Week 8/9 - Bayesian and Uncertainty

Why is reasoning under uncertainty necessary in AI?

Because real-world problems involve partial observability, noisy sensors, uncertain outcomes, and complex dynamics.

What are some alternative methods to probability for handling uncertainty?

Nonmonotonic logic and rules with confidence factors, but these have issues with consistency and combination.

What does a probability model consist of?

A sample space Ω with an assignment of probabilities to each atomic event ω such that ∑P(ω) = 1.

What is a random variable?

A function mapping sample points to a value (e.g., Boolean, discrete, or continuous values).

What is a joint probability distribution?

It specifies the probability of every combination of values for a set of random variables.

How do you compute a marginal distribution from a joint distribution?

By summing the probabilities of atomic events over the irrelevant variables.

What is the formula for conditional probability?

P(A|B) = P(A ∧ B) / P(B), assuming P(B) ≠ 0.

What is Bayes’ Rule?

P(A|B) = P(B|A) × P(A) / P(B). Used to compute diagnostic from causal probability.

What is a common use of Bayes’ Rule in diagnostics?

To compute P(cause|symptom), e.g., P(meningitis|stiff neck).

What is the chain rule of probability?

P(X₁,…,Xn) = Π P(Xi | X₁,…,Xi₋₁)

What is inference by enumeration?

Computing the probability of a query by summing over all atomic events consistent with the query and evidence.

When are two events A and B independent?

If P(A|B) = P(A) or equivalently P(A ∧ B) = P(A) × P(B).

Why is independence useful in probabilistic reasoning?

It reduces the size of the joint distribution from exponential to linear in the number of variables.

What is conditional independence?

X is conditionally independent of Y given Z if P(X|Y,Z) = P(X|Z).

How does conditional independence simplify modeling?

It reduces the number of parameters needed to specify the joint distribution.

What is a naive Bayes model?

A model assuming all effects are conditionally independent given a common cause. Total parameters scale linearly with number of effects.

How was Bayes' rule used in the Air France AF447 case?

Bayesian analysis was used to update the probable wreckage location based on evidence from ocean currents and debris sightings.

What is a Bayesian network?

A graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG).

What components make up a Bayesian network?

Nodes (variables), edges (dependencies), and Conditional Probability Tables (CPTs) for each variable given its parents.

What is the global semantics of a Bayesian network?

The joint probability is the product of the local conditional distributions: P(x₁,…,xₙ) = Π P(xᵢ | Parents(xᵢ)).

What is local semantics in Bayesian networks?

Each node is conditionally independent of its non-descendants given its parents.

How should you order variables when constructing a Bayesian network?

From causes to effects to ensure a compact and semantically meaningful structure.

Why is building a Bayesian network from effects to causes problematic?

It creates unnecessary dependencies and increases the size of CPTs, making the model less compact.

How does a Bayesian network improve compactness?

Instead of storing 2ⁿ entries for a full joint distribution, it stores O(n·2ᵏ) if each variable has ≤ k parents.

What is inference by enumeration in Bayesian networks?

A method that computes posterior probabilities by summing over the full joint distribution using the chain rule.

What is the time and space complexity of enumeration?

Time: O(dⁿ), Space: O(n), where d is the domain size and n is the number of variables.

How does inference by variable elimination work?

It sums out variables from right to left, storing intermediate results (factors) to avoid redundant computation.

What is the pointwise product of factors?

Combining two factors over shared variables by multiplying values for each shared assignment.

What makes a variable irrelevant in a Bayesian network query?

If it is not an ancestor of the query variable or evidence, it is irrelevant and can be omitted.

What is an example of variable irrelevance?

In P(JohnCalls | Burglary), MaryCalls is irrelevant since it's not an ancestor of the query or evidence.

How are Bayesian networks used in car diagnosis?

To identify potential causes like a flat battery or broken starter based on observed variables like oil light and gas gauge.

How are Bayesian networks used in insurance?

To estimate risk factors such as accident or theft likelihood based on driver profile and vehicle data.

What are some common tasks in Bayesian network inference?

Simple queries, conjunctive queries, value of information, sensitivity analysis, and explanation.