Lecture 05 Rule-based Uncertain Reasoning Topics Information Uncertainty Bayesian Inference Model Certainty Factor Model Discussion

Information uncertainty Information can be incomplete, inconsistent, uncertain, or imprecise. Incomplete: missing knowledge Inconsistent: conflicting Knowledge, such as from different experts Uncertainty: lack of exact knowledge Imprecise: ambiguous knowledge, such as terms of often, sometimes, frequently and hardly ever

Information uncertainty

Bayesian inference model Representation (Cause-effect rule) IF H is true THEN E is true {with probability p} Semantics IF event H occurs, THEN the probability that event E will occur is p, or p(E|H) = p. p(H), p(E): prior probability p(EH), p(HE): conditional probability Inference Given p(H) and p(E|H) Find p(H|E).

Bayesian inference model Bayesian rule

Bayesian inference model Single evidence and single hypothesis Single evidence and multiple hypotheses

Bayesian inference model Multiple evidence and multiple hypotheses Multiple evidence and multiple hypotheses under conditional independence

Bayesian inference model Example: Naïve Bayesian Classifier P(Hi) H1 H2 H3 Hm P(Ej|Hi) E1 E2 E3 En

Characteristics of Bayesian inference Humans are hard eliciting probability values consistent with the Bayesian rules. Humans tend to make different assumptions when assessing the conditional and prior probabilities. Bayesian Inference is most appropriate in the domains where reliable statistical data exist, for instance, forecasting. Bayesian inference is of exponential complexity, and thus is impractical for large knowledge bases.

Certainty factor model Representation (diagnostic rule) IF evidence E THEN hypothesis H {cf} Semantics Given that evidence E has occurred, we have cf degree of belief that hypothesis H will happen. { –1 <= cf <= +1} Inference Given cf(E) and cf Find cf(H, E)

Certainty factor model Meaning of certainty factors

Certainty factor model Single evidence and single hypothesis IF evidence E THEN hypothesis H {cf } cf (H, E) = cf (E)  cf Conjunctive rules cf (H, E1E2...En) = min [cf (E1), cf (E2),..., cf (En)]  cf

Certainty factor model Disjunctive rules cf (H, E1E2...En) = max [cf (E1), cf (E2),..., cf (En)]  cf

Certainty factor model Multiple rules conclude with the same hypothesis cfi is the confidence in hypothesis H established by Rule i

Characteristics of Certain factors Certainty factors are used in domains where the probabilities are not known or are too difficult or expensive to obtain, for instance in medicine. The evidential reasoning mechanism can manage incrementally acquired evidence, the conjunction and disjunction of hypotheses, as well as evidence with different degrees of belief.

Discussion More inexact reasoning models Likelihood inference Dempster-Shafer evidential reasoning Fuzzy inference