Bayesian Confidence and Credence, Introduced

Everything so far has treated belief as something a person either has or lacks. That treatment is a simplification, and an obvious one. You believe it will rain this afternoon, and you believe the sun will rise, and you do not hold these with the same grip. The first is a leaning and the second is not, and a theory that assigns both the single label believes has thrown away the difference.

This article introduces the framework that keeps it. The shift is not a refinement of what came before. It changes what the objects of epistemic evaluation are, what the norms governing them look like, and what the problem of induction turns into.

Belief by degrees

The numerical representation needs defending, since nobody experiences their confidence as a decimal. The standard defence is behavioural and runs through betting. If you are willing to pay up to seventy pence for a ticket that pays one pound if it rains, and unwilling to pay more, your betting behaviour reveals a threshold, and that threshold is what the number records. The claim is not that you consciously entertain the figure but that your dispositions determine one.

Frank Ramsey made this argument in 1926 and Bruno de Finetti independently in 1931, and both drew the same striking conclusion from it: the laws of probability are not an optional extra imposed on degrees of belief but a consequence of not being exploitable.

The requirement the argument establishes is called probabilism. Credences must be non-negative, certainty in a logical truth must be 1, and the credence in a disjunction of incompatible propositions must be the sum of the credences in each. The advanced tier examines whether the argument really establishes this, since the betting interpretation has well-known weaknesses and there are non-pragmatic arguments that reach the same conclusion by considering accuracy instead.

Conditioning

Conditional probability is what lets the framework represent learning. Your unconditional confidence that a patient has a disease is one thing; your confidence on the supposition that their test came back positive is another, and the second is what you want when the result arrives.

The rule connecting them is conditionalisation: on learning $e$ and nothing stronger, your new credence in $h$ should equal your old conditional credence in $h$ given $e$. The posterior is the old conditional probability, promoted.

The theorem

The theorem is a trivial consequence of the definition of conditional probability, and its importance is entirely in what it makes visible. It says that how much a piece of evidence should move you depends on three things, and one of them is routinely forgotten.

The likelihood $P(e \mid h)$ measures how expected the evidence is if the hypothesis is true. The prior $P(h)$ measures how plausible the hypothesis was beforehand. And the marginal $P(e)$ measures how expected the evidence was overall, whether or not the hypothesis holds. Evidence confirms a hypothesis to the extent that it is more expected under that hypothesis than it is in general.

The standard illustration is a medical test. A disease affects one person in a thousand. The test detects it in ninety-nine per cent of those who have it and returns a false positive for five per cent of those who do not. Your test is positive. What is the probability that you have the disease?

Work it through with a population of a hundred thousand. A hundred people have the disease, and ninety-nine of them test positive. Of the remaining 99,900, five per cent test positive, which is 4,995 people. So 5,094 positive results, of which 99 are genuine. The probability that a positive result is a true one is about 1.9 per cent.

The answer is famously counterintuitive, and what generates the intuition of near-certainty is neglect of the prior. The test is accurate. It is just that a rare condition starts so far behind that even strong evidence leaves it improbable.

What the framework does to the old questions

The shift to credences changes the shape of nearly everything in the preceding articles.

Justification becomes coherence plus a rule. There is no regress, because credences are not supported by other credences in a chain; they are constrained all at once by the probability axioms, and revised by a rule. The trilemma has no purchase, which is either an elegant dissolution or an evasion depending on where the priors come from.

Induction becomes a question about priors. Given a prior distribution, the theorem determines exactly how observed cases should raise confidence in a generalisation, and the calculation is uncontroversial. What the framework cannot do is tell you which priors to start with, and Hume's problem reappears precisely there: a prior that assigns high probability to a grue-like hypothesis is coherent, updates correctly, and predicts differently. The convergence theorems establish that agents with different priors eventually agree given enough evidence, which is a real result and not a solution, since eventually can be arbitrarily long and the theorems assume the evidence keeps coming.

The relationship between credence and ordinary belief becomes a problem in its own right. The natural thought is that believing something is having a credence above some threshold, and the lottery paradox blocks it. With a million tickets, rationality permits high confidence that each particular ticket will lose, so any threshold below one licenses belief that ticket one loses, and that ticket two loses, and so on for all million. If belief is closed under conjunction, you believe no ticket wins, which you know to be false. Either belief is not a credence threshold, or belief is not closed under conjunction, and neither option is comfortable.

The scope of the programme

Bayesian epistemology is now the dominant framework in formal epistemology, in statistics, and in the philosophy of science, and it is not a general replacement for the epistemology of the preceding articles. It says nothing about knowledge. Gettier cases are not addressed, because the framework has no account of when a high credence amounts to knowing. Safety and sensitivity are not probabilistic conditions, though they have probabilistic cousins.

What it supplies is a theory of rational confidence and its revision, stated precisely enough to prove things about. The two projects are not competitors, and the interesting question, which the advanced tier takes up, is how they fit together.

Next

Two social sources of belief that the framework so far has treated as though they were ordinary evidence: what other people tell us, and what to do when someone just as careful disagrees.