AI_7 Statistical Reasoning

Artificial Intelligence
Chap.5 Uncertainty
Prepared by:
Prof. Khushali B Kathiriya

Outline
 Acting under uncertainty
 Basic probability notation
 The axioms of probability
 Inference using full join distributions.
Prepared by: Prof. Khushali B Kathiriya
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Acting under uncertainty
Prepared by:

Acting under uncertainty
 A agent working in real environment almost never has access to whole
truth about its environment. Therefore, agent needs to work under
uncertainty.
 With knowledge representation, we might write A→B, which means if A is
true then B is true, but consider a situation where we are not sure about
whether A is true or not then we cannot express this statement, this situation
is called uncertainty.
 But when agent works with uncertain knowledge then it might be
impossible to construct a complete and correct description of how its
actions will work.
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Sources of Uncertainty
1. Uncertain input
2. Uncertain knowledge
3. Uncertain output
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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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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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What's the solution for uncertainty?
 Probabilistic reasoning is a way of knowledge representation where we
apply the concept of probability to indicate the uncertainty in knowledge.
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Basic probability notation
Prepared by:

Probability
 Probability can be defined as a chance that an uncertain event will occur.
It is the numerical measure of the likelihood that an event will occur. The
value of probability always remains between 0 and 1 that represent ideal
uncertainties.
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Probability (Cont.)
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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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1. Propositions
 Complex proposition can be formed using standard logical
connectives.
 For example:
1. [(cavity=true) ^ (toothache = false)]
2. [(cavity ^ ~toothache)]
 Random variables:
 Random variables are used to represent the events and objects in the real
world.
 Random variables are like symbols in propositional logic.
 For example:
 P(a)= 1- P(~a)
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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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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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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

5. Conditional Probability
 Conditional probability is a probability of occurring an event when another
event has already happened.
 Let's suppose, we want to calculate the event A when event B has already
occurred, "the probability of A under the conditions of B", it can be written
as:
 Where P(A ^ B)= Joint probability of a and B
 P(B)= Marginal probability of B.
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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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5. Conditional Probability (Cont.)
 𝑃(𝐴|𝐵) = ൗ
𝑃(𝐴 ^ 𝐵)
𝑃(𝐵)
 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

6. Inference using Full joint Distribution
 Probability inference means, computation from observed evidence of
posterior probabilities, for query propositions. The knowledge based
answering the query is represented as full joint distribution.
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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

 One particular common task in inferencing is to extract the distribution over
some subset of variables or a single variable. This distribution over some
variables or single variable is called as marginal probability
(Marginalization/ Summing).
 P(Cavity) = 0.108+ 0.012 + 0.072 + 0.00 8
= 0.2
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 Computing probability of a cavity, given evidence of a toothache is as
follow:
 𝐏 𝐂𝐚𝐯𝐢𝐭𝐲 𝐓𝐨𝐨𝐭𝐡𝐚𝐜𝐡𝐞) = ൘
P(Cavity ^ Toothache)
P (Toothache)
= ൗ
0.108+ 0.012
0.108+ 0.012+0.016 + 0.064
= 0.6
 Just to check also compute the probability that there is no cavity goven
toothache is as follow:
 𝐏 ~ 𝐂𝐚𝐯𝐢𝐭𝐲 𝐓𝐨𝐨𝐭𝐡𝐚𝐜𝐡𝐞) = ൘
P(~ Cavity ^ Toothache)
P (Toothache)
= ൗ
0.016 + 0.064
0.108+ 0.012+0.016 + 0.064
= 0.4
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 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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Bayes’ Rule
Prepared by:

7. Bayes’ Rule
 Bayes' theorem is also known as Bayes' rule, Bayes' law, or Bayesian
reasoning, which determines the probability of an event with uncertain
knowledge.
 In probability theory, it relates the conditional probability and marginal
probabilities of two random events.
 Bayes' theorem was named after the British mathematician Thomas Bayes.
The Bayesian inference is an application of Bayes' theorem, which is
fundamental to Bayesian statistics.
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Refer E-notes for bays’ example

AI_7 Statistical Reasoning

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