Course Notebooks

A browsable index of Univ.AI course notebooks.

Correlation Modeling with LKJ Priors

Modeling correlations using LKJ priors and Cholesky decomposition.

Models correlations in multivariate normal distributions using the LKJ prior. Covers correlation matrix construction, prior sensitivity via the eta parameter, and the…

Two-Component Gaussian Mixture

Identifiability and label-switching in two-component Gaussian mixtures.

variational-inference

Simple 2-component Gaussian mixture model showcasing identifiability and label-switching problems without ordering, then solved with ordered transform. Uses NormalMixture…

Gaussian Mixture Models with ADVI

2D Gaussian mixtures with custom logp, Metropolis sampling, and mini-batch ADVI.

variational-inference

Detailed 2D Gaussian mixture model with marginalization and ADVI. Covers custom logp via DensityDist, MCMC sampling, and ADVI with mini-batching using multivariate normals…

Marginalizing Over Discrete Variables

The log-sum-exp trick for marginalizing out cluster assignments in mixture models.

variational-inference

Demonstrates how to marginalize over discrete cluster assignments in Gaussian mixture models using the log-sum-exp trick. Covers NormalMixture, custom DensityDist with…

Mixture Models and MCMC

Gaussian mixtures, label-switching, and identifiability in MCMC sampling.

Explores Gaussian mixture models with MCMC sampling using Old Faithful data and synthetic 3-component mixtures. Demonstrates label-switching problems, identifiability…

Automatic Differentiation Variational Inference

From CAVI to scalable black-box VI with automatic differentiation.

variational-inference

Covers the progression from CAVI to stochastic CAVI to ADVI, including transformation to unconstrained space, automatic differentiation for gradient computation, and…

Variational Autoencoder

Generative modeling with VAEs on MNIST using PyTorch.

neural-networks

variational-inference

Implements a variational autoencoder (VAE) in PyTorch for the MNIST dataset, covering the encoder-decoder architecture, the reparameterization trick, and the ELBO loss…

Variational Inference with Neural Networks

Bayesian neural network classification using ADVI in PyMC.

neural-networks

variational-inference

Builds a Bayesian neural network for binary classification on the moons dataset using variational inference (ADVI) in PyMC, demonstrating uncertainty quantification in…

Variational Inference

Approximating posteriors through optimization instead of sampling.

variational-inference

Introduction to variational inference: the core idea of turning Bayesian inference into optimization, KL divergence, the ELBO, and coordinate ascent variational inference…

Bayesian Workflow: Zero-Inflated Poisson

Prior selection, simulation-based calibration, and posterior predictive checks for GLMs.

A complete Bayesian workflow applied to a zero-inflated Poisson model of monks producing manuscripts. Covers prior predictive checks, simulation-based calibration…

The Idea Behind the Gaussian Process

From multivariate normals to kernel-based function priors and predictive distributions.

gaussian-processes

Starting from multivariate Gaussian marginals and conditionals, we build intuition for Gaussian Processes by constructing covariance matrices from kernels, drawing prior…

Gaussian Processes and Non-parametric Bayes

From kernel trick to infinite basis functions — the theory behind GP regression.

gaussian-processes

We develop the theoretical foundations of Gaussian Processes: the dual representation of Bayesian linear regression, kernel tricks, Mercer’s theorem, infinite basis function…

Inference for Gaussian Processes

Hyperparameter learning via empirical Bayes and MCMC with PyMC.

gaussian-processes

We tackle GP hyperparameter inference using the marginal likelihood: both Type-2 MLE (empirical Bayes) via optimization and full Bayesian MCMC via PyMC. The sockeye salmon…

Geographic Correlation and Oceanic Tools

Gaussian process covariance captures spatial correlation between island societies.

gaussian-processes

We extend the oceanic tools Poisson regression with a custom GP covariance matrix that models geographic correlation between societies. The inverse-square distance kernel…

GP Recap and Salmon Example

A condensed recap of GP theory applied to sockeye salmon recruitment.

gaussian-processes

A concise review of Gaussian Process fundamentals — MVN conditionals, kernel covariance, and GP regression — applied to the sockeye salmon stock-recruitment dataset with…

Poisson Regression — Model Comparison and Hierarchical Overdispersion

WAIC model comparison, ensemble averaging, and varying-intercepts for overdispersed counts.

