Probabilistic Machine Learning for Finance and Investing

Chapter 2. Analyzing and Quantifying Uncertainty

There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don’t know. But there are also unknown unknowns. There are things we don’t know we don’t know.

—Donald Rumsfeld, Former US Secretary of Defense

The Monty Hall problem, a famous probability brainteaser, is an entertaining way to explore the complex and profound nature of uncertainty that we face in our personal and professional lives. More pertinently, the solution to the Monty Hall problem is essentially a betting strategy. Throughout this chapter, we use it to explain many key concepts and pitfalls in probability, statistics, machine learning, game theory, finance, and investing.

In this chapter, we will solve the apparent paradox of the Monty Hall problem by developing two analytical solutions of differing complexity using the fundamental rules of probability theory. We also derive the inverse probability rule that is pivotal to probabilistic machine learning. Later in this chapter, we confirm these analytical solutions with a Monte Carlo simulation (MCS), one of the most powerful numerical techniques that is used extensively in finance and investing.

There are three types of uncertainty embedded in the Monty Hall problem that we examine. Aleatory uncertainty is the randomness in the observed data (the known knowns). Epistemic uncertainty arises from the lack of knowledge about the underlying ...