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Uncertainty: Understanding Probability and Sources of Uncertainty - Prof. Bruce Draper, Study notes of Computer Science

Colorado State University (CSU)Computer Science
Prof. Bruce Draper

This lecture, from the 22nd programming assignment week, introduces the topic of uncertainty. The lecture covers the concept of uncertainty caused by ignorance and predictive uncertainty, the language of probability, random variables, atomic events, unconditional and conditional probabilities, bayes rule, and the axioms of probability. The document also includes examples and exercises to help students understand the concepts.

Typology: Study notes

Pre 2010

Uploaded on 03/10/2009

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1
Uncertainty
Uncertainty
Lecture #16
Lecture #16
10/23/08
10/23/08
Stepping Back
Stepping Back…
…
We have finished talking about logic
We have finished talking about logic
–
–2
2nd
nd Programming assignment is due a week from
Programming assignment is due a week from
Tuesday
Tuesday
–
–Now would be a good time to ask questions
Now would be a good time to ask questions
We have also finished talking about search
We have also finished talking about search
–
–Although search is a component to everything in this
Although search is a component to everything in this
class
class
Our next topic is uncertainty
Our next topic is uncertainty…
…
–
–Chapter 13 of your text (today)
Chapter 13 of your text (today)
–
–Chapter 14 of your text (Tuesday)
Chapter 14 of your text (Tuesday)
Uncertainty
Uncertainty
Is every proposition true or false?
Sources of Uncertainty
Sources of Uncertainty
Consider the following statements:
Consider the following statements:
1.
1. It rained in Tuscaloosa last Friday
It rained in Tuscaloosa last Friday
2.
2. It will rain in Ft. Collins tomorrow
It will rain in Ft. Collins tomorrow
3.
3. The odds of global nuclear war is 1 in 1
The odds of global nuclear war is 1 in 1
million
million
Uncertainty caused by ignorance
Predictive uncertainty
Refers to a state of knowledge
Refers to a sampling likelihood
Subjective estimate
Subjective uncertainty
The Language of Probability
The Language of Probability
Random variables
Random variables
–
–The equivalent of a proposition in logic
The equivalent of a proposition in logic
–
–Each variable has a set of possible values
Each variable has a set of possible values
Boolean
Boolean
Discrete (finite list of possibilities)
Discrete (finite list of possibilities)
Continuous
Continuous
LoP
LoP (cont.)
(cont.)
An atomic event is the
An atomic event is the complete
complete specification of
specification of
the state of world
the state of world
–
–The world may contain many random variables
The world may contain many random variables
–
–Alternatively, it may contain only one
Alternatively, it may contain only one
–
–An atomic event specifies the value of every random
An atomic event specifies the value of every random
variable
variable
Rain example:
Rain example:
–
–It DID rain in Tuscaloosa and WON
It DID rain in Tuscaloosa and WON’
’T rain in Ft.
T rain in Ft.
Collins.
Collins.
We can assign probabilities to atomic events
We can assign probabilities to atomic events
–
–What is the probability that is rained in Tuscaloosa
What is the probability that is rained in Tuscaloosa
but not Ft. Collins?
but not Ft. Collins?
pf3
pf4

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Download Uncertainty: Understanding Probability and Sources of Uncertainty - Prof. Bruce Draper and more Study notes Computer Science in PDF only on Docsity!

UncertaintyUncertainty

Lecture #16Lecture

Stepping Back…Stepping Back…

We have finished talking about logic We have finished talking about logic

  • – 2 2 ndnd^ Programming assignment is due a week fromProgramming assignment is due a week from TuesdayTuesday
  • – Now would be a good time to ask questionsNow would be a good time to ask questions We have also finished talking about search We have also finished talking about search
  • – Although search is a component to everything in thisAlthough search is a component to everything in this classclass Our next topic is uncertainty… Our next topic is uncertainty…
  • – Chapter 13 of your text (today)Chapter 13 of your text (today)
  • – Chapter 14 of your text (Tuesday)Chapter 14 of your text (Tuesday)

UncertaintyUncertainty

Is every proposition true or false?

