Probabilistic Reasoning and Bayesian Belief Networks

1

• Chapter 12
• Probabilistic Reasoning and Bayesian Belief
  Networks

2
Why probabilistic reasoning?

• Because the world is an uncertain place

3
Uncertainty

• Problem with the standard logical approach
• we do not always know complete truth about the
  environment
• Example
• Leave(t) leave for airport t minutes before
  flight
• Query ?

4
Problems

• Why cant we determine t exactly?
• Partial observability
• road state, other drivers plans
• Uncertainty in action outcomes
• flat tire
• Immense complexity of modeling and predicting
  traffic

5
Problems

• Three specific issues
• Laziness
• Too much work to list all antecedents or
  consequents
• Theoretical ignorance
• Not enough information on how the world works
• Practical ignorance
• If if we know all the physics, may not have all
  the facts

6
What happens with a purely logical approach?

• We either write something which risks falsehood
• Leave(45) will get me there on time
• Or something which leads to conclusions too weak
  to do anything with
• Leave(45) will get me there on time if theres
  no snow and theres no train crossing Airport
  Road and my tires remain intact and there isnt a
  student riot blocking Hudson road and ...
• Leave(1440) might work fine, but then Id have to
  spend the night in the airport

7
Types of Uncertainty

• Uncertainty in prior knowledgeE.g., some causes
  of a disease are unknown and are not represented
  in the background knowledge of a
  medical-assistant agent

8
Types of Uncertainty

• Uncertainty in prior knowledgeE.g., some causes
  of a disease are unknown and are not represented
  in the background knowledge of a
  medical-assistant agent
• Uncertainty in actions E.g., actions are
  represented with relatively short lists of
  preconditions, while these lists are in fact
  arbitrary long

9
Types of Uncertainty

• For example, to drive my car in the morning
• It must not have been stolen during the night
• It must not have flat tires
• There must be gas in the tank
• The battery must not be dead
• The ignition must work
• I must not have lost the car keys
• No truck should obstruct the driveway
• I must not have suddenly become blind or
  paralytic
• Etc
• Not only would it not be possible to list all of
  them, trying would also be very inefficient!

10
Types of Uncertainty

• Uncertainty in prior knowledgeE.g., some causes
  of a disease are unknown and are not represented
  in the background knowledge of a
  medical-assistant agent
• Uncertainty in actions E.g., actions are
  represented with relatively short lists of
  preconditions, while these lists are in fact
  arbitrary long
• Uncertainty in perceptionE.g., sensors do not
  return exact or complete information (locality of
  sensor) about the world a robot never knows
  exactly its position

11
Types of Uncertainty

• Uncertainty in prior knowledgeE.g., some causes
  of a disease are unknown and are not represented
  in the background knowledge of a
  medical-assistant agent
• Uncertainty in actions E.g., actions are
  represented with relatively short lists of
  preconditions, while these lists are in fact
  arbitrary long
• Uncertainty in perceptionE.g., sensors do not
  return exact or complete information (locality of
  sensor) about the world a robot never knows
  exactly its position

12
Methods for handling uncertainty

• Rules with fudge factors
• Leave(25) -gt (0.03) get there on time
• Study for exam -gt(0.8) pass the exam
• Sprinkler -gt(0.99) WetGrass
• WetGrass -gt(0.7) Rain
• Problems with combinations (Sprinkler causes
  Rain?)

13
Solution Probability

• Given the available evidence, Leave(25) will get
  me there on time with probability 0.03
• Probability addresses degree of belief, not
  degree of truth
• Degree of belief changes as evidence about the
  world changes this is different from the WORLD
  changing
• Degree of truth handled by fuzzy logic
• IsSnowing is true to degree 0.2

14
Solution Probability

• Probabilities summarize effects of laziness and
  ignorance
• We will use combinations of probabilities and
  utilities to make decisions

15
Notion of Probability

• The probability of a proposition A is a real
  number P(A) between 0 and 1
• P(True) 1 and P(False) 0

16
Objective Interpretation

• Draw a ball from a bag containing n balls of the
  same size, r red and s yellow.
• The probability that the proposition A the
  ball is red is true corresponds to the relative
  frequency with which we expect to draw a red
  ball ? P(A) r/n

17
Subjective Interpretation

• There are many situations in which there is no
  objective frequency interpretation
• On a windy day, just before paragliding from the
  top of El Capitan, you say there is probability
  0.05 that I am going to die
• You have worked hard on your AI class and you
  believe that the probability that you will get an
  A is 0.9

18
Subjective or Bayesian probability

• We will make probability estimates based on
  knowledge about the world
• P(Leave(45) No Snow) 0.75
• Not assertions about the world
• Probability assessment if the world were a
  certain way
• Probabilities change with new information
• P(Leave(45) No Snow, 5 AM) 0.80
• Analogous to entailment, not truth

19
Making decision under uncertainty

• Suppose I believe the following
• P(Leave(35) gets me there on time ...) 0.04
• P(Leave(45) gets me there on time ...) 0.75
• P(Leave(60) gets me there on time ...) 0.95
• P(Leave(1440) gets me there on time ...)
  0.9999
• Which action do I choose?
• Depends on my preferences for missing flight vs.
  eating in airport, etc.
• Utility theory used to represent preferences
• Decision theory takes into account utility and
  probabilities

