Reasoning Under Uncertainty Questions and Answers

Q: Which independence assumption underlies the noisy-or parameterisation?

Noisy-or gives each parent its own inhibition probability and multiplies them, so the effect is absent only when every present cause is independently inhibited. That reduces a table exponential in the parents to one number per parent. Additive combination, output noise and exclusivity are different models. The saving is why noisy-or appears in almost every large diagnostic network built by hand.

Q: Which set of nodes shields a variable from all the rest?

Conditioned on its parents, its children and its children's other parents, a node is independent of everything else in the network. The co-parents belong because observing a child creates a dependence between its parents. Ancestors, neighbours and connectivity subsets are different sets. Gibbs sampling resamples a node from exactly this set, so the definition is a working tool and not a curiosity.

Q: Which observation makes two otherwise unrelated causes dependent?

A common effect couples its causes once it is observed, so knowing a co-parent's value then changes belief about the node. This is explaining away seen from the blanket's side. Path membership, normalisation and sampling order are different reasons and none of them produces the co-parents. Deriving the blanket from the explaining-away pattern is more durable than memorising the three-part list.

Q: Which property may be read off the graph without any numbers?

The criterion reads conditional independence off the graph alone, without touching the numbers, by checking whether every path between the sets is blocked. Decomposition, tractability and edge reversal are separate questions. Because the answer depends only on structure, it holds for every parameterisation of that structure, which is what makes graphical reasoning worth doing before any table is filled in.

Q: When is a path through a collider blocked?

A collider blocks by default and is unblocked by observing it or any of its descendants, which is the reverse of the rule for chains and forks. That asymmetry is explaining away in graphical form. Observing a parent, or counting descendants, does not enter the rule. Keeping the collider case straight is the whole difficulty of d-separation, and it repays being checked on a three-node example every time.

Q: Which exact method expands a query into a sum of table products?

Enumeration expands the query into a sum of products of table entries and evaluates it term by term, correct but exponential in the number of hidden variables because subexpressions are recomputed. Structure search, sampling and search over query values are different algorithms. Its cost is what motivates variable elimination, so it is worth tracing once on a four-node network to see the repetition appear.

Q: What is a factor in variable elimination?

A factor is a function from assignments of some variables to non-negative numbers, starting as the conditional tables and growing or shrinking as variables are joined and summed out. It need not be a distribution and need not sum to one. Final multipliers, eliminated variables and normalising constants are different things. Thinking of the algorithm as arithmetic on tables keeps the bookkeeping straight.

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.

91.

Which independence assumption underlies the noisy-or parameterisation?

A.Only one of the several causes is ever active at any given time
B.Each cause independently fails to produce the effect
C.The causes combine additively rather than by logical disjunction
D.Noise is added to the output of an ordinary deterministic or gate

Answer: B

Noisy-or gives each parent its own inhibition probability and multiplies them, so the effect is absent only when every present cause is independently inhibited. That reduces a table exponential in the parents to one number per parent. Additive combination, output noise and exclusivity are different models. The saving is why noisy-or appears in almost every large diagnostic network built by hand.

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92.

Which set of nodes shields a variable from all the rest?

A.Every node joined to it by an edge in either direction
B.The smallest subset that renders the network singly connected
C.Parents, children and children's other parents
D.The set of all ancestors of the node together with its descendants

Answer: C

Conditioned on its parents, its children and its children's other parents, a node is independent of everything else in the network. The co-parents belong because observing a child creates a dependence between its parents. Ancestors, neighbours and connectivity subsets are different sets. Gibbs sampling resamples a node from exactly this set, so the definition is a working tool and not a curiosity.

Discuss in Forum

93.

Which observation makes two otherwise unrelated causes dependent?

A.They lie on a path from the node to the rest of the network
B.They are needed to normalise the node's own conditional table
C.Their values must be known before the child can be sampled
D.Observing a shared child makes the parents dependent

Answer: D

A common effect couples its causes once it is observed, so knowing a co-parent's value then changes belief about the node. This is explaining away seen from the blanket's side. Path membership, normalisation and sampling order are different reasons and none of them produces the co-parents. Deriving the blanket from the explaining-away pattern is more durable than memorising the three-part list.

Discuss in Forum

94.

Which property may be read off the graph without any numbers?

A.Whether two sets are independent given a third
B.Whether the network can be decomposed into disconnected parts
C.Whether inference in the network is tractable for a given query
D.Whether the directions of the edges can be reversed without loss

Answer: A

The criterion reads conditional independence off the graph alone, without touching the numbers, by checking whether every path between the sets is blocked. Decomposition, tractability and edge reversal are separate questions. Because the answer depends only on structure, it holds for every parameterisation of that structure, which is what makes graphical reasoning worth doing before any table is filled in.

Discuss in Forum

95.

When is a path through a collider blocked?

A.When the collider has no descendants anywhere in the network
B.When neither the collider nor any descendant is observed
C.When the collider itself is observed but its descendants are not
D.When at least one of the two colliding parents has been observed

Answer: B

A collider blocks by default and is unblocked by observing it or any of its descendants, which is the reverse of the rule for chains and forks. That asymmetry is explaining away in graphical form. Observing a parent, or counting descendants, does not enter the rule. Keeping the collider case straight is the whole difficulty of d-separation, and it repays being checked on a three-node example every time.

Discuss in Forum

96.

Which exact method expands a query into a sum of table products?

A.Enumerating the samples drawn and counting how often the query held
B.Trying each value of the query variable until one fits the evidence
C.Summing the joint over the hidden variables directly
D.Listing every possible network structure and scoring each one

Answer: C

Enumeration expands the query into a sum of products of table entries and evaluates it term by term, correct but exponential in the number of hidden variables because subexpressions are recomputed. Structure search, sampling and search over query values are different algorithms. Its cost is what motivates variable elimination, so it is worth tracing once on a four-node network to see the repetition appear.

Discuss in Forum

97.

What is a factor in variable elimination?

A.A term multiplied into the answer at the very end of the computation
B.A variable that has already been summed out of the expression
C.The number by which the unnormalised answer must be divided
D.A table over a subset of variables

Answer: D

A factor is a function from assignments of some variables to non-negative numbers, starting as the conditional tables and growing or shrinking as variables are joined and summed out. It need not be a distribution and need not sum to one. Final multipliers, eliminated variables and normalising constants are different things. Thinking of the algorithm as arithmetic on tables keeps the bookkeeping straight.

Discuss in Forum

98.

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

99.

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

100.

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

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