We return to the oceanic tools dataset to illustrate model comparison using WAIC, model averaging with Akaike weights, counterfactual posterior predictives, and fighting…

Poisson Regression: Modeling Tool Diversity Across Islands

Centering, correlations, and counterfactual posteriors in Oceanic tool kit evolution.

Analyzes technological diversity across Oceanic island societies using Poisson regression with interaction terms. Shows how parameter centering reduces posterior correlation…

Choosing Priors: From Flat to Weakly Informative

Uninformative, Jeffreys, and regularizing priors for Bayesian inference.

Explores how to choose priors that are robust to transformation and prevent overfitting without strongly biasing inference. Covers flat, improper, Jeffreys, and weakly…

Bayesian Regression with Normal Models

From heights to height-weight relationships using conjugate priors and posterior predictives.

Demonstrates Bayesian regression by modeling height using normal distributions and PyMC. Covers posterior inference, the identifiability problem in regression parameters…

Identifiability in Bayesian Models

How parameter redundancy breaks MCMC convergence — and how priors can help.

Explores how unidentifiable parameters cause MCMC samplers to fail, even when derived quantities converge perfectly. Demonstrates convergence diagnostics that reveal the…

Gelman Schools and Hierarchical Pathology

Diagnosing and fixing funnel pathology in hierarchical models with reparameterization.

Explores the 8-schools hierarchical model to reveal how centered parametrizations create pathological funnel structures that trap HMC samplers. Shows how non-centered…

Hierarchical Bayesian Modeling: The 8 Schools Example

Learning to pool information across groups with non-centered parametrization.

Explores the Gelman Schools hierarchical normal-normal model, a foundational example in Bayesian statistics for estimating treatment effects across multiple groups.…

HMC/NUTS Tuning and Diagnostics

Why centered models diverge and non-centered models don’t — a hands-on HMC exploration.

Explores HMC/NUTS sampling tuning through hierarchical Bayesian models, demonstrating how parametrization, step-size adaptation, and leapfrog length affect sampling…

Data Augmentation

Iterative sampling via hidden variables using the Tanner-Wong algorithm and Gibbs.

Introduces data augmentation as iterative sampling via hidden variables. Contrasts the stochastic Tanner-Wong version with Gibbs sampling, showing how augmenting with latent…

Exploring Hamiltonian Monte Carlo

From typical sets to leapfrog integration: a detailed exploration of HMC and NUTS.

Detailed exploration of Hamiltonian Monte Carlo following Neal and Betancourt. Covers typical set geometry, gradient-based smooth exploration, leapfrog integration, and the…

A Tetchy Gibbs Sampler

Gibbs sampling on a bimodal, highly-correlated posterior with lots of autocorrelation.

Demonstrates Gibbs sampling on a bimodal, highly-correlated posterior distribution. Shows that conditional posteriors are Gaussian despite an ugly joint form, and…

Gibbs Sampling with Conjugate Conditionals

Exploiting conjugate prior pairs for efficient Gibbs sampling.

Demonstrates Gibbs sampling that exploits conjugate prior pairs like Beta-Binomial. Shows how conditional posteriors simplify to known distributions, making each Gibbs step…

The Metropolis-Hastings Algorithm

Sampling from complex distributions with asymmetric proposals and limited support.

Extends the basic Metropolis algorithm to handle asymmetric proposal distributions and limited support domains. Demonstrates MH with a Weibull-like distribution using a…

Metropolis and Support Mismatch

Why combining rejection with Metropolis sampling wastes efficiency.

Demonstrates the efficiency loss when combining rejection sampling with Metropolis by using a proposal distribution with larger support than the target. Shows that a…

Imputation and Convergence

A coal-mine disaster switchpoint model for convergence testing and posterior predictive checks.

Uses a coal-mine disaster switchpoint model to illustrate MCMC convergence testing and posterior predictive imputation. Detects a regulatory change around 1900 with a…

Discrete MCMC

Metropolis sampling from discrete distributions using proposal matrices.

Demonstrates how to use Metropolis MCMC to sample from discrete distributions. Shows that MCMC handles discrete state spaces using proposal matrices that satisfy detailed…

Introduction to Gibbs Sampling

MCMC via alternating conditional sampling from a complex joint distribution.

Gibbs sampling enables sampling from complex joint distributions by alternately sampling from univariate conditionals. Demonstrates the method through a 2D example, deriving…

Markov Chains and MCMC

Stationarity, reversibility, and the mathematical foundation for Metropolis sampling.

