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Introduction to Introduction to ProbabilitiesProbabilities
Farrokh Alemi, Ph.D.Farrokh Alemi, Ph.D.Saturday, February 21, 2004Saturday, February 21, 2004
Probability can quantify Probability can quantify how uncertain we are how uncertain we are about a future eventabout a future event
Why measure uncertainty?Why measure uncertainty?
To make tradeoffs among uncertain To make tradeoffs among uncertain events events
To communicate about uncertaintyTo communicate about uncertainty
What is probability?What is probability?
In the Figure, where are the events that are not “A”?
How to Calculate How to Calculate Probability?Probability?
A
P(A)=
A
P(A)=
Calculus of Probabilities Calculus of Probabilities Helps Us Keep Track of Helps Us Keep Track of Uncertainty of Multiple Uncertainty of Multiple
EventsEvents
Joint probability, probability of Joint probability, probability of either event occurring, revising either event occurring, revising
probability after knew knowledge probability after knew knowledge is available, etc.is available, etc.
Probability of One or Other Probability of One or Other Event OccurringEvent Occurring
P(A or B) = P(A) + P(B) - P(A & B)
Example: Who Will Join Example: Who Will Join Proposed HMO?Proposed HMO?
P(Frail or Male) = P(Frail) - P(Frail & Male) + P(Male)
Probability of Two Probability of Two Events co-occurringEvents co-occurring
Effect of New KnowledgeEffect of New Knowledge
Conditional ProbabilityConditional Probability
Example: Hospitalization rate Example: Hospitalization rate of frail elderlyof frail elderly
Sources of DataSources of Data
Objective frequencyObjective frequency – For example, one can see out of 100 For example, one can see out of 100
people approached about joining an people approached about joining an HMO, how many expressed an intent to HMO, how many expressed an intent to do so? do so?
Subjective opinionsSubjective opinions of experts of experts – For example, we can ask an expert to For example, we can ask an expert to
estimate the strength of their belief that estimate the strength of their belief that the event of interest might happen. the event of interest might happen.
Two Ways to Assess Subjective Two Ways to Assess Subjective ProbabilitiesProbabilities
Strength of Beliefs Strength of Beliefs – Do you think employees will join the Do you think employees will join the
plan? On a scale from 0 to 1, with 1 plan? On a scale from 0 to 1, with 1 being certain, how strongly do you feel being certain, how strongly do you feel you are right?you are right?
Imagined Frequency Imagined Frequency – In your opinion, out of 100 employees, In your opinion, out of 100 employees,
how many will join the plan?how many will join the plan?
Uncertainty for rare, one time events can be
measured
Axioms are always met,but that we want
them to be followed
All Calculus of Probability is All Calculus of Probability is Derived from Three AxiomsDerived from Three Axioms
1.1. The probability of an event is a The probability of an event is a positive number between 0 and 1 positive number between 0 and 1
2.2. One event will happen for sure, so One event will happen for sure, so the sum of the probabilities of all the sum of the probabilities of all events is 1events is 1
3.3. The probability of any two mutually The probability of any two mutually exclusive events is the sum of the exclusive events is the sum of the probability of each.probability of each.
Probabilities provide a Probabilities provide a context in which beliefs context in which beliefs
can be studied can be studied
Rules of probability provide a Rules of probability provide a systematic and orderly systematic and orderly
method method
Partitioning Leads to Bayes Partitioning Leads to Bayes FormulaFormula
P(Joining) = (a +b) / (a + b + c + d) P(Joining) = (a +b) / (a + b + c + d) P(Frail) = (a + c) / (a + b + c + d) P(Frail) = (a + c) / (a + b + c + d) P(Joining | Frail) = a / (a + c) P(Joining | Frail) = a / (a + c) P(Frail | Joining) = a / (a + b)P(Frail | Joining) = a / (a + b) P(Joining | Frail) = P(Frail | Joining) * P(Joining) / P(Frail)P(Joining | Frail) = P(Frail | Joining) * P(Joining) / P(Frail)
Frail elderly
Not frail elderly
Total
Joins the HMO a b a + bDoes not join the HMO c d c + dTotal a + c b + d a + b + c + d
Bayes Formula
Odds Form of Bayes Odds Form of Bayes FormulaFormula
Posterior odds after review of clues =Likelihood ratio associated with the clues * Prior odds
IndependenceIndependence
The occurrence of one event does not The occurrence of one event does not tell us much about the occurrence of tell us much about the occurrence of anotheranother
P(A | B) = P(A)P(A | B) = P(A) P(A&B) = P(A) * P(B) P(A&B) = P(A) * P(B)
Independence SimplifiesIndependence SimplifiesCalculation of Calculation of ProbabilitiesProbabilities
Joint probability can be Joint probability can be calculated from marginal calculated from marginal
probabilitiesprobabilities
Conditional Conditional Independence Simplifies Independence Simplifies
Bayes FormulaBayes Formula
Example of Example of DependenceDependence
P(Medication error ) P(Medication error ) ≠≠ P(Medication error| Long shift)P(Medication error| Long shift)
Conditional IndependenceConditional Independence
P(A | B, C) = P(A | C) P(A&B | C) = P(A | C) * P(B | C)
Conditional Independence Conditional Independence versus Independenceversus Independence
P(Medication error ) P(Medication error ) ≠≠ P(Medication error| Long shift) P(Medication error| Long shift)
P(Medication error | Long shift, Not fatigued) = P(Medication error| Not fatigued)
Can you come up with other examples
Example: What is the odds for Example: What is the odds for hospitalizing a female frail hospitalizing a female frail
elderly?elderly?
