Lecture 7 Probability Update 2024 - Introduction to Artificial Intelligence PDF

CPCS-335 Introduction to Artificial Intelligence Lecture 7: Probability 1 Outlines Random Variable Probability Distributions Joint & Marginal Distributions /Joint Probability Conditional Distributions / Conditional Probability Probabilistic Inferen...

CPCS-335 Introduction to Artificial Intelligence Lecture 7: Probability 1 Outlines Random Variable Probability Distributions Joint & Marginal Distributions /Joint Probability Conditional Distributions / Conditional Probability Probabilistic Inference Inference by Enumeration (Exercises) Dependence Probability /Independence Probability The Product Rule >Bayes' Rule  Random variables A random variable represents an event whose outcome is unknown. i.e. Random variable is some aspect of the world about which we (may) have uncertainty R = Is it raining? T = Is it hot? D = How long will it take to drive to work? We denote Random variables with Capital letters. The range of a random variable is the set of possible values Lowercase letters: values that the random variables can take r ∈ {true, false} or sometimes we write it as {+r, –r} t ∈ {+t, –t} d ∈ [0, ∞)  Probability Distributions The probability distribution of a random variable X gives the probability for each value x in its range Associate a probability with each value; sums to 1 Temperature Weather: P(T) P(W) T P W P hot 0.5 sun 0.6 cold 0.5 rain 0.1 fog 0.3 meteor 0.0  Probability Distributions Temperature Weather: P(T) P(W) T P W P hot 0.5 sun 0.6 cold 0.5 rain 0.1 fog 0.3 meteor 0.0 A distribution is a TABLE of probabilities of values A probability (lower case value) is a single number P(W=rain) = 0.1 P(rain)=0.1 Must have: ∀ X P(X) ≥ 0 and ∑X P(X) = 1  Random variables Example: Traffic on freeway 0.25 Random variable: T = whether there’s traffic Outcomes: T in {none, light, heavy} Distribution: P(T=none) = 0.25, P(T=light) = 0.50, P(T=heavy) = 0.25 0.50 0.25  Probabilistic Inference Probabilistic inference: compute a desired probability from other known probabilities (e.g. conditional from joint) We generally compute conditional probabilities P(airport on time | no accidents) = 0.90 These represent the agent’s beliefs given the evidence Probabilities change with new evidence: P(airport on time | no accidents, 5 a.m.) = 0.95 P(airport on time | no accidents, 5 a.m., raining) = 0.80 Observing new evidence causes beliefs to be updated  Inference in Ghostbusters A ghost is in the grid somewhere Sensor readings tell how close a square is to the ghost On the ghost: usually red 1 or 2 away: mostly orange 3 or 4 away: typically yellow 5+ away: often green Sensors are noisy, but we know P(Color | Distance) Ghostbusters, Revisited Demo