Uncertainty

1
Uncertainty

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

2
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

3
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

4
What happens with a purely logical approach?

• Either risks falsehood
• Leave(45) will get me there on time
• Leads to conclusions to weak to do anything with
• Leave(45) will get me there on time if theres
  no snow and theres no train crossing Route 19
  and my tires remain intact and...
• Leave(1440) might work fine, but then Id have to
  spend the night in the airport

5
Solution Probability

• Given the available evidence, Leave(35) will get
  me there on time with probability 0.04
• Probability address uncertainty, not degree of
  truth
• Degree of truth handled by fuzzy logic
• IsSnowing is true to degree 0.2
• Probabilities summarize effects of laziness and
  ignorance
• We will use combination of probabilities and
  utilities to make decisions

6
Subjective or Bayesian probability

• We will make probability estimates based on
  knowledge about the world
• P(Leave(45) No Snow) 0.55
• Probability assessment if the world were a
  certain way
• Probabilities change with new information
• P(Leave(45) No Snow, 5 AM) 0.75

7
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 ...) 0.55
• 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

8
Axioms of Probability

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

9
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)

10
Probability Distributions

• Probability Distribution gives probability values
  for all values
• P(Weather) lt0.55, 0.05, 0.2, 0.2gt
• must be normalized sum to 1
• Joint Probability Distribution gives probability
  values for combinations of random variables
• P(Weather, Cavity) 4 x 2 matrix

11
Posterior Probabilities

• Conditional or Posterior probability
• P(Cavity Toothache) 0.8
• For conditional distributions
• P(Weather Earthquake)

12
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, Schiller in Mexico) 0.8

13
Definition of Conditional Probability

• Two ways to think about it

14
Definition of Conditional Probablity

• Another way to think about it
• Sanity check Why isnt it just
• General version holds for probability
  distributions
• This is a 4 x 2 set of equations

15
Bayes Rule

• Product rule given by
• Bayes Rule
• Bayes rule is extremely useful in trying to
  infer probability of a diagnosis, when the
  probability of cause is known.

16
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

17
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?

18
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