This is the first installment of a series of writings talking about the difficulties of reasoning under uncertainty. Uncertainty is a fundamentally probabilistic concept; as such, our discussion must begin by examining the very foundations of probability itself. As with any concept as vague yet intuitive as probability, we must conduct our analysis at multiple levels:
We’ll start with the first point, surveying various different meanings of the word probability, before subdividing each meaning into a set of rigorous translations compatible with it; such a combination of a meaning and translation is known as an interpretation of probability. Finally, in order to cover the third point, we will draw out the implications of and paradoxes within each interpretation, as well as comparing their strengths and weaknesses relative to one another. As we’ll see, it turns out that even something as intuitive as probability turns out to be an extremely thorny concept upon further examination, with every interpretation being either limited to an extremely narrow domain or prone to internal contradiction; in short, we’ll discover why probability is difficult.
While I try to stay away from arguments via pure mathematics, seeking to translate them into conceptual terms, some parts are excessively technical. Some calculus, and a basic understanding of sets and set operations, is required to understand most of these.
Table of Contents
The Uncertainty Series
Part 1: Probability is Difficult
Part 2: Statistics is Difficult
Part 3: Causality is Difficult
Part 4: Modeling is Difficult
Part 5: Prediction is Difficult
Part 6: Experimentation is Difficult
Part 7: Science is Difficult
Epilogue: The Past, Present, and Future of Uncertainty
Appendix: Uncertain Appendices
Any introduction to probability will generally begin with a few classical examples — rolling die, flipping a coin, being dealt a certain hand from a deck, and so on. These intuition pumps allow us to examine the features of probability in the most basic cases. To introduce the main interpretations of probability we’ll cover in this essay, let’s go with a particularly contrived example of an uncertain die roll.
I videotape myself rolling a normal, six-sided die, and show this video to a small focus group of people, each of whom has a different idea of what probability means. As the die leaves my hand in the video, I pause it, and ask them what the probability that the die will land on an even number is.
Which justification do we take to be the most accurate? Obviously, the answer depends on what we mean by probability, but what exactly we mean can be very difficult to pin down. Many people have tried, coming up with many different answers, and we'll explore those in turn. Each of these "willing" participants has given a unique account: Carrie the Classical account, Barry the Bayesian account, Frank the Frequentist account, Paul the Propensity account.document.querySelectorAll('.notion-topbar').style.height = "0px"; document.querySelectorAll('.notion-frame').style.height = "auto";