Probability and Uncertainty

one often has to deal with info that is underspecified, incomplete, vague, etc.

logic by itself is not sufficient for these problems.

Vagueness: Fuzzy Set Theory

model theory often based on set theory

fuzzy set theory allows something to be to some degree an element of a set

dominant approach to vagueness (mostly because wtf else can you do)

Fuzzy sets

• universe U, object x ∈ U
• membership function for fuzzy set A is defined to be function $f_A$ from U to [0,1]:
  • $f_A(x)=y$: x is a member of A to degree y
  • $f_A(x)=1$: x is certainly member of A
  • $f_A(x)=0$: x is certainly not a member of A
  • ${x | f_A(x)>0}$: support of A

modifiers (hedges)

operations on fuzzy sets:

• complement: $f_{\sim{A}}(x)=1-f_A(x)$
• union: $f_{A\cup B}(x)=\max(f_A(x),f_B(x))$
• intersection: $f_{A\cap B}(x)=\min(f_A(x), f_B(x))$
• subset: $A\subset B \iff \forall{x}(f_A(x)\leq f_B(x))$

semantics – multivalued fuzzy logic

• v(¬ A) = 1-v(A)
• v(A ∨ B) = max(v(A), v(B))
• v(A ∧ B) = min(v(A), v(B))
• v(A → B) = min(1, 1 - v(A) + v(B))

Fuzzy relations

fuzzy sets can denote fuzzy relations between objects. e.g. approximately equal, close to, much larger than, etc.

fuzzy composition:

$f_{R \circ S} (\langle x,z\rangle ) = \max_{y \in Y} \min(f_R (\langle x,y\rangle ), f_S (\langle y,z\rangle ))$

hands-on example:

Evaluation

• good – flexible, coincides with classical set theory, sever successful applications of fuzzy control
• bad – requires many arbitrary choices, tends to blur differences between probabilistic uncertainty/ambiguity/vagueness

Uncertainties: Probability Theory

General

main interpretations of probability theory:

• optivist (frequentist) probability
  • frequentism: probability is only property of repeated experiments
  • probability of event: limit of relative frequency of occurrence of event, as number of repetitions goes to infinity
• subjective probability
  • Bayesianism: probability is an expression of our uncertainty and beliefs
  • probability of event: degree of belief of idealized rational individual

sample space Ω: set of single outcomes of experiment

event space E: things that have probability (subsets of sample space). if sample space is finite, event space is usually power set.

Axioms of probability

for any event A, B:

• $0 \leq P(A) \leq 1$
• $P(\Omega) = 1$
• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

can derive:

• $P({}) = 0$
• $P(\Omega) = 1$
• $\text{if} A \subset B, P(A) \leq P(B)$

conditional probability (“A given B”):

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Joint probability distributions

for a set of random variables, it gives probability of every atomic even on those random variables.

e.g. P(Toothache, Catch, Cavity):

inference by enumeration:

• for any proposition Φ, sum atomic events where it’s true – $P(\Phi) = \sum_{\omega : \omega \models \Phi} P(\omega)$
• compute conditional probability by selecting cells – e.g. for P(¬ cavity | toothache), select toothache column and (¬ cavity) cells

use Bayes’ rule for opposite conditionals (like finding P(disease | symptom) from P(symptom | disease))

Bayesian networks

simple graphical notation for

• conditional independence assertions
• compact specification of full joint distributions

syntax:

• set of nodes, one per variable
• directed acyclic graph, with a link meaning “directly influences”
• conditional distribution for each node given its parents – $P(X_i | Parents(X_i))$

topology example:

what does it mean?

• weather is independent of other variables
• toothache and catch are conditionally independent given cavity.

Evaluation of probabilities

• good – sound theoretical basis, can be extended to decision-making, some good tools available
• bad – not always computationally easy, need lots of data which may be hard to get