Introduction to Probabilities

1. Introduction to Probabilities Farrokh Alemi, Ph.D. Saturday, February 21, 2004 Updated by Janusz Wojtusiak Fall 2009

2. Probability can quantify how uncertain we are about a future event

3. Why measure uncertainty? • To make tradeoffs among uncertain events  • To communicate about uncertainty

4. What is probability? In the Figure, where are the events that are not “A”?

5. How to Calculate Probability?

6. Calculus of Probabilities Helps Us Keep Track of Uncertainty of Multiple Events Joint probability, probability of either event occurring, revising probability after knew knowledge is available, etc.

7. Probability of One or Other Event Occurring P(A or B) = P(A) + P(B) - P(A & B)

8. Example: Who Will Join Proposed HMO? P(Frail or Male) = P(Frail) - P(Frail & Male) + P(Male)

9. Probability of Two Events co-occurring

10. Effect of New Knowledge

11. Conditional Probability

12. Example: Hospitalization rate of frail elderly

13. Sources of Data • Objective frequency • For example, one can see out of 100 people approached about joining an HMO, how many expressed an intent to do so?   • Subjective opinions of experts  • For example, we can ask an expert to estimate the strength of their belief that the event of interest might happen. 

14. Two Ways to Assess Subjective Probabilities • Strength of Beliefs • Do you think employees will join the plan?  On a scale from 0 to 1, with 1 being certain, how strongly do you feel you are right? • Imagined Frequency • In your opinion, out of 100 employees, how many will join the plan? Uncertainty for rare, one time events can bemeasured

15. All Calculus of Probability is Derived from Three Axioms • The probability of an event is a positive number between 0 and 1 • One event will happen for sure, so the sum of the probabilities of all events is 1 • The probability of any two mutually exclusive events is the sum of the probability of each. Axioms are always met, but that we want them to be followed

16. Probabilities provide a context in which beliefs can be studied Rules of probability provide a systematic and orderly method

17. P(Joining) = (a +b) / (a + b + c + d)  P(Frail) = (a + c) / (a + b + c + d) P(Joining | Frail)  = a / (a + c) P(Frail  |  Joining) = a / (a + b) P(Joining | Frail)  = P(Frail  | Joining)  *  P(Joining) /  P(Frail) Partitioning Leads to Bayes Formula Bayes Formula

18. Odds Form of Bayes Formula Posterior odds after review of clues =Likelihood ratio associated with the clues * Prior odds

19. Independence • The occurrence of one event does not tell us much about the occurrence of another • P(A | B) = P(A) • P(A&B) = P(A) * P(B)

20. Example of Dependence P(Medication error ) ≠ P(Medication error| Long shift)

21. Suppose that one in every fifty patients in a clinic is diagnosed with cancer. You know that all ten patients waiting before you in the line have been diagnosed with cancer. What is probability that you will be diagnosed with cancer?

22. Independence SimplifiesCalculation of Probabilities Joint probability can be calculated from marginal probabilities

23. P(A | B, C) = P(A | C) P(A&B | C) = P(A | C) * P(B | C) Conditional Independence

24. Conditional Independence versus Independence P(Medication error ) ≠ P(Medication error| Long shift) P(Medication error | Long shift, Not fatigued) = P(Medication error| Not fatigued) Can you come up with other examples

25. Conditional Independence Simplifies Bayes Formula

26. Example: What is the odds for hospitalizing a female frail elderly? • Likelihood ratio for frail elderly is 5/2 • Likelihood ratio for Females is 9/10.  • Prior odds for hospitalization is 1/2 Posterior odds of hospitalization=(5/2)*(9/10)*(1/2) = 1.125

27. Verifying Independence • Reduce sample size and recalculate • Correlation analysis • Directly ask experts • Separation in causal maps

28. Verifying Independence by Reducing Sample Size • P(Error | Not fatigued) = 0.50 • P(Error | Not fatigued & Long shift) = 2/4 = 0.50

29. Verifying Conditional Independence Through Correlations • Rab is the correlation between A and B • Rac is the correlation between events A and C • Rcb is the correlation between event C and B • If Rab= Rac Rcb then A is independent of B given the condition C

30. Verifying Independence Through Correlations 0.91 ~ 0.82 * 0.95 

31. Verifying Independence by Asking Experts • Write each event on a 3 x 5 card • Ask experts to assume a population where condition has been met  • Ask the expert to pair the cards if knowing the value of one event will make it considerably easier to estimate the value of the other • Repeat these steps for other populations • Ask experts to share their clustering • Have experts discuss any areas of disagreement • Use majority rule to choose the final clusters

32. Verifying Independence by Causal Maps • Ask expert to draw a causal map • Conditional independence: A node that if removed would sever the flow from cause to consequence Blood pressure does not depend on age given weight

33. Probability of Rare Events • Event of interest is quite rare (less than 5%) • Because of lack of repetition, it is difficult to assess the probability of such events from observing historical patterns.  • Because experts exaggerate small probabilities, it is difficult to rely on experts for these estimates.  • Measure rare probabilities through time to the event

34. Examples for Calculation of Rare Probabilities Probability = 1 / (1+time to event)

35. Take Home Lessons • Probability calculus allow us to keep track of complex sequence of events • Conditional independence helps us simplify tasks • Rare probabilities can be estimated from time to the event