Fundamentals Of Bayesian Statistics

Learn Bayesian Inference, Probability Distributions, MCMC Methods, and Statistical Modeling Step by Step with Examples

What you'll learn

The fundamental differences between Bayesian and frequentist statistics and how they approach probability and inference

Key probability concepts, including marginal and conditional probability, and their role in Bayesian reasoning

Bayes' Theorem and its application in statistical inference

How to specify a prior and understand different types of priors, including Jeffrey’s prior, reference priors, and Zellner’s G-priors

The probability of data given model choice and how it impacts inference

An introduction to probability distributions commonly used in Bayesian data analysis, including Beta, Normal, Poisson, and Gamma distributions

Conjugate priors and their significance in simplifying Bayesian inference

Credible intervals and highest density posterior intervals (HDPI) as Bayesian alternatives to confidence intervals

Objective Bayesian data analysis and its role in making unbiased inferences

Forecasting in Bayesian systems using posterior predictive distributions

Markov Chain Monte Carlo (MCMC) methods, including grid approximations, Metropolis-Hastings sampling, and Gibbs sampling

Hypothesis testing using Bayesian methods, including classical test analogues and pure Bayesian approaches

Hierarchical models and hyperpriors, and how they allow for multi-level Bayesian inference

Bayesian linear regression and its application in predictive modeling

Requirements

No prior knowledge of Bayesian statistics is required, but familiarity with probability concepts will be helpful.

Description

Bayesian statistics is a powerful and intuitive framework for statistical inference, widely used in data science, machine learning, economics, medicine, and many other fields. This course provides a structured and in-depth introduction to Bayesian reasoning, covering fundamental concepts, key mathematical principles, and practical applications.This course provides a complete introduction to the field of Bayesian Statistics. It assumes very little prior knowledge and, in particular, aims to provide explanations of concepts with as little maths as possible. The lectures are designed to be clear, engaging, and practical. Each topic is broken down step by step, ensuring that learners understand both the intuition behind Bayesian methods and the mathematical principles that support them. The course includes real-world examples, case studies, and problem-solving exercises to reinforce key concepts.The course covers the following topics:The fundamental differences between Bayesian and frequentist statistics and how they approach probability and inferenceKey probability concepts, including marginal and conditional probability, and their role in Bayesian reasoningBayes' Theorem and its application in statistical inferenceHow to specify a prior and understand different types of priors, including Jeffrey’s prior, reference priors, and Zellner’s G-priorsThe probability of data given model choice and how it impacts inferenceAn introduction to probability distributions commonly used in Bayesian data analysis, including Beta, Normal, Poisson, and Gamma distributionsConjugate priors and their significance in simplifying Bayesian inferenceCredible intervals and highest density posterior intervals (HDPI) as Bayesian alternatives to confidence intervalsObjective Bayesian data analysis and its role in making unbiased inferencesForecasting in Bayesian systems using posterior predictive distributionsMarkov Chain Monte Carlo (MCMC) methods, including grid approximations, Metropolis-Hastings sampling, and Gibbs samplingHypothesis testing using Bayesian methods, including classical test analogues and pure Bayesian approachesHierarchical models and hyperpriors, and how they allow for multi-level Bayesian inferenceBayesian linear regression and its application in predictive modelingCourse Structure:The course begins with an exploration of Bayesian vs. frequentist statistics, followed by an in-depth discussion of probability distributions, Bayes' theorem, and inference techniques. It introduces priors and posterior distributions, explores conjugate priors and credible intervals, and covers forecasting, hypothesis testing, and Bayesian regression. The course also includes an introduction to MCMC methods, with a focus on Metropolis-Hastings and Gibbs sampling for practical Bayesian computation.Instructor Expertise and Teaching ApproachThis course is developed by an experienced instructor with a strong background in probability, statistics, and data analysis. The instructor emphasizes both theoretical understanding and practical application, making the course suitable for learners who want to develop a strong foundation in Bayesian statistics.A Balanced Mix of Theory and ApplicationWhile the course provides a rigorous introduction to Bayesian methods, it also focuses on their practical implementation. Topics such as posterior and predictive distributions, Bayesian hypothesis testing, and statistical modeling are covered in detail, ensuring learners gain both theoretical insights and hands-on experience with Bayesian approaches.This course is designed to be accessible to learners with a basic understanding of probability and statistics. Whether you are a student, researcher, or professional looking to expand your statistical knowledge, this course will equip you with the tools needed to confidently apply Bayesian reasoning to real-world problems.By the end of this course, you will have a solid understanding of Bayesian inference and the ability to apply Bayesian techniques to real-world problems.No prior knowledge of Bayesian statistics is required, but familiarity with probability concepts will be helpful.Enroll now to start your journey into Bayesian reasoning!

