Master Bayesian Statistics: Thinking in Probabilities

Explore priors in Bayesian statistics, describing how prior beliefs update with data to form the posterior, and distinguish informative, weakly informative, and non-informative priors across real-world contexts.

Apply Bayes' rule to update beliefs by combining prior, likelihood, and evidence, forming the posterior distribution with its normalizing constant and credible intervals.

Explore the likelihood in Bayes theorem, linking prior beliefs with observed data. Understand the difference between probability and likelihood and how the posterior is shaped by data.

Contrast credible intervals and confidence intervals, linking Bayesian posterior distributions and prior beliefs to interval probability; explain when to use each in decision making.

Explore how Bayesian decision making uses posterior distributions to weigh outcomes, utility, and risk, guiding actions with the highest expected value.

Explore hierarchical Bayesian models that borrow strength across groups, using partial pooling to improve estimates for small schools or hospitals, with priors, hyperpriors, and posterior predictive checks and trace plots.

Learn to perform posterior predictive checks by comparing observed data with simulated data from the posterior, identify misfits, and revise priors, likelihood, or model structure. Assess convergence with trace plots, rhat, and effective sample size to ensure reliable Bayesian inferences.

Learn Bayesian regression, where priors and the likelihood yield a posterior over possible lines, capturing uncertainty in slope, intercept, and error variance with credible intervals.

Explore Bayesian networks, probabilistic graphical models with nodes, arrows, and conditional probability tables to map uncertainty in complex systems. See how study time and sleep update beliefs and predict outcomes.

Explore Bayesian A/B testing to compare two designs using priors and posterior distributions, and measure the probability of superiority that version B outperforms A, informing adaptive decisions rather than p-values.

Learn why integrating Bayesian methods into machine learning adds priors, posteriors, and credible intervals for uncertainty quantification, controls overfitting, and guides high-stakes decisions.