Introduces Markov chains as the mathematical foundation for MCMC sampling. Covers stationarity, ergodicity, detailed balance, and the Metropolis algorithm, with a practical…

From Annealing to Metropolis

Understanding detailed balance and the Metropolis algorithm for sampling probability distributions.

Explores the Metropolis algorithm as a method for sampling from probability distributions. Connects detailed balance from simulated annealing to the Metropolis-Hastings…

Hierarchical Bayesian Models

From partial pooling to shrinkage estimation with the rat tumors example.

Introduces hierarchical Bayesian models as a framework for estimating parameters across related units by borrowing statistical strength from collective data. Demonstrates…

The EM Algorithm

Inferring parameters from incomplete data through iterative expectation and maximization.

Introduces the Expectation-Maximization algorithm for maximum likelihood estimation with missing or latent variables. Starts with estimating bivariate normal parameters from…

Generative vs Discriminative Models

Understanding discriminative boundaries vs. generative class models through logistic regression and LDA.

Explores Empirical Risk Minimization versus Bayesian learning, comparing discriminative and generative models. Uses a height/weight classification task to contrast logistic…

Mixture Models and Types of Learning

From generative classifiers to clustering: supervised, unsupervised, and semi-supervised approaches.

Covers the three fundamental types of machine learning through the lens of Gaussian Mixture Models. Demonstrates how GDA works in supervised settings, contrasts with…

Utility, Risk, and Decision Theory

From utility functions to Bayes actions: applying decision theory to regression and model comparison.

decision-theory

Covers decision theory fundamentals, explaining how to formulate and solve optimal decision problems using Bayesian methods. Demonstrates through a stock market prediction…

Regression in PyTorch

Building and training multi-layer perceptrons as universal function approximators.

neural-networks

Demonstrates the Universal Approximation Theorem by implementing MLPs in PyTorch to fit complex 1D functions. Trains neural networks with ReLU and tanh activations using…

Multi-Layer Perceptron for Classification

Training neural networks to classify synthetic data with decision boundaries.

neural-networks

Implements multi-layer perceptrons for classification tasks using PyTorch. Generates synthetic datasets and trains networks to learn non-linear decision boundaries…

How Sigmoids Combine

Visualizing neural network function approximation through hidden layer activations.

neural-networks

Explores how sigmoid activation functions combine in MLPs to create increasingly complex function approximations. Trains neural networks with varying architectures —…

Gradient Descent and SGD

Optimization by following the slope downhill.

Introduces gradient descent for optimization, starting with linear regression as a worked example. Covers batch gradient descent, stochastic gradient descent, mini-batch…

Logistic Regression and Backpropagation

From MLE classification to computing gradients through layers.

Develops logistic regression as a maximum likelihood classification model, then introduces backpropagation as the algorithm for computing gradients through layered…

Bayesian Regression

Putting priors on regression coefficients and updating with data.

Formulates linear regression in the Bayesian framework with conjugate normal priors on the coefficients. Derives the posterior distribution over weights, shows how the…

From the Normal Model to Regression

Building a Bayesian regression from Kung San census data.

Walks through the progression from a normal model to Bayesian linear regression using Howell’s Kung San height and weight data. Implements grid approximation and PyMC3 MCMC…

Lab: The Beta-Binomial Globe Model

Grid approximation, quadratic approximation, and MCMC in practice.

Hands-on lab working through Bayesian inference on the globe-tossing model. Implements grid approximation, Laplace (quadratic) approximation, conjugate prior updates, and…

Importance Sampling

Sample where it matters most to compute integrals efficiently.

Introduces importance sampling as a method for computing expectations and integrals by reweighting samples from a proposal distribution. Shows how choosing a good proposal…

The Inverse Transform

Turning uniform random numbers into any distribution you want.

Introduces the inverse transform method for generating samples from arbitrary distributions using uniform random variables and the inverse CDF. Includes the Box-Muller…

The Normal Model

Conjugate priors for the workhorse of statistics.

Derives the normal-normal conjugate model for Bayesian inference on the mean of a Gaussian with known variance. Shows how the posterior interpolates between prior belief and…

Rejection Sampling

Accept or reject: a simple algorithm for hard distributions.

