Part VII: Probability & Confirmation

The core problem can be stated simply: What is the relationship between evidence and hypothesis? When a scientist performs an experiment and obtains a result, in what sense does that result “support” or “confirm” the theory being tested? The answer turns out to be far more subtle than common sense would suggest.

Consider a simple example. A theory predicts that a certain chemical reaction will produce a blue precipitate. You perform the experiment and observe a blue precipitate. Does this confirm the theory? Intuitively, yes — but notice that many different theories might predict a blue precipitate. The observation is logically consistent with infinitely many hypotheses. The question is how to quantify the degree to which the evidence favours one hypothesis over another.

“Probability is the very guide of life.”— Joseph Butler, The Analogy of Religion (1736)

The Bayesian answer, which has come to dominate philosophy of science, is that confirmation should be understood in terms of conditional probability. Evidence $E$ confirms hypothesis $H$ just in case the probability of $H$ given $E$ is greater than the prior probability of $H$:

This deceptively simple formula, derived from Bayes’ theorem, gives rise to a rich and powerful theory of confirmation — one that can handle the ravens paradox, the grue problem, and many other traditional puzzles. But it also raises new problems: Where do the prior probabilities come from? How do we handle old evidence? Can Bayesian agents with different priors ever converge?

These three chapters trace the arc from the abstract mathematics of probability to the concrete practice of statistical inference in the sciences, showing how philosophical assumptions permeate every stage of the scientific process.

The mathematical theory of probability emerged in the 17th century from correspondence between Pascal and Fermat about games of chance. But its application to scientific reasoning has a longer pedigree. Thomas Bayes, an 18th-century Presbyterian minister, posthumously published the theorem that bears his name in 1763. Pierre-Simon Laplace independently developed and greatly extended Bayesian methods, applying them to astronomical problems and articulating the “classical” interpretation of probability as a ratio of favourable to possible outcomes.

The 20th century saw an explosion of work on probability and confirmation. Rudolf Carnap attempted to develop a purely logical theory of confirmation, while Karl Popper rejected confirmation entirely in favour of falsification. The Bayesian revival, led by figures like Frank Ramsey, Bruno de Finetti, L.J. Savage, and later philosophers like Richard Jeffrey, Jon Dorling, and Colin Howson, restored probabilistic confirmation to the centre of the field.

Meanwhile, the practice of statistics in science developed largely independently of philosophical concerns. R.A. Fisher, Jerzy Neyman, and Egon Pearson developed the frequentist methods that dominated 20th-century science. The tension between frequentist and Bayesian approaches — and the philosophical assumptions underlying each — remains one of the most important and practically consequential debates in the philosophy of science.

Part VII makes extensive use of the probability calculus. The three axioms of probability (Kolmogorov axioms) are:

From these axioms we derive conditional probability and Bayes’ theorem:

This single equation is the engine of Bayesian confirmation theory. Understanding its components — the prior $P(H)$, the likelihood $P(E|H)$, the marginal likelihood $P(E)$, and the posterior $P(H|E)$ — is essential for the chapters that follow.