I recently had the pleasure of reading Bernoulli’s Fallacy: Statistical Illogic and the Crisis of Modern Science. For me, it felt like a professional therapy session, meticulously unpacking how the misapplication of Bernoulli’s famous law of large numbers gave rise to frequentism, that is, an illogical approach to probability.
This book serves as a natural successor to Edwin Thompson Jaynes’ Probability Theory: The Logic of Science. Like Jaynes, Aubrey Clayton argues that probability is not a property of reality but rather a measure of our degree of belief in a hypothesis or proposition; therefore Clayton views probability as epistemological rather than ontological. In this view, probability theory becomes an extension of logic, designed to handle uncertainty with rigour and clarity.
Through the exceptional writing style and deep historical insights, Clayton takes readers on a journey through the history of statistics. Clayton reveals how the obsession with an objective interpretation of probability led to illogical results and disastrous decisions across fields like law. By exploring the contributions of influential figures such as Francis Galton, Karl Pearson, and Ronald Fisher, Clayton demonstrates how they perpetuated the Bernoulli’s fallacy by conflating frequencies with probabilities.
One of the important arguments in the book is the distinction between sampling probabilities and inferential probabilities. Clayton emphasises that the former is insufficient for statistical inference and reasoning. This misalignment, Clayton argues, has contributed to major issues in modern science, including the replication crisis, that is, a phenomenon where many published research findings cannot be replicated.
The solution, Clayton suggests, lies in embracing Bayesian statistics, which allows us to derive inferential probabilities by combining prior knowledge with observed data. This approach not only provides a logical foundation for inference but also addresses many of the shortcomings of frequentist methods.
I thoroughly enjoyed the book’s subject matter, Clayton’s engaging writing style, and candid language. Bernoulli’s Fallacy is a must-read for anyone who works with probability and statistics in any capacity, offering profound insights into the philosophical underpinnings of the field.