Questions: Evidential Support and Confirmation Formalization

Question 1

A disease affects 1 in 1,000 people. A diagnostic test has a likelihood ratio of 50 (it is 50× more likely to return positive if the patient has the disease). You test positive. What is approximately your posterior probability of having the disease?

• A. About 98% — a likelihood ratio of 50 is very strong evidence
• B. About 50% — the likelihood ratio balances the prior uncertainty
• C. About 5% — the extreme rarity of the disease still dominates after updating
• D. Cannot be determined without knowing the false positive rate exactly

Answer: (C)

Using Bayes' theorem: P(disease|+) ≈ (0.001 × 50) / (0.001 × 50 + 0.999 × 1) ≈ 0.05 / 1.05 ≈ 4.8%. A likelihood ratio of 50 is genuinely strong evidence — it moves the probability from 0.1% to about 5%. But the prior was so extreme (1 in 1,000) that even strong evidence leaves the posterior well below 50%. This illustrates why strong evidence is not sufficient for confident belief when the prior is extreme. Option A is the classic base-rate neglect error.

Question 2

What does a likelihood ratio P(e|h) / P(e|¬h) close to 1 tell you about the evidence e?

• A. The evidence strongly confirms h, because both h and ¬h predict it equally
• B. The evidence is equally probable under h and under ¬h — it provides little discriminating force
• C. The posterior probability of h equals the prior probability of h after seeing e
• D. Both B and C — a likelihood ratio of 1 means the evidence leaves credences unchanged

Answer: (D)

Options B and C are both correct, making D the right answer. A likelihood ratio of exactly 1 means P(e|h) = P(e|¬h) — the evidence is no more expected under h than under its negation. By Bayes' theorem, this means the posterior equals the prior: the evidence provides zero discriminating force and leaves belief unchanged. This is why the likelihood ratio is the right measure of evidential strength — it isolates whether the evidence preferentially supports h over ¬h, independent of prior probabilities.

Question 3

Two investigators studying the same hypothesis start with very different prior probabilities but observe the same evidence. They will assign the same likelihood ratio to that evidence.

• T. True
• F. False

Answer: True

The likelihood ratio P(e|h) / P(e|¬h) depends only on the hypothesis h and the evidence e — specifically, on how probable each makes the evidence. It does not depend on the investigators' prior probabilities for h. This is what makes the likelihood ratio an 'objective' measure of evidential strength: two people can rationally disagree about how probable h is while fully agreeing on how much this particular evidence supports h over ¬h.

Question 4

If evidence e confirms hypothesis h — that is, P(h|e) > P(h) — then observing e is sufficient justification to believe h.

• T. True
• F. False

Answer: False

Confirmation (raising probability) is necessary but not sufficient for justified belief. If P(h) = 0.001 and evidence with a likelihood ratio of 10 raises P(h|e) to about 1%, the evidence genuinely confirms h but the posterior is still far too low to justify belief. Justification requires the posterior to exceed some threshold (context-dependent, but typically well above 50% for action). Strong evidence that confirms h can still leave a rational agent doubting h when the prior is sufficiently low.

Question 5

Two confirming observations e1 and e2 each have a likelihood ratio of 20. Explain under what condition their combined likelihood ratio equals 400, and why this condition often fails in practice.

Model answer: The combined likelihood ratio equals LR1 × LR2 = 400 only when e1 and e2 are conditionally independent given h — that is, knowing e1 tells you nothing about e2 beyond what h already tells you, and similarly for ¬h. In that case, P(e1,e2|h) = P(e1|h)P(e2|h), and the ratios multiply. This often fails in practice because many pieces of evidence are correlated: two eyewitness reports of the same event share a common cause (the actual event) and may also share common biases, so they are not independent given h. Treating correlated evidence as independent inflates the combined likelihood ratio and overstates how much the evidence supports h.

This is one of the main ways Bayesian epistemology reveals hidden assumptions in ordinary reasoning. Informal arguments that 'we have lots of evidence for h' can be undermined if all that evidence traces to the same underlying source. The Bayesian framework forces explicitness about dependence structure, preventing double-counting of correlated observations.