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

Probability can quantify how uncertain we are about a future event

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

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

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

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

Probability of Two Events co-occurring

Effect of New Knowledge

Conditional Probability

Example: Hospitalization rate of frail elderly

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

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

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)

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

Conditional Independence Simplifies Bayes

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

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

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

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

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

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

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

Verifying Independence Through Correlations 0.91 ~ 0.82 * 0.95 

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

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

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