An agent assigns credence 0.95 to 'the next flip of this coin will land heads' with no reason to believe the coin is biased. Their credences are coherent (they sum to 1 and satisfy the axioms). A subjective Bayesian says this prior is permissible; an objective Bayesian disagrees. What is the objective Bayesian's objection?
AThe credences violate the normalization axiom, since 0.95 + 0.05 ≠ 1 under the correct reckoning
BThe credences are coherent but violate the principle of indifference, which requires equal probabilities when there is no evidence favoring one outcome
CThe credences are incoherent because Dutch book arguments apply whenever any credence exceeds 0.5
DCredences about future events are never permissible because future states are not yet part of the evidence
The objective Bayesian holds that in cases of symmetrical ignorance — no reason to favor heads over tails — the principle of indifference mandates assigning equal probabilities (0.5 each). Assigning 0.95/0.05 is coherent in the technical sense (satisfies the axioms and cannot be Dutch-booked) but violates the additional rationality constraint that ignorance should be represented by equal distributions. This is precisely the debate: subjective Bayesians say any coherent prior is permissible; objective Bayesians say some coherent priors are nonetheless irrational.
Question 2 Multiple Choice
The principle of indifference and the maximum entropy principle both face which major philosophical challenge?
AThey violate the probability axioms, making them logically self-defeating
BThey require knowing the actual truth before assigning priors, creating circularity
CThey yield different probability assignments depending on how the possibility space is partitioned or described (Bertrand's paradox)
DThey are equivalent to subjective Bayesianism and add no genuinely new constraints
Bertrand's paradox demonstrates the problem: if you are ignorant about a random chord in a circle, should you uniformize over chord endpoints, chord midpoints, or chord lengths? Each partition gives a different 'ignorance prior' for whether the chord is longer than the inscribed triangle's side. The principle of indifference and maximum entropy both depend on a choice of parameterization that is not fixed by the evidence alone — the 'ignorance' they encode is relative to a description, not absolute.
Question 3 True / False
According to subjective Bayesianism, an agent who assigns credence 0.99 to 'the moon is made of cheese' is irrational, because this violates the principle of indifference.
TTrue
FFalse
Answer: False
Subjective Bayesianism imposes only two rationality requirements: (1) credences must satisfy the probability axioms (coherence), and (2) agents must update by conditionalization when new evidence arrives. Any prior satisfying the axioms is permissible — including bizarre ones like 0.99 on a cheese moon. The principle of indifference is an objective Bayesian constraint that subjective Bayesians reject. From the subjective view, what makes an agent irrational is not a wrong starting point but incorrect updating.
Question 4 True / False
An agent whose credences satisfy the probability axioms (coherence) cannot be made to accept a set of bets that guarantees a net loss.
TTrue
FFalse
Answer: True
This is the Dutch book theorem: coherence (satisfying the probability axioms) is necessary and sufficient to avoid Dutch books. An agent with incoherent credences can always be offered a combination of individually acceptable bets that guarantee a loss. Conversely, a coherent agent has no such vulnerability. This is the primary Bayesian argument for why the probability axioms are rationality requirements — not just convenient mathematics.
Question 5 Short Answer
Why do subjective Bayesians hold that different people can rationally start with different priors, and what constraint does rationality actually impose on them?
Think about your answer, then reveal below.
Model answer: Subjective Bayesians argue that rationality is procedural, not about correct starting points. Any prior that satisfies the probability axioms is permissible. The constraint rationality imposes is on updating: when evidence E arrives, agents must update by conditionalization — the posterior probability of hypothesis H is proportional to the prior times the likelihood of E given H. Given enough evidence, agents who began with different (but coherent) priors will converge toward the same posteriors. Rationality lives in the method of updating, not in the initial distribution.
This view prioritizes convergence under evidence: diverse priors are fine because good evidence eventually washes them out. Critics (objective Bayesians) argue that 'enough evidence' may never arrive, making the starting point matter practically even if it doesn't matter in the limit.