Overview

Section 1: Introduction to Bayesian Statistics

Lecture 1 Bayesian statistics syllabus

Lecture 2 Bayesian vs frequentist statistics

Lecture 3 Bayesian vs frequentist statistics probability - part 1

Lecture 4 Bayesian vs frequentist statistics probability - part 2

Section 2: Probability Foundations

Lecture 5 What is a probability distribution?

Lecture 6 What is a marginal probability?

Lecture 7 What is a conditional probability?

Lecture 8 Conditional probability: example breast cancer mammogram - part 1

Lecture 9 Conditional probability: example breast cancer mammogram - part 2

Lecture 10 Conditional probability - Monty Hall problem

Lecture 11 Marginal probability for continuous variables

Lecture 12 Conditional probability for continuous random variables

Section 3: Bayes’ Rule and Its Applications

Lecture 13 Derivation of Bayes' rule

Lecture 14 Bayes' rule - an intuitive explanation

Lecture 15 Bayes' rule in statistics

Lecture 16 Bayes' rule in inference - likelihood

Lecture 17 Bayes' rule in inference - the prior and denominator

Lecture 18 Bayes' rule in inference - example: the posterior distribution

Lecture 19 Bayes' rule in inference - example: forgetting the denominator

Lecture 20 Bayes' rule in inference - example: graphical intuition

Section 4: Exchangeability and Likelihood

Lecture 21 The definition of exchangeability

Lecture 22 Exchangeability and iid

Lecture 23 Exchangeability - what is its significance?

Lecture 24 Bayes' rule denominator: discrete and continuous

Lecture 25 Bayes' rule: why likelihood is not a probability

Lecture 26 Maximum likelihood estimator - short introduction

Lecture 27 Sequential Bayes: Data order invariance

Section 5: Conjugate Priors and Probability Distributions

Lecture 28 Conjugate priors - an introduction

Lecture 29 Bernoulli and Binomial distributions - an introduction

Lecture 30 Beta distribution - an introduction

Lecture 31 Beta conjugate prior to Binomial and Bernoulli likelihoods

Lecture 32 Beta conjugate to Binomial and Bernoulli likelihoods - full proof

Lecture 33 Beta conjugate to Binomial and Bernoulli likelihoods - full proof 2

Lecture 34 Beta conjugate to Binomial and Bernoulli likelihoods - full proof 3

Section 6: Bayesian Inference in Practice

Lecture 35 Bayesian inference in practice - posterior distribution: example Disease prevale

Lecture 36 Bayesian inference in practice - Disease prevalence

Section 7: Predictive Distributions

Lecture 37 Prior and posterior predictive distributions - an introduction

Lecture 38 Prior predictive distribution: example Disease - 1

Lecture 39 Prior predictive distribution: example Disease - 2

Lecture 40 Posterior predictive distribution: example Disease

Section 8: Bayesian Analysis with Normal Distributions

Lecture 41 Normal prior and likelihood - known variance

Lecture 42 Normal prior conjugate to normal likelihood - proof 1

Lecture 43 Normal prior conjugate to normal likelihood - proof 2

Lecture 44 Normal prior conjugate to normal likelihood - intuition

Lecture 45 Normal prior and likelihood - prior predictive distribution

Lecture 46 Normal prior and likelihood - posterior predictive distribution

Lecture 47 Population mean test score - normal prior and likelihood

Section 9: Bayesian Analysis with Poisson and Gamma Distributions

Lecture 48 The Poisson distribution - an introduction - 1

Lecture 49 The Poisson distribution - an introduction - 2

Lecture 50 The gamma distribution - an introduction

Lecture 51 Poisson model: crime count example introduction

Lecture 52 Proof: Gamma prior is conjugate to Poisson likelihood

Lecture 53 Prior predictive distribution for Gamma prior to Poisson likelihood

Lecture 54 Prior predictive distribution (a negative binomial) for gamma prior to Poisson l

Lecture 55 Posterior predictive distribution: a negative binomial for gamma prior to Poisso

Students of statistics, data science, and machine learning,Researchers seeking a strong theoretical foundation in Bayesian methods,Analysts and decision-makers interested in probabilistic reasoning and forecasting,Anyone with a basic understanding of probability and statistics who wants to learn Bayesian statistics step by step