Reasoning Under Uncertainty Questions and Answers

Q: Why is exact inference in Bayesian networks hard in general?

Exact inference is NP-hard in general, though polytrees admit linear-time inference, so the difficulty lives in the loops. Storage, evidence handling and numerical error are real engineering concerns but are not the source of the complexity result. Knowing that the hardness is structural, and that a whole class of networks escapes it, is what makes approximate methods a considered choice rather than a retreat.

Q: Which sampling method draws from the prior and discards mismatches?

Rejection sampling draws complete samples from the prior and throws away those inconsistent with the evidence, estimating the query from what survives. It is correct and hopelessly wasteful when the evidence is unlikely, since almost every sample is discarded. Thresholds, unassigned parents and burn-in belong to other methods. Its inefficiency is precisely the problem likelihood weighting sets out to fix.

Q: Why does likelihood weighting waste none of its samples?

Evidence variables are clamped rather than sampled, and each sample carries the product of the evidence likelihoods as its weight, so no sample is wasted. Performance still degrades when the evidence sits far downstream. Variable weighting, importance regions and relaxed evidence are different ideas. Understanding the weight as an importance-sampling correction explains both why it works and where it fails.

Q: What kind of algorithm is Gibbs sampling?

Gibbs walks a Markov chain over complete assignments, resampling one non-evidence variable at a time from its Markov blanket, and the chain's stationary distribution is the posterior. Direct sampling, exact truncation and mode search are different things. The cost is that consecutive samples are correlated and the chain needs time to mix, which is why burn-in and convergence checks belong to the method.

Q: Which property survives the distorted estimates naive Bayes produces?

The estimated probabilities are badly calibrated, often near zero or one, but the class with the largest score is frequently still the right one, and classification needs only the argmax. Real features are rarely independent, the errors do not cancel exactly, and the prior does not repair them. Separating calibration from discrimination is the general point, and it recurs whenever a model is judged by accuracy alone.

Q: Which numerical object encodes preferences over outcomes?

Utility assigns a number to each outcome so that preferences are respected, and rational choice maximises expected utility. It is defined only up to a positive affine transformation, so absolute values carry no meaning. Evidence value, success probability and cost are different quantities. The representation theorem behind it, deriving utility from a handful of preference axioms, is what stops the number being arbitrary.

Q: Which shape of the money-utility curve produces caution?

A concave utility means each additional unit of money is worth less than the last, so a sure amount is preferred to a gamble of equal expected value. The preference emerges from the shape of the curve, with no change to probabilities and no separate penalty. Variance preference is a symptom rather than the representation. That the same MEU rule covers cautious and reckless agents alike is what makes the framework general. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Q: What is the certainty equivalent of a gamble?

The certainty equivalent is the sure sum whose utility equals the gamble's expected utility, and it falls below the expected value for a risk-averse agent. The gap is the insurance premium. Expected value, avoidance payment and utility rescaling are related but distinct. Computing it once for a concave curve makes the whole vocabulary of risk premiums fall into place.

Q: Which quantity does the normalising constant in Bayes' rule represent?

The denominator is the probability of the evidence summed over every hypothesis, and it merely rescales so the posteriors sum to one. Because it is common to all hypotheses it can be ignored when only the ranking matters, which is why the proportional form of the rule is so often used. Priors, maximum likelihoods and hypothesis counts are different quantities. Treating the constant as bookkeeping rather than as content removes most of the algebra from the rule and leaves the idea visible.

Q: Which comparison does a likelihood ratio express?

The ratio divides the two likelihoods and multiplies the prior odds to give the posterior odds, so it is exactly the weight of evidence. It says nothing about the priors and nothing about how many observations are needed. Working in odds rather than probabilities makes sequential updating a simple product, which is why diagnostic reasoning is usually taught this way. Recognising the ratio as the only part of the evidence that matters is the practical lesson.

81.

Why is exact inference in Bayesian networks hard in general?

A.The problem is NP-hard for arbitrary structures
B.The conditional tables cannot be stored for large networks
C.The evidence cannot be incorporated without recomputing everything
D.Floating point error accumulates too rapidly to be tolerated

Answer: A

Exact inference is NP-hard in general, though polytrees admit linear-time inference, so the difficulty lives in the loops. Storage, evidence handling and numerical error are real engineering concerns but are not the source of the complexity result. Knowing that the hardness is structural, and that a whole class of networks escapes it, is what makes approximate methods a considered choice rather than a retreat.

Discuss in Forum

82.

Which sampling method draws from the prior and discards mismatches?

A.Discarding the earliest samples before the chain has settled
B.Discarding samples that disagree with the evidence
C.Rejecting samples whose probability falls below a fixed threshold
D.Refusing to sample from nodes whose parents remain unassigned

Answer: B

Rejection sampling draws complete samples from the prior and throws away those inconsistent with the evidence, estimating the query from what survives. It is correct and hopelessly wasteful when the evidence is unlikely, since almost every sample is discarded. Thresholds, unassigned parents and burn-in belong to other methods. Its inefficiency is precisely the problem likelihood weighting sets out to fix.