Presents von Neumann’s rejection sampling algorithm for drawing samples from distributions with known functional form. Covers the basic method and an enhanced version that…

Bayesian Statistics

Treating parameters as random variables changes everything.

Introduces Bayesian inference by contrasting it with the frequentist approach. Develops the normal-normal conjugate model, derives the MAP estimate, and shows how MCMC…

The Beta-Binomial Globe Model

Conjugate priors, Bayesian updating, and decision theory on a globe toss.

Develops the beta-binomial conjugate model for estimating the fraction of water on a globe. Covers prior selection, sequential Bayesian updating, loss functions, decision…

Sufficient Statistics and Exchangeability

When less data tells you everything, and order doesn’t matter.

Connects the exponential family to conjugate priors through sufficient statistics. Introduces exchangeability and de Finetti’s theorem, then develops the Poisson-Gamma…

Entropy and Maximum Entropy

Quantifying uncertainty and the distributions it favors.

information-theory

Introduces information entropy as a measure of uncertainty and derives maximum entropy distributions. Shows how constraints on moments lead naturally to the exponential…

Divergence and Deviance

Measuring how far your model is from the truth.

information-theory

Develops KL divergence as a distance between distributions, connects it to cross-entropy and log-likelihood, and introduces deviance as a practical measure for model…

The Significance and Size of Effects

When a drug works, how much does it matter?

Explores significance testing and effect size using a drug vs. placebo experiment. Demonstrates permutation tests, bootstrap confidence intervals, and Cohen’s d as…

Convexity and Jensen’s Inequality

Why the average of a function isn’t the function of the average.

Introduces convex functions and proves Jensen’s inequality. Shows how convexity underpins key results in information theory and probabilistic modeling.

Understanding AIC

An information-theoretic shortcut for model selection.

information-theory

Explains the Akaike Information Criterion from first principles: how deviance, effective parameters, and information theory combine to estimate out-of-sample predictive…

Learning Without Noise

What happens when you fit a model to perfect data.

Introduces the learning problem through a noiseless example: fitting polynomial models to data, measuring in-sample vs out-of-sample error, and understanding how hypothesis…

Learning With Noise

Bias, variance, and the tradeoff that haunts every model.

Extends the learning framework to noisy data, where stochastic noise forces a tradeoff between model complexity and generalization. Derives the bias-variance decomposition…

Regularization

Taming complexity by penalizing parameters.

Introduces regularization as an alternative to model selection for controlling overfitting. Compares Ridge and Lasso regression, showing how a penalty term on coefficients…

Learning Bounds and the Test Set

How to honestly evaluate what your model has learned.

Formalizes out-of-sample error estimation using Hoeffding’s inequality and the train/test split. Walks through scikit-learn’s data pipeline for polynomial regression…

Validation and Cross-Validation

Why one split is never enough.

Explains why a separate validation set is needed to choose model complexity, and how cross-validation averages over multiple splits to give a more robust estimate of…

Maximum Likelihood Estimation

Find the parameters that make your data most probable.

Develops maximum likelihood estimation from first principles, deriving MLEs for the exponential, linear regression, and logistic regression models. Shows how MLE connects…

Basic Monte Carlo

Let randomness do the heavy lifting.

Introduces Monte Carlo methods by estimating the area of a unit circle with random points. Covers the hit-or-miss method and rejection sampling as building blocks for Monte…

Expectations and the Law of Large Numbers

What you expect is what you get — eventually.

Defines expected values, the Law of the Unconscious Statistician (LOTUS), and variance for discrete and continuous distributions. Demonstrates the Law of Large Numbers and…

Monte Carlo Integration

When calculus is hard, sample instead.

Formalizes Monte Carlo integration using uniform random samples and LOTUS, then extends to multidimensional integrals. Uses the Central Limit Theorem to estimate integration…

Sampling and the Central Limit Theorem

Why everything looks normal in the limit.

Builds sampling distributions from repeated coin-flip experiments, showing how the sample mean and its variance behave as sample size grows. Derives the Central Limit…

Distributions Example: Elections

Simulating a presidential election with coin flips.

Simulates the 2012 US presidential election using Bernoulli coin flips for each state based on PredictWise probabilities. Introduces the Binomial distribution, the CLT’s…

Probability

From coin flips to Bayes’ theorem.

Three ways to think about probability: symmetry, models, and long-run frequency, illustrated with coin-flip simulations. Covers the rules of probability, random variables…