Sampling Methods for Uncertainty Quantification

CAP 6938 (Spring 2026)

Department of Computer Science
University of Central Florida
Orlando, FL, USA

Instructor information

Instructor: Ali Siahkoohi

• Email: alisk@ucf.edu
• Office hours: Wednesdays 11–12pm.

See webcourses for the office hour location.

Description

This special topics course introduces modern computational sampling methods for uncertainty quantification in scientific computing and engineering applications. Topics include mathematical and computational principles of classical and contemporary sampling approaches with emphasis on understanding the inner workings of deep generative models, variational inference, and simulation-based inference methods. Students implement sampling algorithms and apply them to uncertainty quantification problems through hands-on programming assignments.

Why take this course?

AI models have recently driven significant advances in various science and engineering domains, yet critical reliability concerns remain: models produce unreliable predictions, lack quantifiable safety guarantees, and their theoretical foundations remain opaque to many practitioners. Current approaches focused on scaling and post-hoc validation cannot systematically address these issues. In this course, you will learn modern uncertainty quantification techniques that not only can be applied widely in various science and engineering domains but also hold the key to designing more reliable AI models. You will go beyond treating generative models, which are key components of modern sampling and uncertainty quantification techniques, as black boxes, understanding and implementing the core mathematical principles that make them work. Through five hands-on programming assignments, you will build these algorithms from the ground up, gaining the deep foundational expertise needed to adapt, debug, and innovate in this rapidly evolving field.

Prerequisites

Students should have background in probability theory, linear algebra, and programming. Familiarity with deep learning basics is recommended but not required.

Textbooks

There is no required textbook for the class. A list of recommended papers will be provided during the course.

Grading

• Programming assignments 60% (5 assignments × 12% each)
• Paper presentation 25%
• Lecture scribing 15% (number of scribing assignments will depend on total class enrollment)
• There will be 2% extra credit for submitting teaching evaluation, if more than 80% of students submitted the evaluation

Course schedule

Course material will be posted on the website before each lecture. The instructor reserves the right to alter this schedule as needed.

Week	Module	Lecture Topics	Resources	Assignments/Notes
Week 1-2	Module 1: Foundations	Uncertainty quantification in scientific computing Review of probability Monte Carlo integration and rejection sampling	What is UQ Rethinking Uncertainty Information Adequacy Illusion You’re Confidently Wrong Confidence Misleads Supervised OOD Detection Mathematics for Machine Learning	MLK Day - No Class on Jan 19 Scribing instructions and template (pdf, zip) Notes on Probability Notes on Linear Algebra Notes on Norms Notes on Calculus Notes on Linear/Quadratic Gradients Notes on Max and Argmax Rejection Sampling Notes Assignment instructions Assignment 1 a1.zip
Week 3-4	Module 2: MCMC	Importance sampling and introduction to MCMC: Metropolis-Hastings Gibbs sampling and convergence diagnostics Gradient-based MCMC (Langevin dynamics and MALA) Hamiltonian Monte Carlo fundamentals	Handbook of Markov Chain Monte Carlo Understanding the Metropolis-Hastings Algorithm Explaining the Gibbs Sampler Markov Chain Monte Carlo Lecture Notes MCMC Slides On Thinning of Chains in MCMC Practical Markov Chain Monte Carlo Stochastic Gradient Langevin Dynamics Preconditioned SGLD Hamiltonian Monte Carlo A Conceptual Introduction to Hamiltonian Monte Carlo MCMC Interactive Gallery	Assignment 1 due Assignment 2 a2.zip
Week 5-7	Module 3: Variational Inference	Introduction to variational inference and KL divergence Reparameterization and the ELBO Stochastic, non-amortized, and amortized variational inference Particle-based variational inference and Stein’s identity Stein Variational Gradient Descent VI vs MCMC: tradeoffs, diagnostics, and hybrid methods	VI Review for Statisticians Advances in VI Auto-Encoding Variational Bayes Monte Carlo Gradient Estimation Reliable Amortized VI Stein Variational Gradient Descent VI with Normalizing Flows Yes, but Did It Work? Evaluating VI Pareto Smoothed Importance Sampling Importance Weighted Autoencoders Practical Posterior Error Bounds from VI Objectives Transport Map Accelerated MCMC	Assignment 2 due Assignment 3 a3.zip
Week 8-10	Module 4: Deep Generative Models	Normalizing flows: theory and architectures Normalizing flows continued Variational autoencoders and amortized inference Score matching and diffusion models Score-based models and flow matching Flow matching continued	Normalizing Flows Survey Real-NVP Masked Autoregressive Flow Universal Approximation for Coupling Flows Householder Flow HINT: Hierarchical Invertible Neural Transport InvertibleNetworks.jl (JOSS) InvertibleNetworks.jl (GitHub) STARFlow Introduction to Variational Autoencoders Denoising Diffusion Probabilistic Models Score Matching with Langevin Dynamics Score-Based Generative Modeling through SDEs Diffusion Objectives as the ELBO Flow Matching for Generative Modeling Stochastic Interpolants Diffusion Meets Flow Matching Improving and Generalizing Flow-Based Generative Models	Spring Break - No Class (Mar 16-20) Assignment 3 due Assignment 4 a4.zip
Week 11-12	Module 5: Simulation-Based Inference	Likelihood-free inference and Approximate Bayesian Computation (ABC) Neural approaches to SBI: NPE, NLE, and NRE Sequential simulation-based inference	The Frontier of Simulation-Based Inference Fast Epsilon-Free Inference Automatic Posterior Transformation Sequential Neural Likelihood Likelihood-Free MCMC with Amortized Ratios Mining Gold from Simulators Simulation-Based Calibration Benchmarking SBI sbi: A Toolkit for SBI Truncated Proposals for Scalable SBI Classifier Two-Sample Tests BayesFlow Information Maximising Neural Networks Neural Approximate Sufficient Statistics Optimal Data Compression for SBI Likelihood Ratios with Calibrated Classifiers ASPIRE: Iterative Amortized Posterior Inference Contrastive Learning for Likelihood-free Inference Mixture Density Networks Flexible Statistical Inference for Mechanistic Models A Kernel Two-Sample Test (MMD)	Assignment 4 due Assignment 5 a5.zip
Week 13-15	Student Paper Presentations	Rethinking aleatoric and epistemic uncertainty Convex potential flows Neural spline flows Black box variational inference Scaling rectified flow transformers Elucidating the design space of diffusion models Energy-based transformers Flow matching on general geometries Adaptive symmetrization of the KL divergence Normalizing flows are capable generative models Unifying summary statistics for ABC Compositional score modeling for SBI Contrastive neural ratio estimation		Assignment 5 due Grading rubric List of papers