AI_7 Statistical Reasoning

Bay's Theorem, Belief Network

1. Artificial Intelligence
Chap.5 Uncertainty
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Prof. KhushaliB Kathiriya

2. Outline
Acting under uncertainty
Basic probability notation
The axioms of probability
Inference using full join distributions.
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3. Artificial Intelligence
Acting under uncertainty
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Prof. KhushaliB Kathiriya

4. Acting under uncertainty
Aagentworkinginrealenvironmentalmostneverhasaccesstowhole
truthaboutitsenvironment.Therefore,agentneedstoworkunder
uncertainty.
Withknowledgerepresentation,wemightwriteA→B,whichmeansifAis
truethenBistrue,butconsiderasituationwherewearenotsureabout
whetherAistrueornotthenwecannotexpressthisstatement,thissituation
iscalleduncertainty.
Butwhenagentworkswithuncertainknowledgethenitmightbe
impossibletoconstructacompleteandcorrectdescriptionofhowits
actionswillwork.
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5. Sources of Uncertainty
1.Uncertain input
2.Uncertain knowledge
3.Uncertain output
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6. Sources of Uncertainty (Cont.)
1.Uncertain input
1.Missing data
2.Noisy data
2.Uncertain knowledge
1.Multiple causes leads to multiple effects
2.Incomplete knowledge
3.Theoretical ignorance
4.Practical ignorance
3.Uncertain output
1.Abduction, induction are uncertain
2.Default reasoning
3.Incomplete deduction inference
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7. Sources of Uncertainty (Cont.)
Uncertainty may be caused by problems with data such as:
1.Missing data
2.Incomplete data
3.Unreliable data
4.Inconsistence data
5.Imprecise data
6.Guess data
7.Default data
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8. What's the solution for uncertainty?
Probabilisticreasoningisawayofknowledgerepresentationwherewe
applytheconceptofprobabilitytoindicatetheuncertaintyinknowledge.
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9. Artificial Intelligence
Basic probability notation
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Prof. KhushaliB Kathiriya

10. Probability
Probabilitycanbedefinedasachancethatanuncertaineventwilloccur.
Itisthenumericalmeasureofthelikelihoodthataneventwilloccur.The
valueofprobabilityalwaysremainsbetween0and1thatrepresentideal
uncertainties.
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11. Probability (Cont.)
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12. Basic probability notation
1.Propositions
2.Atomic events
3.Unconditional (prior) probability
4.Conditional probability
5.Inference using full joint distribution
6.Independence
7.Bayes' rule
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13. Basic probability notation
1. Propositions
Complex propositioncanbeformedusingstandardlogical
connectives.
Forexample:
1.[(cavity=true)^(toothache=false)]
2.[(cavity^~toothache)]
Randomvariables:
Randomvariablesareusedtorepresenttheeventsandobjectsinthereal
world.
Randomvariablesarelikesymbolsinpropositionallogic.
Forexample:
P(a)=1-P(~a)
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14. Basic probability notation
2. Atomic event
An atomic event is a complete specification of the state of the
world about which agent us uncertain.
Example:
If the world consists of cavity and toothache the there are four distinct
atomic events,
1.Cavity= false ^ toothache = True
2.Cavity= false ^ toothache = false
3.Cavity= true ^ toothache = false
4.Cavity= true ^ toothache = true
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15. Basic probability notation
3. Unconditional probability
It is the degree of belief accorded to a proposition in the absence of
any other information.
Written as a P(a)
Example
Ram has cavity
P(cavity=true)=0.1 OR P(cavity)=0.1
When we want to express probabilities of all possible values of a random
variable, then vector of value is used.
P(WEATHER)= <0.7,0.2, 0.08,0.02>
P(WEATHER=sunny)=0.7
P(WEATHER=rain)=0.2
P(WEATHER=cloudy)=0.08
P(WEATHER=cold)=0.02
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16. Basic probability notation
4. Independence
It Is relation between 2 different set of full joint distributions. It is also called as
marginal or absolute independence of the variable.
Independence indicates that whether the 2 full joint distributions affects
probability each other.
The weather is independent of once dental problem.
P(toothache, catch, cavity, weather)= P(toothache, catch, cavity)P(weather)
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Toothache
catch cavity
weather
Toothache
catch
cavity
weather
Decompose into