20
Axioms of Probability

• For any propositions A and B
• Example
• A computer science major
• B born in Iowa

21
Notation and Concepts

• Unconditional probability or prior probability
• P(Cavity) 0.1
• P(Weather Sunny) 0.55
• corresponds to belief prior to arrival of any
  (new) evidence
• Weather is a multivalued random variable
• Could be one of ltSunny, Rain, Cloudy, Snowgt
• P(Cavity) shorthand for P(Cavitytrue)

22
Joint Distribution

• k random variables X1, , Xk
• The joint distribution of these variables is a
  table in which each entry gives the probability
  of one combination of values of X1, , Xk
• Example

23
Joint Distribution Says It All

• P(Toothache) P((Toothache ?Cavity) v
  (Toothache??Cavity))
• P(Toothache ?Cavity)
  P(Toothache??Cavity)
• 0.04 0.01 0.05

24
Joint Distribution Says It All

• P(Toothache v Cavity) P((Toothache ?Cavity) v
  (Toothache??Cavity)
  v (?Toothache ?Cavity)) 0.04 0.01
  0.06 0.11

25
Conditional Probability

• DefinitionP(A?B) P(AB) P(B)
• A?B A?B/B X B/1
• Read P(AB) Probability of A given that we know
  B
• P(A) is called the prior probability of A
• P(AB) is called the posterior or conditional
  probability of A given B

26
Example

• P(Cavity?Toothache) P(CavityToothache)
  P(Toothache)
• P(Cavity) 0.1
• P(CavityToothache) P(Cavity?Toothache) /
  P(Toothache)
  0.04/0.05 0.8

27
Posterior Probabilities

• More knowledge does not change previous
  knowledge, but may render old knowledge
  unnecessary
• P(Cavity Toothache, Cavity) 1
• New evidence may be irrelevant
• P(Cavity Toothache, Paper due Wed.) 0.8

28
Car Example

• Three propositions
• Gas
• Battery
• Starts
• P(BatteryGas) P(Battery)Gas and Battery are
  independent
• P(BatteryGas,?Starts) ? P(Battery?Starts)Gas
  and Battery are not independent given ?Starts

29
Definition of Conditional Probability

• Two ways to think about it

30
Definition of Conditional Probability

• Two ways to think about it

31
Bayes Rule

• P(A ? B) P(AB) P(B) P(BA) P(A)
• Bayes rule is extremely useful in trying to
  infer probability of a diagnosis, when the
  probability of cause is known.

32
Bayes Rule

• P(A ? B) P(AB) P(B) P(BA) P(A)
• Bayes rule is extremely useful in trying to
  infer probability of a diagnosis, when the
  probability of cause is known.

33
Example

• Given
• P(Cavity) 0.1
• P(Toothache) 0.05
• P(CavityToothache) 0.8
• Bayes rule tells
• P(ToothacheCavity) (0.8 x 0.05)/0.1
  0.4

cause
34
Bayes Rule example

• Does my car need a new drive axle?
• If a car needs a new drive axle, with 30
  probability this car jerks around
• P(jerks needs axle) 0.3
• Unconditional probabilites
• P(car jerks) 1/1000
• P(needs axle) 1/10,000
• Then
• P(needs axle jerks) P(jerks needs axle)
  P(needs axle)
  ------------------------------------------
  
  P(jerks)
• (0.3 x 1/10,000) / (1/1000) .03
• Conclusion 3 of every 100 cars that jerk need an
  axle

35
Not dumb question

• Question
• Why should I be able to provide an estimate of
  P(BA) to get P(AB)?
• Why not just estimate P(AB) and be done with the
  whole thing?

36
Not dumb question

• Answer
• Diagnostic knowledge is often more tenuous than
  causal knowledge
• Suppose drive axles start to go bad in an
  epidemic
• e.g. poor construction in a major drive axle
  brand two years ago is now haunting us
• P(needs axle) goes way up, easy to measure
• P(needs axle jerks) should (and does) go up
  accordingly but how to estimate?
• P(jerks needs axle) is based on causal
  information, doesnt change

37
Simple Bayesian Concept Learning (1)

• P (HE) is used to represent the probability that
  some hypothesis, H, is true, given evidence E.
• Let us suppose we have a set of hypotheses H1Hn.
  
• For each Hi
• Hence, given a piece of evidence, a learner can
  determine which is the most likely explanation by
  finding the hypothesis that has the highest
  posterior probability.

38
Simple Bayesian Concept Learning (2)

• In fact, this can be simplified.
• Since P(E) is independent of Hi it will have the
  same value for each hypothesis.
• Hence, it can be ignored, and we can find the
  hypothesis with the highest value of
• We can simplify this further if all the
  hypotheses are equally likely, in which case we
  simply seek the hypothesis with the highest value
  of P(EHi).
• This is the likelihood of E given Hi.

39
Bayesian Belief Networks (1)

• A belief network shows the dependencies between a
  group of variables.
• If two variables A and B are independent if the
  likelihood that A will occur has nothing to do
  with whether B occurs.

• C and D are dependent on A D and E are dependent
  on B.
• The Bayesian belief network has probabilities
  associated with each link. E.g., P(CA) 0.2,
  P(CA) 0.4