Topic 4. Representation and Reasoning with Uncertainty

Transcription

1 Toic 4 Reresetatio ad Reasoig with Ucertaity Cotets 4.0 Reresetig Ucertaity 4. Probabilistic methods Bayesia PART III 4.2 Certaity Factors CFs 4.3 Demster-Shafer theory 4.4 Fuzzy Logic 4. Probabilistic methods Bayesia Alterative form of Bayes Rule Bayes Rule has the form: But to calculate this we eed ad fte is ot available. Sometimes we might ot have but have.g. do t kow has_cold but we do kow seezehas_cold a doctor who oly sees sick eole caot estimate what roortio of the world have colds but ca estimate what roortio of her atiets with colds are seezig A alterative versio of Bayes rule ca be derived. 2

2 Probabilistic methods Bayesia Alterative form of Bayes Rule A alterative versio of Bayes rule ca be derived.. Bayes Rule: 2. We kow: *P + * 3. Substitute ito Bayes Rule I this form we just eed ad Probabilistic methods Bayesia xamle usig Alterative form of Bayes Rule Bayes Rule Alterative Form Give the followig data low oil 0.3 overheat low oil 0.85 overheat low oil 0.2 Use Bayes alterative form to calculate low oil overheat +

3 Toic 4 Reresetatio ad Reasoig with Ucertaity 4. Probabilistic methods Bayesia Multile Sources of videce 5 4. Probabilistic methods Bayesia What haes whe we have more tha oe source of evidece? Sometimes we have multile sources of evidece for our hyothesis.g. if seezig the has_cold 0.8 if has_fever the has_cold 0.3 ow do we combie? 6 3

4 4. Probabilistic methods Bayesia Combiig evidece sources i coditioal robabiliites Method I: Treat as & 2 & 3... * / > & 2 & 3... & 2 & 3 * & 2 & 3 > & 2 & 3... & 2 & 3 * & 2 & 3 + & 2 & 3 BUT: this would requires us to kow 2 + distict robabilities e.g. robability of seezig/o_fever give cold robability of seezig/fever give cold robability of ot_seezig/fever give cold robability of ot_seezig/o_fever give cold etc Probabilistic methods Bayesia Let us cosider the followig roblem: ifectio ai caries 0.52 ifectio ai caries 0.05 ifectio ai caries 0.25 ifectio ai caries 0. ifectio ai caries 0.5 ifectio ai caries 0.2 ifectio ai caries 0.08 ifectioai caries 0.65 caries 0.3 caries 0.7 If we observe ai ad ifectio what is the robability of caries? caries ifectio ai 0.3*0.52/0.3* * If atiet has ai but o ifectio: caries ai ifectio 0.3*0.5/0.3* *

5 5 9 The roblem is that assumig the i are boolea we would eed to kow the value of 2 + robabilities. If we have 0 ossible variables we eed to measure 2048 robabilities. Real roblems ca have hudreds or thousads of variables! Very retty but theoretically itractable Probabilistic methods Bayesia 0 A meas of aroximatig the coditioal robability of a set of evideces. Starts with o evidece ad adds i each evidece source oe at a time. Comlexity is far lower. Alterative formula: Bayes Icremetal Rule ld form: Icremetal form: 4. Probabilistic methods Bayesia

6 4. Probabilistic methods Bayesia Alterative formula: Bayes Icremetal Rule ca be iterreted as a evet that cosists of the simultaeous observatio of a set of evideces I we ca iterret as the observatio of a additioal evidece that is reseted to us after havig observed the set of evideces. Therefore we have total evidece The equatio tells us how it chages our belief i whe we are give a ew evidece. If with the evidece our belief i is whe we observe some ew evidece we should chage our belief to i this way 4. Probabilistic methods Bayesia Bayes Icremetal Rule: Assumig Ideedece Assumig that each evidece source is ideedet: Thus:. 2 6

7 4. Probabilistic Regla de Bayesmethods icremetal Bayesia y suoiedo ideedecia ow do we use this rule? Iitially {}. Thus 2 We are reseted with the first evidece. Udate : { } { } * / 3 We are reseted with the 2d evidece 2. Udate: { 2 } { 2 } * 2 / 2 Problem: ofte the evideces are ot ideedet ad usig this formula will give errors ad 2 it ca give absurd values >. 3 Icremetatl Bayes: ormalised ad assumig ideedece + Iitially {}. - 2 We are reseted with the first evidece. Udate: { } { } * / - 3 We are reseted with the secod evidece 2. Udate: { 2 } { 2 } * 2 / - The solutio is still a estimate but o loger gives absurd values 4 7

8 8 5 xamle: seezig 0.4 cold 0.3 seezig cold 0.75 fever cold 0.2 fever cold 0.7 What is the robability of havig a cold if seezig ad fever? coldseezig 0.56 from rior calculatio coldseezigfever 0.56 *0.7 / 0.56* * /