This course introduced the probabilistic foundations needed to reason about uncertainty in engineering analysis. Starting from conditional probability and Bayes' theorem, the course showed why the direction of conditioning matters, why base rates can dominate apparently strong evidence, and how the marginalisation rule is used to account for the different ways an observation can occur.
The vampire example demonstrated the difference between an analytical Bayesian calculation and a numerical Monte Carlo solution. The NAFEMS Stochastics challenge problem then connected these ideas to engineering reliability.
The final sections showed how probabilistic programming can be used to infer unknown distribution parameters from limited data and estimate the probability of failure. The key message is that the result is not just a single probability of failure: the uncertainty in that probability may itself be important for engineering decision-making.
Congratulations, you have completed the "Probabilistic Foundations of Uncertainty Quantification and Machine Learning" on-demand course!
This session supports several competency statements in the NAFEMS PSE Probabilistic Analysis technical area. In particular, the material relates to the following topics:
PROB-v2-1 Describe why uncertainty matters in engineering analysis.
PROB-v2-3 Describe some of the quantities you might want to calculate using a probabilistic analysis.
PROB-v2-4 Describe some reasons that variation in a model input might occur for a problem relevant to you.
PROB-v2-5 Discuss how you might mathematically describe a quantity that is a random variable.
PROB-v2-9 Explain why you might carry out a sensitivity analysis.
PROB-v2-16 Describe the difference between Frequentist and Bayesian approaches.
PROB-v2-17 Describe measurement uncertainties and their characterisation and how to use them in a stochastic simulation.
PROB-v2-24 Describe the impact of model form uncertainty on predictions.
Review Competencies in the Probabilistic Analysis Technical Area
NAFEMS Stochastics Working Group, Free Access
This short introductory publication explains why uncertainty quantification is needed in engineering simulation. It is a good next step after this course because it places the probability concepts in a broader engineering context: moving from deterministic reserve margins and worst-case assumptions towards quantified variation, robustness and reliability.
NAFEMS Publication by B. Thacker, e-Library download
This publication is aimed at engineers and project managers who are familiar with finite element analysis but new to probabilistic finite element analysis. It explains the practical motivation for combining probability theory with finite element methods, including how probabilistic inputs are defined, what outputs are produced, and how results can be interpreted in engineering applications.
NAFEMS Publication by M. Fortier, e-Library download
This resource is useful for learners who want to understand the organisational and design-process value of stochastic methods. It discusses how stochastic analysis can support robust design, help explore larger regions of the design space, and provide a clearer basis for engineering and management decisions under uncertainty.
Richard McElreath, CRC Press
This is the source of the vampire example used in the course. It is highly recommended for learners who want a deeper conceptual and hands-on introduction to Bayesian modelling.
Stan Development Team
The Stan documentation provides explanations, examples and references for probabilistic programming, Hamiltonian Monte Carlo and the No-U-Turn Sampler. It is the natural next step for learners who want to implement their own models after completing the course.
You can use the PSE Competency Tracker to plan further development against the relevant Probabilistic Analysis competency statements.
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