1. Introduction to Probabilities
Farrokh Alemi, Ph.D.
Saturday, February 21, 2004

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

3. Measure probabilities in order to communicate uncertainty & assess impact of combined events

6. Calculus of Probabilities Helps Us Keep Track of Uncertainty of Multiple Events

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. Partitioning Leads to Bayes Formula
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)
Bayes Formula

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

15. 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)

16. Independence Simplifies Calculation of Probabilities
Joint probability can be calculated from marginal probabilities

17. Conditional Independence Simplifies Bayes

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

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

20. Conditional Independence versus Independence
P(Medication error ) ≠ P(Medication error| Long shift)
P(Medication error | Long shift, Not fatigued) = P(Medication error| Not fatigued)

21. 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

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

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

24. 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

25. Verifying Independence Through Correlations
0.91 ~ 0.82 * 0.95 

26. 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

27. Verifying Independence by Causal Maps
Blood pressure does not depend on age given weight

28. Take Home Lessons
Probability calculus allow us to keep track of complex sequence of events
Conditional independence helps us simplify prediction tasks