人工智能不确定推理推理概述与概率论基础

Artificial IntelligLeabharlann Baidunce
Foundation of probability 概率论基础
Artificial Intelligence
Random events 随机事件
Artificial Intelligence
Artificial Intelligence
Artificial Intelligence
Artificial Intelligence
Frequency definition
If a process or an experiment is repeated a large number of times, n, and if the characteristic, E, occurs m times, then the relative frequency, m/n, of E will be approximately equal to the probability of E. Around 1900, the English statistician Karl Pearson heroically tossed a coin 24,000 times and recorded 12,012 heads, giving a proportion of 0.5005.
Artificial Intelligence
Axioms of probability 概率公理
Axiom 1. 0 ≤ P(A) ≤ 1. Axiom 2. P(S) = 1 Axiom 3. If A and B are mutually exclusive events then P(A ∪ B) = P(A) + P(B) Generalization of Axiom 3 If A1, A2, …, An are mutually exclusive events in a sample space S then P(A1 ∪ A2 ∪ … ∪ An) =P(A1) + P(A2) + … + P(An)
Artificial Intelligence
Joint probability 联合概率
If A and B are any two events then we write P(A given B) or P(A|B) to indicate the probability of A among the subset of cases in which B is known to have occurred.
Chapter 4 Uncertainty Reasoning
不确定推理
Artificial Intelligence
The limitation of FOL FOL的局限性
一阶逻辑作为形式语言存在局限性，其远远不能表示人类 自然语言所能表达的全部知识，而人类所认识的知识类比 自然语言所能表达的知识类还多。因此一阶逻辑所能表示 的知识范围十分有限。从思维推理的角度看，一阶谓词逻 辑所体现的逻辑推理规律，只能用来模拟人类的部分逻辑 思维现象，也远远不能代表人类逻辑思维的全部。因此各 种非规范逻辑的研究很有必要。 FOL过分抽象和简单，二值逻辑(children world)。 经典集合论:
Artificial Intelligence
Joint probability 联合概率
The probability of all events other than an event A is denoted by P(Ac) [Ac stands for “A complement”] or [“A bar”]. Note that P(Ac) = 1 – P(A). If A and B are any two events then we write P(A or B) or P(A ∪ B) to indicate the probability that event A or event B (or both) occurred. If A and B are any two events then we write P(A and B) or P(AB) or P(A ∩ B) to indicate the probability that both A and B occurred.
Artificial Intelligence
Classical definition
If an event can occur in N mutually exclusive, equally likely ways, and if m of these possess characteristic E, then the probability P(E) is equal to m/N. Example: What is the probability of rolling a total of 7 on two dice?
a, A a A or a A
Artificial Intelligence
4.1 Overview
“Uncertainty reasoning” refers in a general way to problems discussed by that subset of the AI community interested in representing and reasoning with knowledge that cannot be expressed as certainties. The range of problems discussed runs the gamut from fundamental philosophical inquiry into the nature of uncertainty and how (if at all) it can be measured and modelled, to practical performance issues arising from the (automatic) construction of real-world models and making inferences from such models.
A ∪ B is the entire shaded area
Artificial Intelligence
Conditional probability 条件概率
Prior probability (先验概率): the probability before we consider any additional knowledge：P(A). The conditional (or Posterior 后验) probability of an event A given B (i.e. given that B has occurred) is denoted P(A | B).
1.Representation issue(表示问题) ‫ ٭‬Evidence uncertainty (E, C(E)) ‫ ٭‬Rule uncertainty (EH, f(H,E)) 2.Computation issue（计算问题） ‫ ٭‬Evidence combination ‫ ٭‬Propagation algorithm of uncertainty ‫ ٭‬The composition of conclusion uncertainty 3.Semantic issue（语义问题） ‫ ٭‬The meaning of above representation & computation
Artificial Intelligence
不确定性是普遍存在的，是现实世界的本质属性。 人类的实践和认识活动并不仅仅在于追求确定性， 更重要的是消除不确定性。 不确定性渗透在人类的思想、行动之中，是各种理 论的前提和基础。 不确定性根源于客观事物或过程自身和认识主体以 及认识主体与客体之间的复杂关系中，具有本体论 与认识论的含义。
Artificial Intelligence
An Objectivist uses the classical and/or relative frequency definition to determine probabilities. A Subjectivist uses all three methods to assess the likelihood of an event.
p
Artificial Intelligence
Personal probability or subjective probability 个人概率 或 主观概率
It has reference to reasonableness of belief or expectation. Personal assessment of which sports team will win a match. Personal assessment of which is more likely to provide a cure – traditional healing or modern medicine. Subjective probability can be determined by comparison or gambling.
Artificial Intelligence
Uncertainty in Rules
Rule
Premise
Conclusion
Error
Parameter
Evidence combination
Error
Parameter
Artificial Intelligence
4.1.2 The fundamental issues of uncertainty reasoning
Artificial Intelligence
4.1.1 The sources of uncertainty
Objective uncertainty ‫ ٭‬Randomness (随机性)
▪ Coin tossing, die tossing, lotteries, etc.
‫ ٭‬Fuzziness (模糊性)
Artificial Intelligence
4.1.3 The classification of uncertainty reasoning method
pure probalitym ethod subjectiveBayes on theroy confirm ati m ethod num berical Evidencetheory Model m ethod Fuzzy Logic Roughset classical Logic m ethod non num berical reasoning non m onotonic Control m ethod
The meaning of probability
Probability provides a measure of uncertainty associated with the occurrence of events or outcomes – Models and theory Probability - meaning ‫ ٭‬classical ‫ ٭‬frequentist ‫ ٭‬subjective (personal)
▪ Tall, beautiful, young, etc.
Subjective uncertainty ‫ ٭‬Ambiguity (歧义性) ‫ ٭‬Incompleteness (不完全) ‫ ٭‬Inexactness (不精确) ‫ ٭‬Unreliability (不可靠) ‫ ٭‬Inconsistency (不一致)