1. PROBABILITY

2. Uncertainty  Let action A t = leave for airport t minutes before flight from Logan Airport  Will A t get me there on time ? Problems : 1. Partial observability ( road state, other drivers ' plans, etc.). 2. Noisy sensors ( traffic reports ). 3. Uncertainty in action outcomes ( flat tire, etc.). 4. Immense complexity of modeling and predicting traffic.

3. Uncertainty  Let action A t = leave for airport t minutes before flight from Logan Airport  Will A t get me there on time ? A purely logical approach either : 1) risks falsehood : “ A 120 will get me there on time,” or 2) leads to conclusions that are too weak for decision making : “ A 120 will get me there on time if there ' s no accident on I -90 and it doesn ' t rain and my tires remain intact, etc., etc.” ( A 1440 might reasonably be said to get me there on time but I ' d have to stay overnight in the airport … )

4. Methods for handling uncertainty  Default logic :  Assume my car does not have a flat tire  Assume A 120 works unless contradicted by evidence  Issues : What assumptions are reasonable ? How to handle contradiction ?  Probability  Model agent ' s degree of belief  Given the available evidence, A 120 will get me there on time with probability 0.4

5. Probability  Probabilistic assertions summarize effects of  laziness : failure to enumerate exceptions, qualifications, etc.  ignorance : lack of relevant facts, initial conditions, etc.  Subjective probability :  Probabilities relate propositions to agent ' s own state of knowledge e. g., P ( A 120 | no reported accidents ) = 0.6  These are not assertions about the world, but represent belief about the whether the assertion is true.  Probabilities of propositions change with new evidence :  e. g., P ( A 120 | no reported accidents, 5 a. m.) = 0.15

6. Making decisions under uncertainty  Suppose I believe the following : P ( A 135 gets me there on time |...) = 0.04 P ( A 180 gets me there on time |...) = 0.70 P ( A 240 gets me there on time |...) = 0.95 P ( A 1440 gets me there on time |...) = 0.9999  Which action to choose ?  Depends on my preferences for missing flight vs. airport cuisine, etc.  Utility theory is used to represent and infer preferences  Decision theory = utility theory + probability theory

7. Quick Question  You go to the doctor and are tested for a disease. The test is 98% accurate if you have the disease. 3.6% of the population has the disease while 4% of the population tests positive.  How likely is it you have the disease ?

8. Syntax  Basic element : random variable  Boolean random variables  e. g., Cavity ( do I have a cavity ?)  Discrete random variables  e. g., Weather is one of  Continuous random variables  represented by probability density functions ( PDFs ), valued [0,1]

9. Syntax  Elementary propositions constructed by assignment of a value to a random variable  e. g., Weather = sunny, Cavity = false  ( abbreviated as sunny or  cavity )  Complex propositions formed from elementary propositions and standard logical connectives  e. g., ( Weather = sunny  Cavity = false )  e. g., ( sunny   cavity )

10. Syntax  Atomic event :  A complete specification of the state of the world.  A complete assignment of values to variables.  E. g., if the world consists of only two Boolean variables Cavity and Toothache, then there are 4 distinct atomic events : Cavity = false  Toothache = false Cavity = false  Toothache = true Cavity = true  Toothache = false Cavity = true  Toothache = true  Atomic events are mutually exclusive and exhaustive

11. Axioms of probability ( Kolmogorov ’ s Axioms )  For any propositions A, B 1. 0 ≤ P ( A ) ≤ 1 2. P ( true ) = 1 and P ( false ) = 0 3. P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )

14. Priors  Prior or unconditional probabilities of propositions correspond to belief prior to arrival of any ( new ) evidence. e. g., P ( cavity ) = P ( Cavity = true )= 0.1 P ( Weather = sunny ) = 0.72 P ( cavity  Weather = sunny ) = 0.072

15. Distributions  Probability distribution gives probabilities of all possible values of the random variable.  Weather is one of  P ( Weather ) =  ( Normalized, i. e., sums to 1. Also note the bold font..)

16. Joint Probability The entire table sums to 1.

17. Inference by Enumeration

19. Conditional Probability

20. Inference by Enumeration

21. Conditional Probability

22. Absolute Independence

23.  Absolute independence powerful but rare  For n independent biased coins, O (2 n ) →O ( n ) 2*2*2*4=32 entries 2*2*2+4=12 entries

24. Conditional independence  If I have a cavity, the probability that the probe catches in it doesn ' t depend on whether I have a toothache :  P ( catch | toothache, cavity ) = P ( catch | cavity )  The same independence holds if I haven ' t got a cavity :  P ( catch | toothache,  cavity ) = P ( catch |  cavity )  Catch is conditionally independent of Toothache given Cavity :  P ( Catch | Toothache, Cavity ) = P ( Catch | Cavity )  Equivalent statements : P ( Toothache | Catch, Cavity ) = P ( Toothache | Cavity ) P ( Toothache, Catch | Cavity ) = P ( Toothache | Cavity ) P ( Catch | Cavity )

25. Bayes ' Rule

26. Much easier to calculate/estimate than the part on the left!

27. Bayes Rule : Example Note: probability of meningitis still very small!

28. Bayes ' Rule and conditional independence

29. Naïve Bayes Model

30. Summary  Probability is a formalism for uncertain knowledge  Basic probability statements include prior probabilities and conditional probabilities  The full joint probability distribution specifies probability of each complete assignment of values to variables.  Usually too large to create and use in its explicit form.  Absolute independence between subsets of random variables allows the full joint distribution to be factored into smaller joint distributions.  Absolute independent rarely occurs in practice.  Bayes ’ rule allows unknown probabilities to be computed from known conditional probabilities.  Applying Bayes ’ rule with many variables runs into the same scaling problem as above.  Conditional independence, brought about by direct causal relationships in the domain, may allow the full joint distribution to be factored into smaller, conditional distributions.  Naïve Bayes model assumes conditional independence of all effect variables.