Reasoning Under Uncertainty Questions and Answers

Q: What is a likelihood?

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.

Q: Bayes' rule states \( P(H|E) = P(E|H)P(H)/P(E) \). What does the denominator do?

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.

Q: A test is 99 per cent accurate for a disease affecting 1 in 10000. What is the chance a positive result is correct?

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.

Q: What is the base rate fallacy?

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.

Q: What does it mean for two variables to be independent?

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.

Q: What is conditional independence?

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.

Q: Why is conditional independence more useful than full independence?

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.

Q: How many numbers specify a joint distribution over 20 binary variables?

Each combination of values needs its own probability:\[ 2^{20} - 1 \approx 10^{6} \]so about a million numbers, which could neither be stored nor estimated from any realistic data set. Conditional independence reduces that to a few dozen. Working out the size of the unstructured representation is what makes the case for structure concrete rather than abstract.

Q: What does a Bayesian network represent?

The network encodes a joint distribution as a product of conditional distributions, one per node given its parents, with the graph asserting the conditional independences. It is often drawn causally because that gives the sparsest structure, but the mathematics requires only that the independences hold. Rules with certainties and pairwise correlations are different formalisms. Separating what the network asserts from why it is usually drawn that way avoids claiming more than the formalism supports.

Q: What does each node of a Bayesian network carry?

The conditional probability table gives one distribution for every combination of the parents' values, and the product of all such tables is the joint. A node with no parents carries its marginal. Correlations and causal strengths are different quantities. That the local tables multiply to give the global distribution is the whole content of the factorisation, and it is what makes local elicitation sufficient.

21.

What is a likelihood?

A.The probability of the hypothesis given the evidence
B.The prior probability of the hypothesis being true
C.The probability that the hypothesis is likely to hold
D.The probability of the evidence given the hypothesis

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.

Discuss in Forum

22.

Bayes' rule states \( P(H|E) = P(E|H)P(H)/P(E) \). What does the denominator do?

A.Normalises so the posterior probabilities sum to one
B.Corrects for the reliability of the evidence gathered
C.Removes the influence of the prior from the calculation
D.Accounts for hypotheses that have not been considered

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.

Discuss in Forum

23.

A test is 99 per cent accurate for a disease affecting 1 in 10000. What is the chance a positive result is correct?

A.About 90 per cent
B.About 1 per cent
C.About 99 per cent
D.About 50 per cent

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.

Discuss in Forum

24.

What is the base rate fallacy?

A.Confusing the rate of an event with its total frequency
B.Using an outdated base rate that no longer applies
C.Ignoring the prior probability when interpreting evidence
D.Assuming the base rate is the same in every population

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.

Discuss in Forum

25.

What does it mean for two variables to be independent?

A.The two variables never take the same value at once
B.The two variables have the same probability distribution
C.Neither variable causes the other to take its value
D.Knowing one tells you nothing about the other

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.

Discuss in Forum

26.

What is conditional independence?

A.Two variables independent once a third is known
B.Independence that holds only under certain conditions of the model
C.Two variables that are independent of a third but not of each other
D.Independence that has been assumed rather than demonstrated

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.

Discuss in Forum

27.

Why is conditional independence more useful than full independence?

A.It guarantees that the joint distribution is uniform
B.It holds far more often while still permitting factorisation
C.It is easier to verify from data than full independence
D.It applies to continuous variables where independence does not

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.

Discuss in Forum

28.

How many numbers specify a joint distribution over 20 binary variables?

A.Four hundred
B.Twenty
C.About a million
D.Forty

Answer: C

Each combination of values needs its own probability:

\[ 2^{20} - 1 \approx 10^{6} \]

so about a million numbers, which could neither be stored nor estimated from any realistic data set. Conditional independence reduces that to a few dozen. Working out the size of the unstructured representation is what makes the case for structure concrete rather than abstract.

Discuss in Forum

29.

What does a Bayesian network represent?

A.The causal mechanisms operating in the domain
B.A set of rules with attached certainty factors
C.The correlations observed between every pair of variables
D.A joint distribution factorised by conditional independence

Answer: D

The network encodes a joint distribution as a product of conditional distributions, one per node given its parents, with the graph asserting the conditional independences. It is often drawn causally because that gives the sparsest structure, but the mathematics requires only that the independences hold. Rules with certainties and pairwise correlations are different formalisms. Separating what the network asserts from why it is usually drawn that way avoids claiming more than the formalism supports.

Discuss in Forum

30.

What does each node of a Bayesian network carry?

A.A distribution over its values given each combination of parent values
B.The marginal probability of each of its own values
C.The correlation between it and each of its parents
D.The causal strength with which its parents influence it

Answer: A

The conditional probability table gives one distribution for every combination of the parents' values, and the product of all such tables is the joint. A node with no parents carries its marginal. Correlations and causal strengths are different quantities. That the local tables multiply to give the global distribution is the whole content of the factorisation, and it is what makes local elicitation sufficient.

Discuss in Forum

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