Reasoning Under Uncertainty Questions and Answers

Q: Why does naive Bayes work despite its false assumption?

Correlated features cause the probabilities to be badly calibrated, often close to zero or one, but the class with the highest score is frequently still the right one, and only the ordering matters for a classification decision. The assumption rarely holds, the errors do not cancel, and training does not correct it. That a model can be wrong in its outputs and right in its decisions is a distinction worth drawing whenever a probability is used only to choose.

Q: What is a utility function in decision theory?

Utility converts a preference ordering over outcomes into numbers that may be combined with probabilities to compute an expectation. It is determined only up to a positive linear transformation, which is why only ratios of differences are meaningful. Probabilities, costs and action rankings are different quantities. That a number can be attached to a preference at all follows from axioms about consistent choice rather than being assumed. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Q: What does the principle of maximum expected utility prescribe?

The principle weights each outcome by its probability and selects the action with the greatest average, which follows from a small set of axioms about consistent preference. Minimax over worst cases is a more conservative alternative, and the other options ignore either probability or the less likely outcomes. A decision rule derived from axioms about preference is on far firmer ground than one adopted because it seems reasonable. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.

Q: Why is a utility function determined only up to a linear transformation?

Adding a constant or multiplying by a positive number leaves every comparison of expected utilities unchanged, so the scale carries no meaning beyond the ordering and the relative sizes of differences. Arbitrary transformations would change decisions, and normalisation is a convention. Knowing which transformations leave a quantity's meaning intact tells you exactly how much information it carries, which is a useful check on any numerical measure.

Q: What is risk aversion in utility terms?

Concavity means each additional unit is worth less than the last, so a certain amount is preferred to a gamble of equal expected value. That is how attitude to risk is expressed without abandoning expected utility maximisation. Outcome counts, refusal and probability distortion are different phenomena, the last being a documented human tendency rather than a modelling device. Absorbing an apparent departure from a theory into the shape of one of its functions is a move that preserves the theory's structure.

Q: What is a decision network?

Adding rectangular decision nodes and a diamond utility node to a Bayesian network gives a single structure over which the maximum expected utility action can be computed. It unifies the probabilistic and the decision theoretic parts of a problem. Multi agent networks, decision only graphs and model selection are different notions. Extending a representation with two node types, rather than building a separate formalism, is an economical way to add decision making to a probabilistic model.

Q: How is a decision network evaluated?

Setting the decision node to each of its values in turn, computing the posterior over the utility node's parents and taking the expectation, gives the value of each action. The highest is chosen. Sampling is one way to compute the expectations rather than the procedure, and decisions are maximised rather than summed out. Distinguishing the variables that are summed from the one that is maximised is the whole difference between inference and decision making.

Q: What is the value of information?

An observation is worth acquiring only if it might change the decision, and its value is the expected gain in utility from acting on it, compared against the cost of obtaining it. Entropy reduction is a related information measure that ignores the decision. Cost is what the value is compared against. That an observation which cannot change the decision is worthless, however informative, is the point most easily missed and the most useful in practice.

Q: When is the value of information zero?

If the same action is optimal whatever the observation turns out to be, learning it changes nothing and its value is zero. Independence from utility is a sufficient condition but not the general criterion, certainty of confirmation is a special case, and cost is compared against the value rather than making it zero. Asking whether an answer would change what one does, before paying to obtain it, is a discipline that applies far beyond formal decision theory.

Q: Why is the value of information never negative?

Since the agent could act as though it had not observed, the observation can never make it worse off in expectation, so the value is zero or positive. The result assumes the agent is rational and the model correct; a mistaken model can certainly be harmed by evidence. Uncertainty reduction, bounds and axioms are not the argument. A result proved by noting that one option is always available is a pattern worth recognising, since it establishes a bound with almost no work.

31.

Why does naive Bayes work despite its false assumption?

A.The errors introduced by the assumption cancel exactly
B.The classifier corrects the assumption during training
C.Classification needs the ranking of classes rather than exact probabilities
D.The independence assumption turns out to hold in most real data

Answer: C

Correlated features cause the probabilities to be badly calibrated, often close to zero or one, but the class with the highest score is frequently still the right one, and only the ordering matters for a classification decision. The assumption rarely holds, the errors do not cancel, and training does not correct it. That a model can be wrong in its outputs and right in its decisions is a distinction worth drawing whenever a probability is used only to choose.

Discuss in Forum

32.

What is a utility function in decision theory?

A.The probability that a given outcome will occur
B.The cost of performing a particular action
C.A ranking of the actions available to the agent
D.A numerical measure of how desirable each outcome is

Answer: D

Utility converts a preference ordering over outcomes into numbers that may be combined with probabilities to compute an expectation. It is determined only up to a positive linear transformation, which is why only ratios of differences are meaningful. Probabilities, costs and action rankings are different quantities. That a number can be attached to a preference at all follows from axioms about consistent choice rather than being assumed. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Discuss in Forum

33.

