Bayesianism in Scientific Confirmation — Epoche C1
For decades, a prevailing sentiment within the philosophy of science, particularly among frequentist statisticians and Popperian falsificationists, has cast a shadow of doubt over the Bayesian approach to statistical inference. The core of the critique lies in the perceived subjectivity inherent in the selection of prior probabilities. Critics argue that if scientific knowledge is to be objective and universally accessible, its foundational methods cannot rest on personal degrees of belief, which vary from one researcher to another. This essay sets out that critique in its strongest form, then argues that it fails — not because priors are harmless, but because their selection can be, and in mature scientific practice routinely is, constrained by evidence. The position defended here is sometimes presented as a recent discovery; it is in fact the accumulated result of half a century of work in what is now called the objective-Bayesian tradition, from Harold Jeffreys and E. T. Jaynes through Colin Howson, Peter Urbach, John Earman and Jon Williamson. The Machinery: What a Bayesian Actually Computes The argument cannot be followed without the formula at its centre, so we begin there. For any two events or propositions $A$ and $B$, the conditional probability of $A$ given $B$ is defined as $P(A \mid B) = P(A \cap B)/P(B)$: the share of the probability of $B$ that is also probability of $A$. Writing this definition twice, once in each direction, and eliminating the joint term $P(A \cap B)$ yields Bayes' theorem, a two-line consequence of the definition and not an additional assumption: $$P(H \mid E) = \frac{P(E \mid H)\, P(H)}{P(E)}.$$ Read with $H$ a hypothesis and $E$ a piece of evidence, each factor has a name and a role. $P(H)$ is the prior : how credible the hypothesis was before the evidence arrived. $P(E \mid H)$ is the likelihood : how strongly the hypothesis predicts the evidence. $P(E)$ is the expectedness of the evidence overall, computed by the law of total probability as $P(E) = P(E \mid H)P(H) + P(E \mid \neg H)P(\neg H)$ — a weighted average over the hypothesis being true or false. And $P(H \mid E)$ is the posterior : the credibility of the hypothesis after the evidence is taken into account. On the Bayesian account, evidence $E$ confirms $H$ exactly when the posterior exceeds the prior, and the size of the boost is given by dividing through: $$\frac{P(H \mid E)}{P(H)} = \frac{P(E \mid H)}{P(E)}.$$ This little identity does most of the work in what follows: a hypothesis gains credibility in proportion to how much better it predicted the evidence than the field of alternatives collectively did. The rival school, frequentism, refuses the first step. For a frequentist, probability is the long-run relative frequency of an outcome in repeated trials; a hypothesis is simply true or false, not the kind of thing that has a probability at all. Inference then proceeds not by updating beliefs but by controlling error rates — designing tests, in the tradition of Jerzy Neyman and Egon Pearson, that would rarely reject a true hypothesis or accept a false one. Karl Popper's falsificationism reaches a similar hostility by a different road: Popper argued that a strictly universal law, since it makes claims about infinitely many instances, must receive prior probability zero, and a glance at Bayes' theorem shows that a zero prior stays zero no matter what evidence arrives. Science, he concluded, cannot confirm theories at all; it can only fail to refute them, a status he called corroboration. Both schools converge on the same accusation: the prior $P(H)$ is a number the Bayesian must invent, and whatever is built on an invented number inherits its arbitrariness. The Thesis: Subjectivity as Disqualification Stated carefully, the traditional argument runs: prior probabilities represent an investigator's initial degrees of belief; degrees of belief legitimately differ between investigators looking at the same evidence; therefore two impeccable Bayesians can reach different posteriors from identical data; therefore Bayesian conclusions are facts about persons, not about the world, and are unsuitable for grounding objective scientific knowledge. The concern is most acute where findings must command public trust — medical research, climate science, the epidemic models that guided lockdown policy in 2020 — because in such settings the public argument turns precisely on whether the modellers' input assumptions smuggled in a preferred conclusion. The critique is not a caricature: the founding subjectivists conceded its premise. Frank Ramsey and Bruno de Finetti, who in the 1920s and 1930s showed that degrees of belief must obey the probability axioms on pain of accepting a 'Dutch book' — a set of bets guaranteeing a loss — insisted that the axioms constrain only the coherence of beliefs, not their content. Two coherent agents may start anywhere. The Antithesis: Evidence-Based Constraints on Priors The counter-argument has three layers, in ascending order of strength. First, priors wash out. A pair of theorems circumscribes how much the starting point can matter in the long run. The merging-of-opinions theorem of David Blackwell and Lester Dubins (1962) says that if two agents' priors agree on what is impossible — formally, each assigns probability zero only where the other does — then as shared evidence accumulates, their predictions about future observations converge, with probability one. Doob's earlier consistency theorem gives the single-agent version: under mild conditions, a Bayesian whose prior does not rule out the truth will concentrate her posterior on the true hypothesis as data accumulate. These results must be stated honestly: they are asymptotic, they require the truth not to have been assigned zero prior probability, and with small samples the prior genuinely matters. But they already blunt the disqualification argument, because they show that Bayesian disagreement is a diminishing function