Discuss in Forum

83.

Why does likelihood weighting waste none of its samples?

A.It samples more often from the regions of higher probability
B.It rejects fewer samples by relaxing the evidence slightly
C.It fixes the evidence and weights each sample by its likelihood
D.It weights each variable by how likely it is to affect the query

Answer: C

Evidence variables are clamped rather than sampled, and each sample carries the product of the evidence likelihoods as its weight, so no sample is wasted. Performance still degrades when the evidence sits far downstream. Variable weighting, importance regions and relaxed evidence are different ideas. Understanding the weight as an importance-sampling correction explains both why it works and where it fails.

Discuss in Forum

84.

What kind of algorithm is Gibbs sampling?

A.A direct sampling method that draws from the prior distribution
B.An exact method that becomes approximate only when truncated
C.A search method that finds the most probable explanation
D.A Markov chain Monte Carlo method

Answer: D

Gibbs walks a Markov chain over complete assignments, resampling one non-evidence variable at a time from its Markov blanket, and the chain's stationary distribution is the posterior. Direct sampling, exact truncation and mode search are different things. The cost is that consecutive samples are correlated and the chain needs time to mix, which is why burn-in and convergence checks belong to the method.

Discuss in Forum

85.

Which property survives the distorted estimates naive Bayes produces?

A.Ranking the classes survives the distorted probabilities
B.The independence assumption turns out to hold in most real datasets
C.The errors introduced by the assumption cancel out exactly
D.The prior over classes corrects for the dependence between features

Answer: A

The estimated probabilities are badly calibrated, often near zero or one, but the class with the largest score is frequently still the right one, and classification needs only the argmax. Real features are rarely independent, the errors do not cancel exactly, and the prior does not repair them. Separating calibration from discrimination is the general point, and it recurs whenever a model is judged by accuracy alone.

Discuss in Forum

86.

Which numerical object encodes preferences over outcomes?

A.The cost incurred by an agent while carrying out a plan
B.A numerical measure of preference over outcomes
C.A function measuring how useful a piece of evidence proves to be
D.The probability that an action achieves the goal it was chosen for

Answer: B

Utility assigns a number to each outcome so that preferences are respected, and rational choice maximises expected utility. It is defined only up to a positive affine transformation, so absolute values carry no meaning. Evidence value, success probability and cost are different quantities. The representation theorem behind it, deriving utility from a handful of preference axioms, is what stops the number being arbitrary.

Discuss in Forum

87.

Which shape of the money-utility curve produces caution?

A.By preferring the action with the lower variance of outcomes
B.By adding a penalty term to the expected utility of each action
C.By a concave utility function over money
D.By reducing the probabilities assigned to the very good outcomes

Answer: C

A concave utility means each additional unit of money is worth less than the last, so a sure amount is preferred to a gamble of equal expected value. The preference emerges from the shape of the curve, with no change to probabilities and no separate penalty. Variance preference is a symptom rather than the representation. That the same MEU rule covers cautious and reckless agents alike is what makes the framework general. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Discuss in Forum

88.

What is the certainty equivalent of a gamble?

A.The expected monetary value of the gamble computed in the usual way
B.The amount an agent would pay to avoid facing the gamble at all
C.The utility of the gamble expressed in monetary units of measure
D.The sure amount an agent would accept instead

Answer: D

The certainty equivalent is the sure sum whose utility equals the gamble's expected utility, and it falls below the expected value for a risk-averse agent. The gap is the insurance premium. Expected value, avoidance payment and utility rescaling are related but distinct. Computing it once for a concave curve makes the whole vocabulary of risk premiums fall into place.

Discuss in Forum

89.

Which quantity does the normalising constant in Bayes' rule represent?

A.The total probability of the evidence
B.The prior probability assigned to the hypothesis under consideration
C.The largest likelihood among all of the competing hypotheses
D.The number of hypotheses over which the belief must be spread

Answer: A

The denominator is the probability of the evidence summed over every hypothesis, and it merely rescales so the posteriors sum to one. Because it is common to all hypotheses it can be ignored when only the ranking matters, which is why the proportional form of the rule is so often used. Priors, maximum likelihoods and hypothesis counts are different quantities. Treating the constant as bookkeeping rather than as content removes most of the algebra from the rule and leaves the idea visible.

Discuss in Forum

90.

Which comparison does a likelihood ratio express?

A.How many times the evidence would have to repeat to be conclusive
B.How much better one hypothesis explains the evidence than another
C.How much more probable one hypothesis is than another before any evidence
D.How far the posterior has moved away from the prior after updating

Answer: B

The ratio divides the two likelihoods and multiplies the prior odds to give the posterior odds, so it is exactly the weight of evidence. It says nothing about the priors and nothing about how many observations are needed. Working in odds rather than probabilities makes sequential updating a simple product, which is why diagnostic reasoning is usually taught this way. Recognising the ratio as the only part of the evidence that matters is the practical lesson.

Discuss in Forum

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