Bayes Theorem

1
Bayes Theorem
Public Administration and Policy PAD634 Judgment
and Decision Making Behavior

• Thomas R. Stewart, Ph.D.
• Center for Policy Research
• Rockefeller College of Public Affairs and Policy
• University at Albany
• State University of New York
• T.STEWART_at_ALBANY.EDU

2
A problem
P(Test resultEvent)
Outcome 1

Test

-
Outcome 2
The data are organized like this
Event

Outcome 3
-
Test
Outcome 4
-
P(EventTest result)
Outcome 1

Event

-
Outcome 2
But to make a decision you need
Test

Outcome 3
-
Event
Outcome 4
-
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Bayes Theorem flips probabilities
A is the event we are interested in, e.g., a
disease. X is the evidence, e.g., a test
result.

• Conditional probability
• P(AX) means probability of A given X
• By symmetry P(XA)P(A) P(AX)P(X)

See next two slides for expansion of P(X) in
denominator.
Bayes theorem
means not A
4
Expansion of P(A)
A
X
Occurrence of X includes events X and A and X
and not A
5
Expansion of P(A) for the denominator of Bayes
Theorem
Therefore
Bayes theorem
6
Likelihood ratio form
Arkes, H. R., Mellers, B. A. (2002). Do juries
meet our expectations? Law and Human Behavior,
26(6), 625-639.
7
Bayesian belief updating
p0 is the prior probability of the event A Odds0
p0/(1-p0) are the prior odds in favor of A lrX
P(XA)/P(Xnot A) is the likelihood ratio for
new data X Odds1 p0/(1-p0)lrX are the
posterior odds in favor of A p1 Odds1/(1
Odds1) posterior probability of event A
p(AX)
Belief updating spreadsheet
8
Dealing with causality

• Case 1
• X represents some data or evidence. It provides
  a clue to whether event A will occur (prediction)
  or has or is occurring (diagnosis). It is not
  causal.
• Examples
• Doctor observes redness in the ear
• Social worker observes unclean house.

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Dealing with causality

• Case 2
• X represents some event that influences the
  likelihood that event A will occur. It has a
  causal influence
• Examples
• Federal Reserve chairman says tax cut not a good
  idea.
• Asian stock markets fall sharply.

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What is easier to estimate?

• Both require prior probability of A
• Case 1
• Probability of X given A
• Probability of X given not A
• Case 2
• Probability of A given X
• Impact multiplier (Odds multiplier)