
videos to help them make a strong argument for their ideas
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In the succeeding semesters, course assignments organically develop into independent
research projects
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Everyone benefits from better planning their poster session, so they
may devote more time to a poster that piques their interest based on the introduction video,
for example
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Bayesian estimation of future realized volatility, Bayesian inference using
Python, Bayesian nonparametric models, Bayesian time series, and Bayesian variable and
model selection are some of the past and present independent study topics resulting from this
Bayesian statistics course
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In the succeeding semesters, course assignments organically develop into independent
research projects
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In these autonomous studies, students participate in nearly every
stage of applied statistics research, including literature reviews, dataset collection and analysis,
technique implementation, outcome analysis, and report writing in the form of journal articles
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Working with pupils on subjects that interest them also exposes the teacher to fresh ideas and
study fields
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It contains case studies, DataCamp modules, and
paper discussions (online and in-person), with the last two reflecting the two distinctive and
significant aspects of our course
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As
previously stated, the first third and a half of the semester are spent assigning assignments
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The final two-fifths of the semester would have seen the students
mastering the fundamental skills and information
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We so substitute open-ended case studies for conventional "closed-ended"
homework, and we have purposefully decided to make case studies a component of
participation rather than being formally assessed
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Conclusion
While prior Bayesian research experience is preferred to teach an undergraduate-level
Bayesian course, we consider that graduate-level Bayesian statistics course experience is
sufficient for prospective instructors

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