Posterior odds of
hospitalization=
Likelihood ratio
associated with being frail elderly
*
Likelihood ratio
associated with being
female
*Prior odds of
hospitalization
Likelihood ratio for frail elderly is 5/2Likelihood ratio for frail elderly is 5/2 Likelihood ratio for Females is 9/10. Likelihood ratio for Females is 9/10. Prior odds for hospitalization is 1/2Prior odds for hospitalization is 1/2
Posterior odds of hospitalization=(5/2)*(9/10)*(1/2) = 1.125
Verifying IndependenceVerifying Independence
Reduce sample size and recalculateReduce sample size and recalculate Correlation analysisCorrelation analysis Directly ask expertsDirectly ask experts Separation in causal mapsSeparation in causal maps
Verifying Independence by Verifying Independence by Reducing Sample SizeReducing Sample Size
P(Error | Not fatigued) = 0.50P(Error | Not fatigued) = 0.50 P(Error | Not fatigue & Long shift) = 2/4 = P(Error | Not fatigue & Long shift) = 2/4 =
0.50 0.50
CaseMedication
error Long shift Fatigue1 No Yes No2 No Yes No3 No No No4 No No No5 Yes Yes No6 Yes No No7 Yes No No8 Yes Yes No9 No No Yes10 No No Yes11 No Yes Yes12 No No Yes13 No No Yes14 No No Yes15 No No Yes16 No No Yes17 Yes No Yes18 Yes No Yes
Verifying Conditional Verifying Conditional Independence Through Independence Through
CorrelationsCorrelations
RRabab is the correlation between A and B is the correlation between A and B
RRacac is the correlation between events A is the correlation between events A and Cand C
RRcbcb is the correlation between event C is the correlation between event C and Band B
If RIf Rabab= R= Racac R Rcb cb then A is independent of then A is independent of B given the condition CB given the condition C
Verifying Independence Verifying Independence Through CorrelationsThrough Correlations
0.91 0.91 ~~ 0.82 * 0.95 0.82 * 0.95
Case Age BP Weight1 35 140 2002 30 130 1853 19 120 1804 20 111 1755 17 105 1706 16 103 1657 20 102 155
Rage, blood pressure = 0.91
Rage, weight = 0.82
R weight, blood pressure = 0.95
Rage, blood pressure
=
0.91
0.82 * 0.95 = R
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Verifying Independence by Verifying Independence by Asking ExpertsAsking Experts
Write each event on a 3 x 5 cardWrite each event on a 3 x 5 card Ask experts to assume a population where Ask experts to assume a population where
condition has been met condition has been met Ask the expert to pair the cards if knowing the Ask the expert to pair the cards if knowing the
value of one event will make it considerably value of one event will make it considerably easier to estimate the value of the other easier to estimate the value of the other
Repeat these steps for other populations Repeat these steps for other populations Ask experts to share their clusteringAsk experts to share their clustering Have experts discuss any areas of Have experts discuss any areas of
disagreement disagreement Use majority rule to choose the final clustersUse majority rule to choose the final clusters
Verifying Independence by Verifying Independence by Causal MapsCausal Maps
Ask expert to draw a causal mapAsk expert to draw a causal map Conditional independence: A node that if Conditional independence: A node that if
removed would sever the flow from cause removed would sever the flow from cause to consequence to consequence
Blood pressure does not depend on age given weight
Probability of Rare EventsProbability of Rare Events
Event of interest is quite rare (less Event of interest is quite rare (less than 5%)than 5%)– Because of lack of repetition, it is difficult Because of lack of repetition, it is difficult
to assess the probability of such events to assess the probability of such events from observing historical patterns. from observing historical patterns.
– Because experts exaggerate small Because experts exaggerate small probabilities, it is difficult to rely on probabilities, it is difficult to rely on experts for these estimates. experts for these estimates.
Measure rare probabilities through Measure rare probabilities through time to the event time to the event
Examples for Calculation of Examples for Calculation of Rare ProbabilitiesRare Probabilities
Probability = 1 / (1+time to event)
ISO 17799 word Frequency of event Calculation
Rare probability
Negligible Once in a decade =1/(1+3650) 0.0003
Very low 2-3 times every 5 years =2.5/(5*365) 0.0014
Low <= once per year =1/365 0.0027
Medium <= once per 6 months =1/(6*30) 0.0056
High <= once per month =1/30 0.0333
Very high => once per week =1/7 0.1429
Take Home LessonsTake Home Lessons
Probability calculus allow us to keep Probability calculus allow us to keep track of complex sequence of eventstrack of complex sequence of events
Conditional independence helps us Conditional independence helps us simplify taskssimplify tasks
Rare probabilities can be estimated Rare probabilities can be estimated from time to the eventfrom time to the event