Artificial Intelligence Reasoning under Uncertainty Syllabus

Reasoning under Uncertainty syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Reasoning under Uncertainty in Artificial Intelligence, not a summary of it.

1. 1

   Quantifying Uncertainty

   3 topics

   • Acting under Uncertainty
   • Basic Probability Notation
   • Bayes' Rule and Its Use
2. 2

   Probabilistic Reasoning

   3 topics

   • Bayesian Networks
   • Exact Inference
   • Approximate Inference
3. 3

   Reasoning over Time

   4 topics

   • Time and Uncertainty
   • Hidden Markov Models
   • Kalman Filters
   • Dynamic Bayesian Networks
4. 4

   Making Decisions

   4 topics

   • Utility Theory
   • Decision Networks
   • Value of Information
   • Sequential Decision Problems

Reasoning under Uncertainty flashcards for Artificial Intelligence

19 of 55 cards from the Reasoning under Uncertainty deck — real questions with worked answers.

1. What does it mean for an agent to act under uncertainty, and why can't a purely logical (categorical) agent handle it?

   Acting under uncertainty means choosing actions when the agent's knowledge does not uniquely determine the world state, action outcomes, or observations. A logical agent fails because it would need to enumerate an impractically large (often infinite) set of antecedents and consequents to be complete, and complete categorical rules rarely exist for real domains (e.g., medical diagnosis).

2. State the Maximum Expected Utility (MEU) principle for a rational agent under uncertainty.

   A rational agent should choose the action that maximizes expected utility: $$a^{*} = \arg\max_{a} \sum_{s'} P(s' \mid a, e)\, U(s'),$$ where $e$ is the evidence, $P(s'\mid a,e)$ is the probability of outcome state $s'$, and $U(s')$ is its utility.

3. Distinguish a prior (unconditional) probability from a posterior (conditional) probability.

   A prior probability $P(X)$ is the degree of belief in proposition $X$ in the absence of any other information. A posterior probability $P(X \mid e)$ is the belief in $X$ after evidence $e$ has been observed. They are related by conditioning: $P(X\mid e) = \frac{P(X, e)}{P(e)}$.

4. Write the product rule and the definition of conditional probability.

   Conditional probability: $P(a \mid b) = \frac{P(a \wedge b)}{P(b)}$ for $P(b) > 0$. The product rule rearranges this as $$P(a \wedge b) = P(a \mid b)\,P(b) = P(b \mid a)\,P(a).$$

5. What is a full joint probability distribution, and how do you answer any query from it?

   A full joint distribution assigns a probability to every atomic event (every complete assignment of all random variables). Any query is answered by marginalization (summing out) and normalization: $$P(X \mid e) = \alpha \sum_{y} P(X, e, y),$$ where $y$ ranges over the unobserved variables and $\alpha$ normalizes.

6. State the three axioms of probability (Kolmogorov's axioms).

   1) $0 \leq P(a) \leq 1$ for any proposition $a$. 2) $P(\text{true}) = 1$ and $P(\text{false}) = 0$. 3) Inclusion–exclusion: $P(a \vee b) = P(a) + P(b) - P(a \wedge b)$.

7. Define marginalization (the summing-out rule) for probability distributions.

   Marginalization computes the distribution over a subset of variables by summing over the others: $$P(X) = \sum_{y} P(X, y),$$ often written using conditioning as $P(X) = \sum_{y} P(X \mid y)\,P(y)$.

8. What does it mean for two variables $X$ and $Y$ to be independent, and how does independence simplify the joint?

   $X$ and $Y$ are independent iff $P(X \mid Y) = P(X)$ (equivalently $P(Y\mid X)=P(Y)$). Then the joint factorizes: $$P(X, Y) = P(X)\,P(Y).$$ This reduces representation size from exponential to a product of smaller tables.

9. Define conditional independence and give its factorization.

   $X$ and $Y$ are conditionally independent given $Z$ iff $P(X \mid Y, Z) = P(X \mid Z)$. Equivalently $$P(X, Y \mid Z) = P(X \mid Z)\,P(Y \mid Z).$$ Conditional independence is the key structural property exploited by Bayesian networks.

10. State Bayes' rule and name each term.

    $$P(h \mid e) = \frac{P(e \mid h)\,P(h)}{P(e)}.$$ $P(h)$ is the prior, $P(e\mid h)$ is the likelihood, $P(e)$ is the evidence/marginal likelihood, and $P(h\mid e)$ is the posterior.

11. Why is Bayes' rule useful in diagnosis, and what is the normalized two-hypothesis form?

    It lets you compute a hard-to-assess diagnostic probability $P(\text{cause}\mid\text{effect})$ from an easier-to-assess causal likelihood $P(\text{effect}\mid\text{cause})$ plus priors. Normalized form: $$P(H \mid e) = \alpha\, P(e \mid H)\,P(H),\quad \alpha = \frac{1}{\sum_{h} P(e\mid h)P(h)}.$$

12. How does Bayes' rule combine with conditional independence for multiple pieces of evidence (naive Bayes)?

    If effects $e_1,\dots,e_n$ are conditionally independent given the cause, then $$P(\text{Cause} \mid e_1,\dots,e_n) = \alpha\, P(\text{Cause}) \prod_{i} P(e_i \mid \text{Cause}).$$ This is the naive Bayes model, requiring only $O(n)$ parameters instead of $O(2^n)$.

13. Define a Bayesian network and its two components.

    A Bayesian network is a directed acyclic graph (DAG) where: (1) nodes are random variables, and (2) each node has a conditional probability table (CPT) giving $P(X_i \mid \text{Parents}(X_i))$. Directed edges encode direct probabilistic dependence.

14. Give the chain-rule factorization that a Bayesian network represents.

    A Bayes net encodes the full joint as the product of local conditionals: $$P(x_1, \dots, x_n) = \prod_{i=1}^{n} P(x_i \mid \text{parents}(X_i)).$$

15. How many independent parameters does a CPT for a Boolean node with $k$ Boolean parents require?

    It requires $2^{k}$ independent numbers (one per combination of parent values, giving the probability that the node is true). A full joint over all $n$ Booleans would need $2^{n}-1$ numbers, so local structure yields large savings when $k \ll n$.

16. State the two equivalent semantics for what a Bayesian network asserts (numerical vs. topological).

    Numerical (global) semantics: the net defines the full joint as the product of local CPTs. Topological (local) semantics: each variable is conditionally independent of its non-descendants given its parents. The two are equivalent.

17. What is a node's Markov blanket, and what independence does it provide?

    A node's Markov blanket consists of its parents, its children, and its children's other parents. A node is conditionally independent of all other nodes in the network given its Markov blanket.

18. State the general form of an exact inference (variable elimination) query in a Bayes net.

    $$P(X \mid e) = \alpha \sum_{y} \prod_{i} P(x_i \mid \text{parents}(X_i)),$$ where $X$ is the query variable, $e$ is evidence, and $y$ ranges over hidden variables. $\alpha$ normalizes the result to sum to 1.

19. How does the variable-elimination algorithm improve on enumeration, and what are its two core operations?

    Variable elimination avoids repeated subcomputations by carrying out summations right-to-left and storing intermediate results as factors. Its two operations are pointwise product of factors and summing out (marginalizing) a variable from a product of factors.

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