Questions: Bayesian Confirmation Theory

Question 1

A geologist observes a rock formation consistent with Hypothesis A (continental drift) and Hypothesis B (fixed continents). The observation has probability 0.90 given A and 0.80 given B. Does this observation strongly confirm Hypothesis A over B?

• A. Yes — the observation is more likely given A (0.90) than given B (0.80), confirming A substantially
• B. No — the observation confirms A only if P(observation|A) = 1; anything less counts as weak evidence
• C. No — the likelihood ratio is only 0.90/0.80 = 1.125, providing only weak relative confirmation for A over B
• D. Whether the observation confirms A depends solely on whether P(A) > 0.5 before the observation

Answer: (C)

Bayesian confirmation is comparative: evidence confirms A over B to the extent that A predicts it better than B does, measured by the likelihood ratio P(E|A)/P(E|B) = 0.90/0.80 = 1.125. A ratio barely above 1 means A is only marginally a better predictor of this observation than B — both hypotheses predict it quite well. This is the subtlety of Bayesian evidence: it is not just 'does the hypothesis predict this?' but 'does it predict it substantially better than the alternative?' Option D confuses prior probability with the logic of evidential updating.

Question 2

According to Bayesian confirmation theory, evidence E confirms hypothesis H when...

• A. E is logically entailed by H (i.e., H → E is a valid implication)
• B. Observing E raises the probability of H: P(H|E) > P(H)
• C. E could not have occurred if H were false: P(E|¬H) = 0
• D. The scientist predicted E before the observation was made

Answer: (B)

The Bayesian definition is probabilistic: E confirms H if and only if P(H|E) > P(H) — the posterior probability exceeds the prior. This is broader than logical entailment (option A) and different from conclusive evidence (option C, which requires P(E|¬H) = 0 — decisive disconfirmation of alternatives). The framework accommodates partial, graded confirmation: E can raise H's probability a little or a lot. Option D (prior prediction) has nothing to do with the formal definition — surprising evidence often confirms more strongly than expected evidence.

Question 3

According to Bayesian confirmation theory, observing a green apple technically provides some confirmation for the hypothesis 'all ravens are black.'

• T. True
• F. False

Answer: True

This is the Bayesian resolution of Hempel's ravens paradox. 'All ravens are black' is logically equivalent to 'All non-black things are non-ravens.' A green apple is a non-black non-raven, consistent with this equivalent statement. Bayes' theorem confirms this technically: a green apple very slightly raises P(all ravens are black) because it eliminates one non-black object from the sample space without revealing a non-black raven. The Bayesian response is that this confirmation is real but negligible — the sample space of non-black things is enormous, so each instance provides only infinitesimal update.

Question 4

Evidence confirms a hypothesis independently of competing hypotheses — the degree of confirmation depends mainly on the two-place relationship between E and H.

• T. True
• F. False

Answer: False

Bayesian confirmation is inherently comparative. P(H|E) = P(E|H)P(H)/P(E), and P(E) = P(E|H)P(H) + P(E|¬H)P(¬H) — the denominator explicitly depends on how well alternatives to H also predict E. A hypothesis that uniquely predicts E gains massive confirmation when E is observed; a hypothesis that predicts E no better than its competitors gains almost none. Evidence is always evidence relative to alternatives: 'data confirms theory' is always implicitly a comparison.

Question 5

Why does a single counterexample decisively refute a universal hypothesis ('all ravens are black'), while many confirming instances raise its probability only gradually?

Model answer: A universal hypothesis assigns probability zero to any counterexample — 'all ravens are black' strictly rules out white ravens, so P(white raven observed | all ravens are black) = 0. When a white raven is observed, Bayes' theorem gives P(H|E) = 0 × P(H)/P(E) = 0, regardless of the prior. Confirming instances (black ravens), by contrast, are quite common even under alternative hypotheses, so the likelihood ratio P(E|H)/P(E|¬H) is only slightly above 1 for each black raven, producing tiny probability increments. The asymmetry is structural: a universal claim creates exactly one condition that destroys it completely, while confirmation requires ruling out every possible counterexample.

This asymmetry has deep implications for scientific methodology. Popper's falsificationism is the philosophical response: since confirmation accumulates slowly while falsification strikes decisively, focusing on bold predictions that could refute a theory is epistemically efficient. The Bayesian framework explains *why* this asymmetry exists — it follows directly from the mathematics of zero likelihood under a universal claim.