Bayes Theorem (Bonus!)

M. Drew LaMar
September 9, 2020

“I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive.”

- Plato

Medical testing

Quote: “It’s easy to think of medical tests as black and white. If the test is positive, you have the disease; if it’s negative, you don’t. Even good clinicians sometimes fall into that trap.”

- Harriet Hall

Bayes Theorem to the rescue!!!

Example 5.9: Detection of Down Syndrome

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Question: What is Pr[DS | +]?

Discuss: What probabilities are given to us here?

Example 5.9: Detection of Down Syndrome

       result
status       -      +    Sum
  DS        40     60    100
  no DS  94905   4995  99900
  Sum    94945   5055 100000

Sensitivity: Pr[+ | DS] = 60/100 = 0.6 [True positive rate]

Specificity: Pr[- | no DS] = 94905/99900 = 0.95 [True negative rate]

False positive rate: Pr[+ | no DS] = 1-0.95 = 0.05

False negative rate: Pr[- | DS] = 1-0.6 = 0.4

Example 5.9: Detection of Down Syndrome

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Discuss: Graphically, what is:

  Pr[DS]

  Pr[+ | DS]

  Pr[+ | no DS]

  Pr[DS | +]

Example 5.9: Detection of Down Syndrome

Example 5.9: Detection of Down Syndrome (Visual Check)

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Example 5.9: Detection of Down Syndrome (Calculate)

Discuss: Pr[DS | +] = ???

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Example 5.9: Detection of Down Syndrome (Calculate)

Discuss: Pr[DS | +] = ???

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Example 5.9: Detection of Down Syndrome (Calculate)

Discuss: Pr[DS | +] = ???

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Effect of disease rarity

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Question: What happens to Pr[DS | +] when the proportion of the population with the disease goes to zero?

Example 5.9: Detection of Down Syndrome (Calculate)

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