Questions: Conditionalization and Bayesian Updating

Question 1

An agent assigns prior credences: P(rain) = 0.3, P(cloudy|rain) = 0.9, P(cloudy|no rain) = 0.2. She then observes that it is cloudy. What should her new credence in rain be?

• A. 0.9 — because it is almost always cloudy when it rains
• B. 0.3 — a single observation shouldn't change her prior
• C. Approximately 0.66 — computed via Bayes' theorem as P_old(rain|cloudy)
• D. 0.27 — the joint probability P(rain ∧ cloudy)

Answer: (C)

Conditionalization says P_new(rain) = P_old(rain|cloudy). By Bayes' theorem: P(rain|cloudy) = P(cloudy|rain)·P(rain)/P(cloudy) = (0.9×0.3)/(0.9×0.3 + 0.2×0.7) = 0.27/0.41 ≈ 0.659. Option A confuses the likelihood P(cloudy|rain) with the posterior — the likelihood is an input to the update, not the result. Option B incorrectly treats the prior as fixed against evidence. Option D gives the joint probability, the numerator before normalizing — not the conditional.

Question 2

Two agents disagree: P₁(H) = 0.1 and P₂(H) = 0.9. Both conditionalize faithfully on the same stream of shared evidence. What will happen to their credences over time?

• A. Their posteriors will immediately agree after the first piece of evidence
• B. Their posteriors will never converge because their priors are so far apart
• C. Their posteriors will tend to converge as the amount of shared evidence grows
• D. Conditionalization cannot be applied when agents have such different priors

Answer: (C)

The convergence property of Bayesian updating states that agents who start with different priors but observe the same evidence and conditionalize faithfully will converge in credence, given sufficient evidence. The convergence is asymptotic — not immediate (option A) and not impossible (option B). The exception: if either agent assigns prior probability exactly 0 to H, no amount of evidence can move that credence, since conditionalizing on any evidence always multiplies a zero prior by a finite likelihood ratio and returns zero.

Question 3

Under conditionalization, if P(p | e) = P(p), then learning e with certainty leaves your credence in p unchanged.

• T. True
• F. False

Answer: True

If P(p|e) = P(p), then p and e are probabilistically independent in the prior — knowing e tells you nothing new about p. The conditionalization rule sets P_new(p) = P_old(p|e) = P_old(p), so credence is unchanged. This is not a pathological case — it is the correct behavior. For example, learning that it rained in Tokyo should not change your credence that Shakespeare wrote Hamlet; they are independent, and rational updating reflects that.

Question 4

The problem of old evidence shows that conditionalization is fundamentally flawed and should be abandoned as a model of rational belief updating.

• T. True
• F. False

Answer: False

The problem of old evidence identifies a genuine scope limitation: if you already assign certainty to e (P(e) = 1), then conditionalizing on e leaves all credences unchanged — which seems wrong when you later realize e bears on a new hypothesis you just formulated. But most Bayesian epistemologists treat this as a problem requiring refinement or supplementation of the framework (e.g., counterfactual priors), not as a refutation. The conditionalization rule is well-supported as the correct update when you learn genuinely new information with certainty. The old evidence problem applies only in the specific case of hypothesis formation after the evidence is already known.

Question 5

Explain the 'renormalization' picture of conditionalization: what does it mean to say you eliminate impossible worlds and redistribute probability mass?

Model answer: Your prior is a probability distribution across all logically possible worlds — each world gets some credence that reflects how likely you think that world is actual. When you learn that e is true, you know you are not in any world where e is false, so you set those worlds' probabilities to zero (eliminate them). The remaining worlds — those where e holds — now sum to P(e) < 1, not 1. To restore a valid probability distribution summing to 1, you divide each remaining world's probability by P(e) (renormalize). After this operation, a world w gets probability P_old(w)/P(e) if e holds in w, and 0 otherwise. Your credence in any proposition p then equals the sum of these rescaled probabilities for worlds where p holds — which is exactly P_old(p|e), confirming that conditionalization is precisely this elimination-and-rescaling procedure.