Why Uncertainty? Sensors have noise, and consequences have probability.

Probability Review

Probability: Measure of uncertainty

• Frequentists: Objective probability computed by frequency of events
• Bayesians: Subjective probability defined based on beliefs, update on new evidence

Random Variable: Has a domain and probability distribution for each V in D

• Notation
• $P(A\lor B) = P(A) + P(B) - P(A \land B)$

Prior: $P(X)$ Unconditional probability of X

Posterior: $P(X|Y)$ Conditional probability of X given Y

Joint Distribution: Multiple random variables, combination of outcomes

• Distribution size: $d^n$
• Sum Rule: $P(A \land B) = \sum_{c\in C} P(A \land B \land c)$
• Product Rule: $P(A | B) = P(A \land B) / P(B)$
• Chain Rule: $P(A \land B \land C) = P(A | B \land C) \cdot P(B|C) \cdot P(C)$
• Bayes’ Rule: $P(A|B)=P(B|A) P(A) / P(B)$ $=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}$
• Example

Bayesian Networks

Probabilistic Inference: Determine a wanted prob given other probs

Bayes Net: Compact version of a joint distribution, created using un/conditional independence