Reasoning Under Uncertainty Questions and Answers

Answer: A

Probability assigns a degree of belief to each proposition and provides consistent rules for updating those beliefs as evidence arrives. It does not remove the uncertainty, which is a property of the agent's information rather than something to be eliminated, and it guarantees good decisions on average rather than correct ones. Adopting a calculus whose rules follow from a few axioms about consistent preference, rather than an invented scheme, is what gives the approach its foundation.

Answer: B

The prior is the degree of belief in the absence of the evidence in question, and conditioning on evidence turns it into the posterior. It is prior relative to particular evidence rather than absolutely, since every prior is itself a posterior with respect to earlier information. Evidence probability and model versions are different notions. Recognising that the distinction is relative to a body of evidence, rather than absolute, avoids a good deal of confusion.

Answer: C

The posterior is what Bayes' rule computes: the probability of the hypothesis given the evidence, combining the prior with how likely the evidence would be under that hypothesis. The probability of evidence given hypothesis is the likelihood, which is a different quantity pointing the other way. Timing and completeness are unrelated. Keeping the two directions distinct is the single most important discipline in applying the rule. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Answer: D

The likelihood points from hypothesis to evidence, and is usually the direction in which knowledge is available, since one knows how often a disease produces a symptom. Bayes' rule converts it into the posterior, which points the other way and is what a diagnosis requires. Confusing the two is the base rate fallacy in its usual form. That the available knowledge and the wanted conclusion point in opposite directions is precisely why the rule is needed.

Answer: A

The denominator is the total probability of the evidence, summed over every hypothesis, and serves only to make the posteriors sum to one. When comparing hypotheses it may be omitted, since it is the same for all of them, which is why the rule is often written as a proportionality. Reliability, prior removal and unconsidered hypotheses are different concerns. Recognising a term as a normalising constant tells you at once when it can be ignored.

Answer: B

Among ten thousand people one has the disease and tests positive, while about a hundred of the healthy test positive falsely:

\[ \tfrac{1}{1 + 100} \approx 0.01 \]

so the posterior is about 1 per cent. Ignoring the rarity of the disease and reporting the test's accuracy instead is the base rate fallacy, and it is committed constantly in medicine and in law.

Answer: C

The fallacy reports the likelihood as though it were the posterior, so a ninety nine per cent accurate test for a rare condition is taken to mean a positive result is ninety nine per cent reliable. The prior is what makes the true figure so much lower. The other options describe different errors about base rates. An error that arises from conflating two conditional probabilities pointing in opposite directions is best cured by writing both down explicitly.

Answer: D

Independence means the conditional probability equals the unconditional one, so the joint factorises into the product of the marginals. It is a strong condition that rarely holds between interesting variables. Mutual exclusion, identical distributions and absence of causation are different notions, the last being related but distinct. Independence is what would make a joint distribution small, and its rarity is why the weaker conditional form matters so much.

Answer: A

Two symptoms may be correlated in the population yet independent once the disease is known, because the disease explains the correlation. That pattern is extremely common and is what makes Bayesian networks compact. Occasional independence, third variable relations and assumptions are different notions. A weaker condition that holds far more often, and buys almost as much, is a better foundation than a strong one that rarely applies. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: B

Interesting variables are rarely independent, but very often become so once a common cause is known, and that is enough to factorise the joint into small pieces. Verification is not easier, continuity is irrelevant and uniformity does not follow. Finding the weakest assumption that still delivers the structural benefit is a recurring move, and here it is what makes probabilistic reasoning over many variables possible at all. Checking the units of each quantity catches a wrong formula before the arithmetic begins.