There is a formula, discovered by an 18th-century English minister named Thomas Bayes and published posthumously in 1763, that tells you how to update a belief in the light of new evidence. It is not long: P(H | E) = P(E | H) × P(H) / P(E). The formula is, technically, a theorem of probability theory — something that follows from the axioms by a few lines of algebra. But Bayes' theorem is more than a theorem. It is a prescription: when you learn E, this is how your belief in H should change. That normative reading — not just "this follows mathematically" but "this is how a rational agent updates" — is what makes Bayesianism a philosophy as much as a method.
The pieces of the formula are worth naming carefully. P(H) is your prior probability — how likely you thought H was before seeing the evidence. P(E | H) is the likelihood — how probable the evidence would be if H were true. P(E) is the marginal probability of the evidence — how probable it is under all hypotheses. P(H | E) is the posterior — your updated belief after seeing the evidence. Posterior equals prior times likelihood, normalized.
An example helps. Suppose you take a medical test for a rare disease that affects 1 in 1000 people. The test is 99% accurate in both directions. You test positive. How worried should you be? Intuition says: the test is 99% accurate, so there's a 99% chance you have the disease. Bayes' theorem says otherwise. Run the formula: posterior ≈ 9%. The disease is still rare even after a positive test, because the prior was so low. Most positive results come from the 1% false positive rate applied to the 999-in-a-thousand healthy people, which swamps the true positives from the rare sick ones.
This example is not just a calculation. It reveals a systematic way that human intuition fails. People are bad at incorporating base rates — the prior — into their judgments. We focus on the likelihood ("the test is 99% accurate") and neglect the prior ("but the disease is very rare"). Psychologists Tversky and Kahneman documented this as "base rate neglect," one of the most robust biases in human cognition. Bayes' theorem is, among other things, a correction to this bias.
The deeper philosophical content comes from the question of what probabilities are. In the frequentist interpretation, probability is a fact about the world: the long-run frequency of an event in repeated trials. On this view, it makes no sense to assign a probability to a one-time event or a hypothesis that is either true or false. The frequentist cannot say "the probability that this defendant is guilty is 0.85." Either he is or he isn't.
The Bayesian interpretation says probability is a degree of belief, a measure of an agent's uncertainty given their information. "P(defendant is guilty) = 0.85" is a coherent statement — it describes a state of informed uncertainty. And Bayes' theorem tells you how that uncertainty should change when you get new information. Prior plus evidence yields posterior. Posteriors become priors for the next round. Rational belief is a continuous update process.
What makes Bayesianism powerful is that it handles uncertainty about things that aren't repeatable. Science involves forming beliefs about theories — and theories are either true or false, they don't have frequencies. Frequentist statistics struggles here; it can tell you "if the null hypothesis were true, results this extreme would occur 3% of the time," but it cannot tell you the probability that the null hypothesis is true. Bayesian inference can: it takes your prior over hypotheses and updates it based on the data.
The objection to Bayesianism is the subjectivity of the prior. Two rational agents can have different priors and, given the same evidence, reach different posteriors. The Bayesian response is that priors are not arbitrary; they should represent your actual state of information before the evidence. And with enough evidence, priors wash out — agents who start with very different priors will converge as evidence accumulates. The subjectivity is initial, not permanent.
I find myself drawn to the Bayesian picture not because it resolves all questions about probability but because it takes seriously the thing that actually matters: what you should believe, given what you know, and how you should revise those beliefs as you learn more. The alternative — refusing to commit to any probability until you have a large sample of repeated trials — is often not available. You have to act. You have to decide. Bayes' theorem is a tool for doing that rationally, for encoding your uncertainty precisely enough that it can be updated by evidence. It does not eliminate subjectivity. It disciplines it.