Probability

Uncertainty:

• observed variables (evidence)

• unobserved variables

• model

  • agent knows something about how the known variables relate to the unknown variables

Random variables:

• some aspect of the world about which we (may) have uncertainty

  • R = is it raining?

  • T = is it hot or cold?

  • D = how long does it take to drive to work?

  • L = where is the ghost?

• We denote with capital letters

• Have domains

  • R = {true, false}

  • T = {hot, cold}

  • D = {0, infinity}

  • L = {(0,0), (2, 3)}

Probability Distributions

• associate a probability with each value

• unobserved random variables have distributions

• ex. P(T): T P Where P(Hot) = 0.5

  Hot 0.5

  Cold 0.5

Joint Distributions

• over a set of random variables: X₁, X_{2, …} X_n

• specifies a real number for each assignment (or outcome)

  • P(X₁ = x₁, X₂ = x_{2 …} X_n = x_{n OR} P(x₁, x_{2 …,} x_n)

• Size of distribution if n variables with domain size d?

  • dⁿ

Probabilistic Models

• a joint distribution over a set of random variables

• (Random) variables with domains

• assignments are called outcomes

• say whether assignments are likely

• normalized: sum to 1

Events:

• a set of E outcomes

  • P(E) = ∑ P(x₁, x_{2 …,} x_n)

• from a joint distribution, we can calculate the probability of any event

Marginal Distributions

• sub-tables which eliminate variables

• Marginalization (summing out)

  • combining collapsed rows by adding

Conditional Probabilities

• simple relation between joint and conditional probabilities

• P(a | b) = P(a, b) / P(b)

Conditional Distributions

• probability distributions over some variables given fixed values of others

• ex. P(W | T = hot)

  • capital letter is an array of all the values

Normalization Trick

• Step 1: select the joint probabilities matching the evidence

• Step 2: normalize the selection (make it sum to 1) by dividing

Probabilistic Inference:

• compute a desired probability from other known probabilities

• generally compute conditional probabilities

• probabilities change with new evidence

The Product Rule

• P(y)P(x | y) = P(x, y)

The Chain Rule

• P(x₁, x_2, x₃) = P(x₁)P(x₂ | x₁)P(x₃ | x₁ ⋅ x₂)

Bayes Rule

• P(x, y) = P(x | y)P(y) = P(y | x)P(x)

  • P(x | y) = [ P(y | x) / P(y) ] * P(x)