1. Probabilistic Reasoning

This introductory section explains how probability can be used to model uncertainty in engineering analysis. Instead of treating uncertain inputs, observations or model results as single fixed values, we describe what we know and what we do not know using probabilities.

The central idea is that a probabilistic model represents how we believe data are generated. Once data are observed, we can use statistical inference to update what we believe about the unknown state of reality.

"There are known unknowns, and there are unknown unknowns."

— Donald Rumsfeld, former U.S. Secretary of Defense

Tools used in this course

The course uses a small set of freely available tools for probabilistic modelling, statistical computing and technical documentation.

• R is an open-source programming language widely used for statistical computing, data analysis and graphics.
• The Tidyverse is a collection of R packages that makes data handling, transformation and plotting more convenient.
• Stan is a probabilistic programming language used to define and solve more complex statistical models.
• Quarto and RStudio are used to create reproducible documents and course material.

These tools allow us to move from simple probability calculations to numerical solutions of more realistic uncertainty quantification problems.

Conditional probability and inference

A key point in probabilistic reasoning is that the direction of conditioning matters. The probability of observing some data if an assumption is true is not the same as the probability that the assumption is true after observing the data.

• P(data | assumption) asks: if this assumption were true, how likely would this observation be?
• P(assumption | data) asks: after seeing this observation, how plausible is the assumption?

This distinction is the reason why Bayes' Theorem is so important. It provides a way to combine prior knowledge, observed data and a probabilistic model to update our belief about an unknown quantity.

For example, a surprising observation may be unlikely under a simple null hypothesis. However, that does not automatically mean the alternative explanation is likely. We also need to consider base rates, prior knowledge and the other ways the observation could have occurred.