Sources of Uncertainty Sources of Uncertainty

Consider the following statements:Consider the following statements:

1.1. It rained in Tuscaloosa last FridayIt rained in Tuscaloosa last Friday

2.2. It will rain in Ft. Collins tomorrowIt will rain in Ft. Collins tomorrow

3.3. The odds of global nuclear war is 1 in 1The odds of global nuclear war is 1 in 1

millionmillion

Uncertainty caused by ignorance

Predictive uncertainty

Refers to a state of knowledge

Refers to a sampling likelihood

Subjective estimate Subjective uncertainty

The Language of Probability The Language of Probability

Random variablesRandom variables

  • – The equivalent of a proposition in logicThe equivalent of a proposition in logic
  • – Each variable has a set of possible valuesEach variable has a set of possible values Boolean Boolean Discrete (finite list of possibilities) Discrete (finite list of possibilities) Continuous Continuous

LoP (cont.)LoP(cont.)

An atomic event is the An atomic event is the completecomplete specification ofspecification of the state of worldthe state of world

  • – The world may contain many random variablesThe world may contain many random variables
  • – Alternatively, it may contain only oneAlternatively, it may contain only one
  • – An atomic event specifies the value of every randomAn atomic event specifies the value of every random variablevariable Rain example:Rain example:
  • – It DID rain in Tuscaloosa and WONIt DID rain in Tuscaloosa and WON’’T rain in Ft.T rain in Ft. Collins.Collins. We can assign probabilities to atomic events We can assign probabilities to atomic events
  • – What is the probability that is rained in TuscaloosaWhat is the probability that is rained in Tuscaloosa but not Ft. Collins?but not Ft. Collins?

Unconditional ProbabilitiesUnconditional Probabilities

TheThe unconditionalunconditional oror priorprior probability of anprobability of an

atomic event is our state of belief in theatomic event is our state of belief in the

event prior to any specific informationevent prior to any specific information

  • – P(rainP(rain in Tuscaloosa) = .3in Tuscaloosa) =. It rains a lot in AlabamaIt rains a lot in Alabama……..
  • – P(rainP(rain in Ft. Collins) = .1in Ft. Collins) =. Not so much in ColoradoNot so much in Colorado……

Conditional ProbabilitiesConditional Probabilities

The conditional probability of an atomic The conditional probability of an atomic

event is its probability,event is its probability, given that thegiven that the

values of some of the random variable arevalues of some of the random variable are

known.known.

Example: Example:

((P(FtP(Ft. Collins = rain | Tuscaloosa = rain) = ?. Collins = rain | Tuscaloosa = rain) =?

BayesBayes RuleRule

P ( ) b

P a b

P a b

BayesBayes Rule (II)Rule (II)

The previous equation holds for P(b The previous equation holds forP(b) > 0) > 0

  • – What happens whenWhat happens when P(bP(b) = 0?) = 0?

Bayes rule can be rewritten as the product Bayesrule can be rewritten as the product

rule:rule:

P ( a ∧ b ) = P ( a | b ) P ( ) b

The Axioms of Probability The Axioms of Probability

All of probability theory can be derivedAll of probability theory can be derived

from 3 axioms:from 3 axioms:

  • – Axiom 1:Axiom 1:^ ∀∀a,a, 0^0 ≤≤^ P(aP(a))^ ≤≤^11
  • – Axiom 2:Axiom 2: P(trueP(true) = 1,) = 1, P(falseP(false) = 0) = 0
  • – Axiom 3:Axiom 3: P(aP(a v b) =v b) = P(aP(a) +) + P(bP(b)) –– P(a^bP(a^b))

Derivable from above:Derivable from above:

  • – Lemma 1:Lemma 1: ΣΣP(aP(a) = 1) = 1 Summed over all possible values of a Summed over all possible values of a What does P(a) = .3 actually mean?

Probability as a Venn DiagramProbability as a Venn Diagram

A

B

P(a v b) = P(av b) = P(aP(a) +) + P(bP(b)) –– P(a^bP(a^b))

Simple Bayesian InferenceSimple Bayesian Inference

What is the probability of a cavity, given aWhat is the probability of a cavity, given a

toothache and that the dentists probetoothache and that the dentists probe

caught?caught?

  • – i.e. what isi.e. what is P(cavityP(cavity | toothache, catch)?| toothache, catch)?
  • – P(toothacheP(toothache ^ catch) = 0.124^ catch) = 0.
  • – P(cavityP(cavity ^ toothache ^ catch) = .108^ toothache ^ catch) =.

( )

( ) P ( ) b

Pa b P a b

∧ | =. 871

. 124 . 108 =