1. Uncertain Reasoning CPSC 315 – Programming Studio Spring 2009 Project 2, Lecture 6

2. Reasoning in Complex Domains or Situations Reasoning often involves moving from evidence about the world to decisions Systems almost never have access to the whole truth about their environment Reasons for lack of knowledge Cost/benefit trade-off in knowledge engineering Less likely, less influential factors often not included in model No complete theory of domain Complete theories are few and far between Incomplete knowledge of situation Acquiring all knowledge of situation is impractical

3. Forms of Uncertain Reasoning Partially-believed domain features E.g. chance of rain = 80% Probability (focus of today’s lecture) Other (we will return to this) Partially-true domain features E.g. cloudy =.8 Fuzzy logic (outside scope of this class)

4. Making Decisions to Meet Goals Decision theory = Probability theory + Utility theory Decisions – the outcome of system’s reasoning, actions to take or avoid Probability – how system reasons Utility – system’s goals / preferences

5. 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?

6. Quick Question 2 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 7% of the population tests positive. How likely is it you have the disease?

7. Basics of Probability Unconditional or prior probability Degree of belief of something being true in absence of any information P (cavity = true) = 0.1 or P (cavity) = 0.1 Implies P (not cavity) = 0.9

8. Basics of Probability Unconditional or prior probability Can be for a set of values P (Weather = sunny) = 0.7 P (Weather = rain) = 0.2 P (Weather = cloudy) =.08 P (Weather = snow) =.02 Note: Weather can have only a single value – system must know that rain and snow implies clouds

9. Basics of Probability Conditional or posterior probability Degree of belief of something being true given knowledge about situation P (cavity | toothache) = 0.8 Mathematically, we know P (a | b) = P (a ^ b) / P (b) Requires system to know unconditional probability of combinations of features This knowledge becomes exponential relative to the size of the feature set

10. Bayes’ Rule Remember: P (a | b) = P (a ^ b) / P (b) Can be rewritten P (a ^ b) = P (a | b) * P (b) Swapping a and b features yields P (a ^ b) = P (b | a) * P (a) Thus P (b | a) * P (a) = P (a | b) * P (b) Rewriting we get Bayes’ Rule P (b | a) = P (a | b) * P (b) / P (a)

11. Reasoning with Bayes’ Rule Bayes’ Rule P (b | a) = P (a | b) * P (b) / P (a) Example Let’s take P (disease) = 0.036 P (test) = 0.04 P (test | disease) = 0.98 P (disease | test) = ?

12. Reasoning with Bayes’ Rule Bayes’ Rule P (b | a) = P (a | b) * P (b) / P (a) Example P (disease) = 0.036 P (test) = 0.04 P (test | disease) = 0.98 P (disease | test) = ? = P (test | disease) * P (disease) / P (test) = 0.98 * 0.036 / 0.04 = 88.2 %

13. Reasoning with Bayes’ Rule What if test has more false positives Still 98% accurate for those with disease Example P (disease) = 0.036 P (test) = 0.07 P (test | disease) = 0.98 P (disease | test) = ? = P (test | disease) * P (disease) / P (test) = 0.98 * 0.036 / 0.07 = 50.4 %

14. Reasoning with Bayes’ Rule What if test has more false negatives Now 90% accurate for those with disease Example P (disease) = 0.036 P (test) = 0.04 P (test | disease) = 0.90 P (disease | test) = ? = P (test | disease) * P (disease) / P (test) = 0.90 * 0.036 / 0.04 = 81 %

15. Combining Evidence What happens when we have more than one piece of evidence Example: toothache and tool catches on tooth P (cavity | toothache ^ catch) = ? Problem: toothache and catch are not independent If someone has a toothache there is a greater chance they will have a catch and vice-versa

16. Independence of Events Independence of features / events Features / events cannot be used to predict each other Example: values rolled on two separate die Example: hair color and food preference Probabilistic reasoning works because systems divide domain into independent sub-domains Do not need the exponentially increasing data to understand interactions Unfortunately, non-independent sub-domains can still be huge (have many interacting features)

17. Conditional Independence What happens when we have more than one piece of evidence Example: toothache and tool catches on tooth P (cavity | toothache ^ catch) = ? Conditional independence Assume indirect relationship Example: toothache and catch are both caused by cavity but not any other feature Then P (toothache ^ catch | cavity) = P (toothache | cavity) * P (catch | cavity)

18. Conditional Independence This let’s us say P (toothache ^ catch | cavity) = P (toothache | cavity) * P (catch | cavity) P (cavity | toothache ^ catch) = ? = P (toothache ^ catch | cavity) * P (cavity) = P (toothache | cavity) * P (catch | cavity) * P (cavity) Avoids requiring system to have data on all permutations Difficulty: How true? What about a chipped or cracked tooth?

19. Human Reasoning Studies show people, without training and prompting, do not reason probabilistically People make incorrect inferences when confronted with probabilities like those of the last few slides If asked for all prior and posterior probabilities then they will posit systems with rather large inconsistencies

20. Human Reasoning Studies show people, without training, do not reason probabilistically Some systems have used non-probabilistic forms of uncertain reasoning Qualitative categories rather than numbers Must be true, highly likely, likely, some chance, unlikely, virtually impossible, impossible Rules for how these combine based on human reasoning Value depends on where belief values come from If belief values from external evidence about world then use probability If belief values provided by user then non-probabilistic approach may do better