Artificial Intelligence Basics Uncertainty Reasoning

At present, in artificial intelligence, the main mathematical tools for dealing with uncertainty problems include probability theory and fuzzy mathematics. Currently commonly used mathematical methods for uncertainty reasoning mainly include likelihood reasoning based on probability (Plausible Reasoning), fuzzy inference based on fuzzy mathematics...

Edited at 2022-11-07 23:27:18

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Artificial Intelligence Basics Uncertainty Reasoning

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Outline

Uncertainty Reasoning

matching algorithm

Subjective Bayesian approach

In subjective Bayesian methods, knowledge is represented by the following production rule: IF E THEN (LS,LN) H. (LS, LN) respectively measure the degree of support of evidence (premise) E for conclusion H and the degree of support of ~E for H. The value range of LS and LN is [0, ∞), and their specific values are determined by domain experts.

O(H|E) =LS × O(H)

When E is true, LS can be used to update the prior probability O(H) of H to its posterior probability O(H|E); When E is false, LN can be used to update the prior probability O(H) of H to its posterior probability O(H|~E);

O(H|~E)=LN×O(H)

The credibility of the full evidence depends on the partial evidence, expressed as P(E|S). If all the evidence is known, then E=S, and P(E|S)=P(E). Among them, P(E) is the prior likelihood of evidence E, P(E|S) is the trust in E after knowing partial knowledge S in the full evidence E, and is the posterior likelihood of E.

(1)The evidence must be true

(2) The evidence is definitely false

(3) The evidence is neither true nor false P(H|S)=P(H|E)×P(E|S) P(H|┐E)×P(┐E|S)

①P(E|S)=1 When P(E|S)=l, P(┐E|S)=0. then there is

②P(E|S)=0 When P(E|S)=0, P(┐E|S)=1. then there is

③P(E|S)=P(E) When P(E|S)=P(E), it means that E has nothing to do with S. It can be obtained from the total probability formula: P(H|S)=P(H|E)×P(E|S) P(H|┐E)×P(┐E|S) =P(H|E)×P(E) P(H|┐E)×P(┐E) =P(H)

④When P(E|S) is other values, the value of P(H|S) can be obtained through the piecewise linear interpolation function of the above three special points.

Update algorithm

Due to the appearance of evidence E, P (H) becomes P(H|E)

The Bayes method is to study the use of evidence E to update the prior probability P(H) to the posterior probability P(H|E)

probabilistic reasoning

Multiplication theorem P(AB)=P(A)P(B|A)

Suppose A, B1, B2,…,Bn are some events, P(A)>0, B1, B2,…,Bn are disjoint with each other, P(Bi)>0, i=1, 2,…, n, then For k=1, 2, …, n,

Total probability formula: Suppose B1, B2,...Bn are disjoint with each other, and P(Bi)>0, i=1,2,...,n, then for any event A, P(A)=∑iP(Bi)P (A|Bi)

Priori probability. Refers to the probability of occurrence of each event determined based on historical data or subjective judgment. This type of probability has not been experimentally confirmed and belongs to the probability before testing.

Posterior probability. Refers to the probability obtained by obtaining new additional information through surveys or other methods, and using Bayes' formula to modify the prior probability.

total probability formula

Conditional probability of subtopic event B

If E Then H, where E is the premise and H is the conclusion. Then the conditional probability P(Hi|E) represents the probability of Hi when E occurs, which can be used as the degree of certainty of the conclusion Hi when evidence E occurs.

Synthesis Algorithm for Conclusion Uncertainty

Uncertainty Transfer Algorithm

Uncertainty Expression of Evidence

What plays a key role in probabilistic reasoning is the so-called Bayes formula, which is also the basis of the subjective Bayes method.

The relationship between probability and probability

Uncertainty representation of knowledge

Reasoning using Bayes' formula