What does the principle of maximum expected utility prescribe?

A.Choosing the action with the highest probability weighted utility
B.Choosing the action whose worst outcome is least bad
C.Choosing the action with the highest utility in the most likely outcome
D.Choosing the action that maximises utility regardless of probability

Answer: A

The principle weights each outcome by its probability and selects the action with the greatest average, which follows from a small set of axioms about consistent preference. Minimax over worst cases is a more conservative alternative, and the other options ignore either probability or the less likely outcomes. A decision rule derived from axioms about preference is on far firmer ground than one adopted because it seems reasonable. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.

Discuss in Forum

34.

Why is a utility function determined only up to a linear transformation?

A.The axioms permit any transformation of the utilities whatever
B.Only differences and ratios of differences affect the decision
C.The absolute scale cannot be measured by any experiment
D.Utilities must be normalised to lie between zero and one

Answer: B

Adding a constant or multiplying by a positive number leaves every comparison of expected utilities unchanged, so the scale carries no meaning beyond the ordering and the relative sizes of differences. Arbitrary transformations would change decisions, and normalisation is a convention. Knowing which transformations leave a quantity's meaning intact tells you exactly how much information it carries, which is a useful check on any numerical measure.

Discuss in Forum

35.

What is risk aversion in utility terms?

A.A refusal to accept any action with an uncertain outcome
B.A tendency to overestimate the probability of bad outcomes
C.A utility function concave in the quantity of interest
D.A preference for actions with fewer possible outcomes

Answer: C

Concavity means each additional unit is worth less than the last, so a certain amount is preferred to a gamble of equal expected value. That is how attitude to risk is expressed without abandoning expected utility maximisation. Outcome counts, refusal and probability distortion are different phenomena, the last being a documented human tendency rather than a modelling device. Absorbing an apparent departure from a theory into the shape of one of its functions is a move that preserves the theory's structure.

Discuss in Forum

36.

What is a decision network?

A.A network representing the decisions of several agents
B.A network whose nodes are the decisions available to an agent
C.A network used to decide which Bayesian model to adopt
D.A Bayesian network extended with decision and utility nodes

Answer: D

Adding rectangular decision nodes and a diamond utility node to a Bayesian network gives a single structure over which the maximum expected utility action can be computed. It unifies the probabilistic and the decision theoretic parts of a problem. Multi agent networks, decision only graphs and model selection are different notions. Extending a representation with two node types, rather than building a separate formalism, is an economical way to add decision making to a probabilistic model.

Discuss in Forum

37.

How is a decision network evaluated?

A.For each decision, compute the expected utility and take the best
B.By sampling the network until the decision stabilises
C.By eliminating the decision nodes as ordinary variables
D.By maximising the utility node's value directly

Answer: A

Setting the decision node to each of its values in turn, computing the posterior over the utility node's parents and taking the expectation, gives the value of each action. The highest is chosen. Sampling is one way to compute the expectations rather than the procedure, and decisions are maximised rather than summed out. Distinguishing the variables that are summed from the one that is maximised is the whole difference between inference and decision making.

Discuss in Forum

38.

What is the value of information?

A.The probability that the observation will change the decision
B.The expected improvement in utility from observing a variable
C.The reduction in entropy that an observation produces
D.The cost of performing the observation in question

Answer: B

An observation is worth acquiring only if it might change the decision, and its value is the expected gain in utility from acting on it, compared against the cost of obtaining it. Entropy reduction is a related information measure that ignores the decision. Cost is what the value is compared against. That an observation which cannot change the decision is worthless, however informative, is the point most easily missed and the most useful in practice.

Discuss in Forum

39.

When is the value of information zero?

A.When the observation is certain to confirm what is believed
B.When the cost of observing exceeds the possible gain
C.When the observation cannot change the action chosen
D.When the variable observed is independent of the utility

Answer: C

If the same action is optimal whatever the observation turns out to be, learning it changes nothing and its value is zero. Independence from utility is a sufficient condition but not the general criterion, certainty of confirmation is a special case, and cost is compared against the value rather than making it zero. Asking whether an answer would change what one does, before paying to obtain it, is a discipline that applies far beyond formal decision theory.

Discuss in Forum

40.

Why is the value of information never negative?

A.Observations always reduce the uncertainty in the model
B.The utility function is bounded below by construction
C.Negative values are excluded by the axioms of probability
D.The agent may always ignore the observation it receives

Answer: D

Since the agent could act as though it had not observed, the observation can never make it worse off in expectation, so the value is zero or positive. The result assumes the agent is rational and the model correct; a mistaken model can certainly be harmed by evidence. Uncertainty reduction, bounds and axioms are not the argument. A result proved by noting that one option is always available is a pattern worth recognising, since it establishes a bound with almost no work.

Discuss in Forum

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