Reliabilism in Bayesian Credence Updating — Epoche C1
From Beliefs to Credences: Two Frameworks Two independent apparatuses meet in this note, and each needs a plain introduction before their combination can be assessed. The first is reliabilism, a theory of epistemic justification developed most influentially by Alvin Goldman. Traditional epistemology analysed knowledge as justified true belief, until Edmund Gettier's famous 1963 counterexamples showed that a person can hold a justified true belief and still, intuitively, fail to know — because the truth of the belief arrived by luck rather than through anything creditable in how it was formed. Reliabilism responds by relocating justification from the believer's reasons to the believer's processes : a belief is justified, on Goldman's account, if and only if it was produced by a cognitive process that yields a high ratio of true to false beliefs across its normal range of use. Vision in good light and careful arithmetic are reliable processes; wishful thinking and reading tea leaves are not, and a belief formed by an unreliable process remains unjustified even on the occasions when it happens to be true. The second apparatus is the Bayesian model of partial belief. Rather than treating belief as all-or-nothing, Bayesians represent an agent's confidence in a proposition as a credence : a number between 0 (certain falsehood) and 1 (certain truth). Two norms govern the ideal Bayesian agent. Synchronically, her credences at any moment should obey the axioms of probability — a demand defended pragmatically by the Dutch book argument (an agent whose credences violate the axioms will accept a set of bets that guarantees her a loss) and non-pragmatically by James Joyce's 1998 accuracy argument, which proves that any non-probabilistic credence assignment is "dominated": there exists a probabilistic assignment that is closer to the truth however the world turns out. Diachronically, she should revise by conditionalisation : on learning evidence $E$, her new credence in a hypothesis $H$ should equal her old conditional credence $P(H \mid E)$, computed by Bayes' theorem, $P(H \mid E) = P(E \mid H)P(H)/P(E)$, where $P(H)$ is the prior (her credence before the evidence), $P(E \mid H)$ is the likelihood (how strongly the hypothesis predicts the evidence), and the result $P(H \mid E)$ is the posterior . The question of this note is what reliabilist justification looks like when the objects of assessment are not binary beliefs but these continually updated credences — and the thesis is that the assessment must attach to the updating process itself, not to the accuracy of its outputs, on pain of mistaking epistemic luck for justification. The Limits of Outcome-Focused Reliabilism in Bayesian Contexts The classical Chinese proverb, '千里之堤，潰於蟻穴' (a thousand- li dike collapses through an ant-hole) — a saying descended from the Legalist philosopher Han Feizi, who observed that a thousand- zhang embankment is breached through the burrows of insects — offers a precise analogy. A dike is not sound because it happens to be holding today; it is sound if its structure would withstand the range of floods it may meet. Likewise, a posterior credence is not justified because it happens to sit near the truth today; it is justified if the process that produced it would place credences well across the range of situations the agent may meet. For binary beliefs, reliabilism already says something like this. But credences introduce a subtlety: a single credence of 0.8 is not "true" or "false" in the way a belief is, so the truth-ratio definition of reliability does not carry over directly. The natural extension, developed in recent literature by Jeff Dunn and Weng Hong Tang, is calibration : a credence-forming process is reliable to the degree that, among all the propositions to which it assigns credence around $x$, the proportion that are true is close to $x$. A weather forecaster is well calibrated if it rains on about 70% of the days she announces "70% chance of rain". With this notion in hand, the thesis can be stated exactly: justification for a posterior credence requires that the whole pipeline — prior formation, likelihood assessment, and the update rule — be well calibrated as a process, not that the final number land near the truth on this occasion. Prioritising Process Reliability over Outcome Accuracy To see why each stage matters independently, consider them in order. Reliability of prior elicitation. Bayes' theorem is a valve, not a source: it transmits whatever quality the prior possesses. The point is not hypothetical, because the empirical psychology of probability judgement has documented systematic unreliability in exactly this stage. Tversky and Kahneman's research programme on judgemental heuristics showed that people estimate probabilities by shortcuts — judging likelihood by how representative or how mentally available an outcome is — and thereby neglect base rates, the background frequencies that should anchor a prior. A concrete measurement: in 1978 Casscells, Schoenberger and Graboys asked sixty students and staff at Harvard Medical School for the probability that a patient has a disease, given a disease prevalence of 1 in 1,000 and a test with a 5% false-positive rate. Bayes' theorem gives $P(H \mid E) = \frac{1 \times 0.001}{1 \times 0.001 + 0.05 \times 0.999} \approx 0.02$: about 2%, because in a thousand people the test flags roughly fifty healthy ones alongside the single genuine case. The most common answer given was 95%, and only a small minority answered near 2%. A clinician who forms her prior by such intuition and then updates impeccably will produce a posterior that is precisely, reliably wrong; and if on some occasion her answer happens to be accurate, that accuracy is luck, not justification. Reliability of evidence integration. The update step itself has two failure modes that outcome-checking cannot distinguish. One is computational: misapplying the theorem — most commonly by confusing $P