A Comprehensive Guide to Bayesian Statistics

Explore how statistical inference uses sample data to estimate population properties, addressing population size challenges and the goals of parameter estimation and data protection.

Explore how probabilities arise from well-defined experiments and the sample space, contrast frequentist inference with Bayesian updates via Bayes' theorem as unknowns become random variables with new data.

Compare the frequentist and Bayesian approaches to inference, highlighting how probability is defined, fixed versus random parameters, and the role of prior knowledge in updating beliefs.

Explore how Bayesian thinking updates beliefs with evidence by assigning prior distributions to events and parameters, then applying probability rules to infer uncertain outcomes in marketing, science, and everyday life.

Bayesian statistics ripple through physics, analytics, psychology, business, and AI, enabling interpretations of quantum mechanics, spam filtering, and DNA decoding. It informs finance forecasting and market research under uncertainty.

Apply Bayes' theorem to relate conditional probabilities, derive the posterior from prior and likelihood, and illustrate with rain given a cloudy morning.

Apply Bayesian reasoning to covid-19 screening by linking prior probability with sensitivity. Observe how sequential testing updates beliefs from 2 percent to 50 percent, then 98 percent, guided by specificity.

Learn how prior and posterior distributions update beliefs about a parameter by combining the likelihood with the prior under Bayes' theorem, with normalization to form the posterior.

Explore how Bayesian thinking updates beliefs by combining initial priors with new evidence, using Bayes' theorem to assess claims with observations and data.

Engage with the attached simple activity before continuing to the next sections to ensure your Bayesian journey runs smoothly.

Work through the attached practice exercise featuring new medical problems, try it out, and consult the provided solutions if you encounter challenges.

Explore how prior probability drives Bayesian inference, how posterior probability updates with new data, and why parameter estimation helps when a prior is unknown, using a bowling analogy.

Learn how Bayesian statistics blend prior beliefs with new data to improve forecasts and manage uncertainty in business, as shown by COVID-19 updates.

Explore the three priors—informative, weakly informative, and uninformative—and learn how to choose priors using the principle of indifference and simple distributions like normal and uniform to shape posterior inferences.

Compute posterior distributions for theta under Bernoulli and binomial data, using priors from uniform to beta; recognize natural conjugate priors and beta-binomial updates.

Derive Jeffreys non-informative prior from the Fisher information and illustrate with a normal model showing a uniform prior when sigma is known.

Apply your Bayesian statistics knowledge by working through the attached practice exercise, tackling new medical problems, and compare your results with the provided solutions.

Explore point estimation and interval estimation for parameter estimation, including maximum likelihood estimation and Bayesian inference, which uses credible intervals rather than confidence intervals.

Using a diabetes example, this lecture shows how the posterior distribution yields a point estimate and a 90% credible interval for diabetes rate, illustrating how width communicates confidence.

Compare confidence intervals and credible intervals, clarifying misinterpretations in frequentist versus Bayesian views and explaining how interval interpretation relies on repeated sampling versus posterior distribution.

Compute the confidence interval for a population mean with known variance using the sample mean, standard error, and z critical values, illustrated with data scientist salaries.

Compute the credible interval with a lower and upper bound around a point estimate using standard error and credible value. State that a 95 percent credible interval has posterior probability 0.95.

Explore computing a credible interval for the posterior mean with normal data and normal prior, derive the posterior, and compare bayesian credible intervals to frequentist confidence intervals.

Practice the attached exercise to apply Bayesian statistics to new medical problems, and compare your results with the provided solutions.

Analyze how Bayes factor informs hypothesis testing, showing how prior odds and posterior odds from data X update beliefs about the null hypothesis and its alternative.

Interpret the base factor using the Jefferys scale to classify evidence for or against a hypothesis, and apply the reciprocal when comparing opposite hypotheses.

Explore Bayes factors for competing hypotheses, distinguishing simple from composite, and compute them by comparing likelihoods, priors, and integrating over parameter spaces.

Engage with the attached practice exercise featuring new medical problems, try it out, and consult the provided solutions if you encounter challenges.

Explore how Bayesian decision theory compares multiple decisions, quantifies information, and uses loss functions and priors to compute a posterior, guiding optimal choices under uncertainty.

Explore decision theory basics, including loss matrices, decision rules, and risk minimization, illustrated with spam classification to connect actions, data, and expected loss in Bayesian inference.

Compare two cancer screening hypotheses using Bayesian decision theory to minimize posterior expected loss, weighing false negatives and false positives.

Explore types of loss functions, including the squared error loss used in regression, zero-one loss for binary classification, and absolute error loss that grows with the prediction error.

Explore loss functions in real-world bayesian statistics, where the true barometer is unknown; use prior and posterior distributions to define expected loss and derive bayesian point estimates.

Explore decision making with frequentist and Bayesian approaches on a simple insurance problem, comparing 10% vs 20% Greensmith boys using a five-sample binomial likelihood and posterior probabilities.

Explore Bayesian expected loss, Bayes estimate, and frequentist risk, showing how posterior risk guides base actions and base estimates by minimizing the expected loss over data.

Admissibility guides decision rules in the frequentist framework by comparing risk, considering domination, and identifying non-dominated rules as admissible.

Define the Bayes risk as the expected loss under the prior and data, and show that the Bayes rule minimizes base risk by choosing the action with minimal posterior loss.

Apply the normal form of analysis to derive the Bayes estimate and Bayes risk. Compute risk via loss function and expectation, and optimize the decision rule under a normal model.

Use extensive form of analysis to minimize posterior expected loss and derive Bayes estimates. Squared loss yields posterior mean; absolute loss yields posterior median; examples include normal and Poisson.

Practice applying Bayesian methods with new medical problems using the attached exercise, tackle challenges, and compare your results with the provided solutions.

Review bayesian statistics by contrasting frequentist and bayesian inference, and summarizing priors, posteriors, interval and credible intervals, bayesian hypothesis testing, and loss and risk in decision theory.

Examine Bayesian defense and critique, showing how Bayes' theorem uses a prior to shape hypothesis probabilities, decisions, and uncertainty, while noting subjective priors and philosophical objections.

Explore how Bayesian methods power data science, machine learning, and artificial intelligence across business, psychology, physics, applied science, ecology, life sciences, and conservation, with practical applications.

Learn how Bayesian reasoning guides updating beliefs with evidence, avoiding strong opinions when evidence is weak, and basing priors on mental models while staying sensitive to reality.

Take the self evaluation to test your Bayesian learning with the attached quiz and reinforce key concepts.

Celebrate completing a journey through Bayesian statistics and its applications, and access additional reading material and an Excel-based demonstration for practical understanding.

Express gratitude for completing the course, invite questions and suggestions in the comments, request reviews, and encourage sharing to improve the material.

Receive access to additional statistics and data science resources via the attached link in this bonus lecture.