Two philosophers start with very different prior credences about a historical claim — one at 0.1, the other at 0.9. Both then examine a large body of evidence and update correctly by conditionalization. What does Bayesian theory predict about their posterior credences?
ATheir posteriors remain far apart, since radically different priors cannot be overcome by shared evidence
BTheir posteriors converge toward the same value as evidence accumulates — a result known as 'washing out of priors'
CTheir posteriors average out to roughly 0.5 because they started symmetrically
DTheir posteriors are undefined without knowing the prior probability of the evidence itself
With sufficient shared evidence and correct conditionalization, different priors tend to converge toward the same posterior — a result called 'washing out of priors.' This is why Bayesian subjectivity about prior probabilities doesn't collapse into relativism: given enough data, rational agents who started with different priors will end up agreeing. However, in realistic conditions with limited evidence, prior choice can still dominate outcomes — which is why the choice of priors remains a genuine philosophical challenge for Bayesianism.
Question 2 Multiple Choice
Your credence that it will rain tomorrow is 0.7, and your credence that it will NOT rain tomorrow is also 0.7. What is epistemically problematic about this?
ABoth credences should be 0.5 to remain epistemically neutral about uncertain events
BHaving credences above 0.5 for both options indicates overconfidence and violates epistemic humility
CThese credences violate the probability axioms — complementary events must sum to 1 — making you vulnerable to a Dutch book: a set of bets guaranteeing a net loss no matter the outcome
DCredences must be derived from frequency data; assigning 0.7 without data is unjustified
The probability axioms require that P(A) + P(¬A) = 1. If your credences add to 1.4, a clever bookmaker can construct bets that you individually find fair but which guarantee you lose money overall, regardless of whether it rains. This is the Dutch book argument: incoherent credences make you exploitable with certainty — a sure loss, which is irrational by any practical standard. The argument doesn't prove that rational agents are actually probabilistic reasoners; it shows that they must be, on pain of guaranteed loss.
Question 3 True / False
Bayesian epistemology is a descriptive theory of how humans naturally reason under uncertainty.
TTrue
FFalse
Answer: False
Bayesian epistemology is a normative theory — it specifies how credences should be structured and updated for rational belief revision, not how people actually reason. Empirical research in cognitive psychology shows that humans systematically violate probabilistic norms (base-rate neglect, anchoring, conjunction fallacy). Bayesianism is a standard of rational epistemic conduct that actual reasoners often fall short of, in the same way that logic sets standards for valid inference that humans routinely violate.
Question 4 True / False
An agent whose credences satisfy the probability axioms at all times cannot be offered a Dutch book — a set of bets guaranteeing a net loss regardless of how the world turns out.
TTrue
FFalse
Answer: True
Probabilistic coherence (satisfying the axioms) is precisely the condition that protects against Dutch books. The converse — that violating the axioms opens you to a sure-loss betting scheme — is the substance of the Dutch book theorem. This is why the Dutch book argument is offered as a pragmatic justification for requiring probabilistic credences: coherence is not just aesthetically pleasing, it is the minimum condition for not being rationally exploitable.
Question 5 Short Answer
The subjectivity of prior probabilities seems to make Bayesian epistemology relativistic — two perfectly rational agents can hold different credences about the same claim. How do Bayesians typically respond to this objection?
Think about your answer, then reveal below.
Model answer: Bayesians respond that prior subjectivity does not entail relativism because priors 'wash out' as evidence accumulates. Two agents who start with different priors but conditionalize correctly on the same evidence will converge toward the same posterior credences, given sufficient data. Bayesianism is not claiming that any prior is equally valid — it is claiming that the updating rule (conditionalization) is uniquely rational, and that this rule plus shared evidence produces convergence over time. The remaining worry is that in evidence-poor situations, priors can dominate — a genuine limitation the theory acknowledges.
This is one of the most important objections to Bayesianism and the 'washing out' response has real force but also real limits. In highly contested empirical domains with limited evidence, agents with dramatically different priors can conditionalize correctly and still reach opposite conclusions. Recognizing this limit is part of understanding Bayesianism rather than just memorizing its claims.