A researcher designs an experiment whose result E is almost equally likely whether hypothesis H is true or false — P(E|H) ≈ P(E|¬H) ≈ 0.9. She observes E. What happens to her credence in H by Bayesian lights?
AHer credence rises substantially — she observed evidence that the hypothesis predicts
BHer credence barely changes — the Bayes factor P(E|H)/P(E|¬H) ≈ 1 means E is nearly neutral
CHer credence drops — if ¬H also predicts E, that undermines H
DShe must assign H a credence of exactly 0.9 to match the likelihood
The strength of evidence is determined by the Bayes factor P(E|H)/P(E|¬H), not by P(E|H) alone. When both the hypothesis and its negation predict E nearly equally well, observing E tells you almost nothing about which is true — the ratio is close to 1 and the posterior barely shifts from the prior. This is why the common intuition 'the experiment confirmed the hypothesis because the prediction came true' can be misleading: what matters is whether the hypothesis predicted E *better than the alternatives*, not just whether it predicted E at all. Trivially predictable results are weak evidence.
Question 2 Multiple Choice
You observe a green apple. According to Bayesian confirmation theory, does this observation confirm the hypothesis 'All ravens are black'?
ANo — observations of non-ravens are logically irrelevant to hypotheses about ravens
BYes, but only infinitesimally — the Bayes factor is barely above 1 because a green apple is almost equally likely whether or not all ravens are black
CYes, and as strongly as observing a black raven — both are positive instances of the logically equivalent contrapositive
DNo — confirmation only counts when you observe a direct instance of the subject class
Bayesianism dissolves the raven paradox by treating confirmation as a matter of degree, not a binary yes/no. Logically, 'All ravens are black' is equivalent to 'All non-black things are non-ravens,' so a green apple (non-black, non-raven) is a positive instance. Does it confirm? Technically yes, but the Bayes factor is barely above 1 — knowing you've seen a green apple is almost equally likely whether all ravens are black or not, because the world contains overwhelmingly more non-ravens than ravens. The paradox arose from a binary notion of confirmation; the quantitative Bayesian approach shows the green apple confirms only negligibly while a black raven confirms substantially.
Question 3 True / False
According to Bayesian confirmation theory, evidence E confirms hypothesis H if and only if observing E raises your credence in H above its prior value.
TTrue
FFalse
Answer: True
This is the Bayesian definition of confirmation: E confirms H iff P(H|E) > P(H). It is elegant because it is both mathematically precise and intuitively appealing — learning E is good news for H if and only if E makes H more likely than it was before. The definition also has a natural companion: E disconfirms H iff P(H|E) < P(H), and E is neutral iff P(H|E) = P(H). The quantitative framework then lets us ask how much confirmation E provides, answering with the Bayes factor rather than just yes/no.
Question 4 True / False
If two scientists begin with very different prior probabilities for the same hypothesis, Bayesian updating cannot bring their posteriors into agreement regardless of how much evidence accumulates.
TTrue
FFalse
Answer: False
Under mild conditions (both scientists assign non-zero prior probability to the true hypothesis and share the same evidence), Bayesian agents will converge to the same posterior as evidence accumulates, regardless of their starting priors. This is the 'washing out of priors' result. The evidence eventually overwhelms any finite prior — only a prior of exactly 0 (assigning zero probability to a hypothesis) prevents convergence, because updating a zero prior produces a zero posterior no matter what. This convergence property is one of Bayesianism's strongest responses to the worry about subjective priors in science.
Question 5 Short Answer
Why are trivially predictable experimental results weak evidence for a hypothesis, from a Bayesian perspective? Use the concept of the Bayes factor in your explanation.
Think about your answer, then reveal below.
Model answer: The strength of evidence E for hypothesis H is measured by the Bayes factor: P(E|H)/P(E|¬H) — the ratio of how likely E is if H is true to how likely E is if H is false. If E is trivially predictable, both H and ¬H (and every other competitor) assign it high probability, so P(E|H) ≈ P(E|¬H), and the Bayes factor ≈ 1. A factor of 1 means E does not shift the odds between H and its competitors at all. Evidence confirms H strongly only when H predicts E substantially better than its competitors — when P(E|H) is much larger than P(E|¬H). The discriminating power of evidence comes not from whether H predicts E, but from whether H predicts E *specifically*, in a way that other hypotheses do not.
A classic example: every hypothesis predicts that 'the sun will rise tomorrow,' so observing the sunrise is not evidence for or against any particular hypothesis. By contrast, a specific quantitative prediction that comes true — and that a competing hypothesis would not have made — has a high Bayes factor and genuinely discriminates between theories.