17. Artificial Intelligence
Basic probability notation
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Prof. KhushaliB Kathiriya

18. Basic probability notation
5. Conditional Probability
Conditionalprobabilityisaprobabilityofoccurringaneventwhenanother
eventhasalreadyhappened.
Let'ssuppose,wewanttocalculatetheeventAwheneventBhasalready
occurred,"theprobabilityofAundertheconditionsofB",itcanbewritten
as:
Where P(A ^ B)= Joint probability of a and B
P(B)= Marginal probability of B.
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19. Basic probability notation
5. Conditional Probability (Cont.)
If the probability of A is given and we need to find the probability of B,
then it will be given as:
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20. Basic probability notation
5. Conditional Probability (Cont.)
??????(&#3627408436;|&#3627408437;)= ൗ
??????(&#3627408436;^&#3627408437;)
??????(&#3627408437;)
P(B) =30/100 = 0.3
P(A ^ B) = 20/100 = 0.2
P(A|B)=0.2/0.3 = 0.67
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50 20 30

21. Basic probability notation
6. Inference using Full joint Distribution
Probabilityinferencemeans,computationfromobservedevidenceof
posteriorprobabilities,forquerypropositions.Theknowledgebased
answeringthequeryisrepresentedasfulljointdistribution.
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Toothache ~Toothache
Catch ~Catch Catch ~Catch
Cavity 0.108 0.012 0.072 0.008
~Cavity 0.016 0.064 0.144 0.576

22. Basic probability notation
6. Inference using Full joint Distribution
Oneparticularcommontaskininferencingistoextractthedistributionover
somesubsetofvariablesorasinglevariable.Thisdistributionoversome
variablesorsinglevariableiscalledasmarginalprobability
(Marginalization/Summing).
P(Cavity)=0.108+0.012+0.072+0.008
=0.2
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23. Basic probability notation
6. Inference using Full joint Distribution
Computingprobabilityofacavity,givenevidenceofatoothacheisas
follow:
????????????????????????&#3627408418;????????????????????????????????????&#3627408417;????????????&#3627408417;??????)= ൘
P(Cavity^Toothache)
P(Toothache)
= ൗ
0.108+0.012
0.108+0.012+0.016+0.064
=0.6
Justtocheckalsocomputetheprobabilitythatthereisnocavitygoven
toothacheisasfollow:
??????~??????????????????&#3627408418;????????????????????????????????????&#3627408417;????????????&#3627408417;??????)= ൘
P(~Cavity^Toothache)
P(Toothache)
= ൗ
0.016+0.064
0.108+0.012+0.016+0.064
=0.4
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24. Basic probability notation
6. Inference using Full joint Distribution
Notice that in these 2 calculations the term 1/P (toothache) remains
constant, no matter which value of cavity we calculate. With this notation
we can write above two questions in one.
P(Cavity | Toothache)
= ∞ P (Cavity, Toothache)
= ∞ [P(Cavity, Toothache, Catch) + P(~Cavity, Toothache, ~Catch)]
= ∞ [<0.108 , 0.016> + <0.012 , 0.064>]
= ∞ [<0.12, 0.08>] = [<0.6, 0.4>]
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25. Artificial Intelligence
Bayes’ Rule
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Prof. KhushaliB Kathiriya

26. Basic probability notation
7. Bayes’ Rule
Bayes'theoremisalsoknownasBayes'rule,Bayes'law,orBayesian
reasoning,whichdeterminestheprobabilityofaneventwithuncertain
knowledge.
Inprobabilitytheory,itrelatestheconditionalprobabilityandmarginal
probabilitiesoftworandomevents.
Bayes'theoremwasnamedaftertheBritishmathematicianThomasBayes.
TheBayesianinferenceisanapplicationofBayes'theorem,whichis
fundamentaltoBayesianstatistics.
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Refer E-notes for bays’ example