Reasoning Under Uncertainty Questions and Answers

Answer: C

Independent observations contribute independent likelihood ratios, and the posterior odds are the prior odds times their product, which is why a sequence of weak clues can amount to strong evidence. Averaging, reversion to the prior and replacement all describe update rules that violate the axioms. Seeing the accumulation as a product also explains why an assumption of independence that is false leads to overconfidence so quickly.

Answer: D

Treating correlated observations as independent multiplies their ratios as though each carried fresh information, so the posterior moves much further than it should. The ranking usually survives, normalisation is unaffected, and there is no drift back to the prior. This is the practical hazard behind naive Bayes' poor calibration, and it is also why a network with a missing edge is worse than one with a spurious edge in most applications.

Answer: A

A zero prior multiplies to zero whatever the likelihood, so a hypothesis excluded at the outset can never be resurrected by data. That is why smoothing is applied to estimated tables, and why Cromwell's rule advises against certainty. Normalisation still works, sampling is unaffected and no recomputation is forced. The rule matters most where counts are sparse, which in practice is almost everywhere.

Answer: B

Laplace smoothing adds a pseudocount to every cell before normalising, which keeps every probability strictly positive at a small cost in accuracy where data are plentiful. It amounts to a Dirichlet prior over the table. Discarding features loses information, rescaling distorts the ratios and neighbour averaging assumes an ordering that categorical values do not have. The Bayesian reading is what tells you how large the pseudocount should be.

Answer: C

Entropy is the expected surprise, maximal for the uniform distribution and zero when one outcome is certain. It counts outcomes only indirectly, through their probabilities. Distance from uniform is relative entropy, and the largest probability is the mode. Entropy underpins information gain in decision trees and the reduction criterion used to choose which variable to observe, so it links this topic to learning and to decision making alike.

Answer: D

Information gain is the entropy before the test minus the expected entropy after it, averaged over the test's possible outcomes. It is the information-theoretic cousin of the value of information, which weights by utility rather than by entropy. Example counts, outcome probabilities and costs enter other criteria. Noting that both quantities are non-negative for the same structural reason connects the two halves of this topic neatly.

Answer: A

A dynamic Bayesian network repeats a slice of variables at each time step with edges across slices, and the hidden Markov model is the special case with one hidden state per slice. Factoring the state into several variables often makes a model far more compact. Decision networks, repeated classifiers and polytrees are different objects. Seeing that model as a degenerate case explains why its algorithms carry over with only bookkeeping changes.

Answer: B

First-order means the future is conditionally independent of the past given the present state, which is what makes the recursive update possible. Stationarity is a separate assumption, observability distinguishes a Markov chain from a hidden one, and the observations are not independent unconditionally. Keeping the three assumptions apart matters, because a model may well satisfy one and fail another.

Answer: C

Any finite-memory dependence can be folded into a richer state, restoring the Markov property at the cost of a larger state space. That trade is the standard move whenever the assumption fails. More samples reduce variance rather than bias, time-varying transitions address non-stationarity and direct observation is usually impossible. Recognising the enlargement as a general technique rather than a trick makes many modelling problems routine.

Answer: D

The Kalman filter keeps a Gaussian belief, updating its mean and covariance in closed form for linear dynamics with Gaussian noise. Under those assumptions it is exact and cheap. Sample populations belong to particle filters, discretised tables to grid filters, and a point estimate with an error bound is not a distribution. Knowing where its assumptions fail is what tells you when to reach for a particle filter instead.