## Probability

## AI, Decision Making, and Probability

About decision-making

Uncertainty, likelihood, and probability

Decision-theoretic methods

## Probability

Idea:

*chance*and*likelihood*are important concepts for real reasoningMethod: assign probabilities to events and combinations of events

Reason from model using calculation

This is a general plan:

`Evidence -----------------> Conclusions | ^ | | v | Model -----------------> Extrapolation`

## Probability of Events

Domain: events

*pr(E)*is probability that event*E*happens*pr(E) = #E / (#E + #¬E)*For a coin

*pr(H) = #H / (#H + #T) = 1 / 2*For a pair of dice:

*pr(R7) = #R7 / #R**#R7 = #{(1, 6), (2, 5), (3, 4), …} = 6**#R = #{D×D} = 36**pr(R7) = 6 / 36 = 1/6*Check with computer program [prob7](http://github.com/pdx-cs-ai/prob7]

## Probability of Logical Situations

Domain: propositional formula

*pr(p)*is probability of logical combination of events*p*is a prop formula

Priors and conditionals

*pr(p|q)*is prob of*p*given*q*(easy to get backward)

## Axioms Of Probability

equivalence: if

*p ≡ q*then*pr(p) = pr(q)*range:

*0 ≤ pr(p) ≤ 1*negation:

*pr(¬p) = 1 - pr(p)*conjunction:

*pr(p ∧ q) = pr(p) pr(q|p) = pr(q) pr(p|q)*

## Derived Probability Rules

disjunction:

*pr(p ∨ q) = pr(¬(¬p ∧ ¬q)) = 1 - pr(¬p ∧ ¬q)*Bayes's Rule:

`pr(q) pr(p|q) = pr(p ∧ q) = pr(p) pr(q|p) pr(q) pr(p|q) = pr(q|p) pr(p) pr(p|q) = pr(q|p) pr(p) / pr(q)`

## Independence

When

*p*and*q*are conditionally independent*pr(p|q) = pr(p)*By Bayes's Rule

`pr(q|p) = pr(p|q) pr(q) / pr(p) = pr(p) pr(q) / pr(p) = pr(q)`

In this case, conjunction gets easier

*pr(p ∧ q) = pr(p|q) pr(q) = pr(p) pr(q)*

When

*p*and*q*are strictly independent*pr(p ∧ q) = 0*In this case, disjunction gets easier

`pr(p ∨ q) = 1 - pr(¬p ∧ ¬q) = 1 - pr(¬p) pr(¬q) = 1 - (1 - pr(p))(1 - pr(q)) = 1 - (1 - pr(p) - pr(q) - pr(p) pr(q)) = pr(p) + pr(q) + pr(p) pr(q) = pr(p) + pr(q)`

## Bayes's Rule

Given

evidence

*E*with*prior probability**pr(E)*hypothesis

*H*with pp*pr(H)*probability

*pr(E|H)*of the evidence given that the hypothesis holds

We want

- probability
*pr(H|E)*that the hypothesis holds given the evidence

- probability
By Bayes's Rule

*pr(H|E) = pr(E|H) pr(H) / pr(E)*

## The Medical Example

H = "You have Glaubner's Disease"

E = "Reaper's Test is positive"Rare disease:

*pr(H) = 1e-6*Low false positive rate:

*pr(E|¬H) = 1e-4*Perfect false negative rate:

*pr(E|H) = 1*`pr(H|E) = pr(H) pr(E|H) / pr(E) = pr(H) / pr(E∧H ∨ E∧¬H) = pr(H) / (pr(E|¬H) pr(¬H) + pr(E|H) pr(H)) = 1e-6 / (1e-4 × (1-1e-6) + 1e-6) ~= 1e-6 / 1e-4 = 1e-2 = 0.01`

IRL the false negative rate will be nonzero too, so you will not learn a ton from the test either way

Been there

## Bayesian Belief Networks

Bayes Net (BBN, influence diagram) : indicate which priors and conditionals have significant influence in practice

`Cloudy / \ Sprinkler Rain \ / Wet Grass`

Usually reason one of two ways:

causal/top-down:

`p(W|C) = p(W|S∨R) = 1 - p(W|¬S) p(¬S|C) p(C) × p(W|¬R) p(¬R|C) p(C)`

diagnostic/bottom-up:

*p(C|W)*

Polytrees: special case for easy computation

Problem: everything depends on everything else

Need to know impossible number of prior and conditional probabilities to conclude anything

Maybe

*learn*probabilities?

## Is Your Probabilistic Model Meaningful?

Difference between 0.5 and "don't know" and "don't care"

MYCIN and probabilities

*vs*"likelihoods"Consequence of Cox's Theorem: under reasonable assumptions, any labeling of logical sentences with real numbers will be consistent with probability

Measurement issues; numeric issues