Bayesian analysis with a discrete distribution

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Bayesian analysis with a discrete distribution

• Source Stattrek.com

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Bayes Theorem

• Probability of event A given event B depends
  not only on the relationship between A and B but
  on the absolute probability of A not concerning
  B and the absolute probability of B not
  concerning A.
• Start with --P(AknB)P(Ak).P(BAk)

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Bayes Theorem

• If A1, A2, A3An are mutually exclusive events of
  sample space S and B is any event from the same
  sample space and P(B)gt0. then,
• P(AkB) P(Ak). P(BAk)
  
• P(A1). P(BA1)P(A2). P(BA2) P(An).
  P(BAn)

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Sufficient conditions for Bayes

• The sample space is partitioned into a set of
  mutually exclusive events A1, A2, . . . , An .
  
• Within the sample space, there exists an event B,
  for which P(B) gt 0.
• The analytical goal is to compute a conditional
  probability of the form P( Ak B ).
• We know at least one of the two sets of
  probabilities described below.
• P( Ak n B ) for each Ak
• P( Ak ) and P( B Ak ) for each Ak

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Eg. of Discrete Distribution

• Marie is getting married tomorrow, at an outdoor
  ceremony in the desert. In recent years, it has
  rained only 5 days each year.

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Eg. of Discrete Distribution

• The weatherman has predicted rain for tomorrow.
• When it actually rains, the weatherman correctly
  forecasts rain 90 P( B A1 )of the time.
• When it doesn't rain, he incorrectly forecasts
  rain 10 P( B A2 ) of the time.
• What is the probability that it will rain on the
  day of Marie's wedding (P(A1))?
• A2does not rain on wedding.

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Values for calculation

• P( A1 ) 5/365 0.0136985 It rains 5 days out
  of the year.
• P( A2 ) 360/365 0.9863014 It does not rain
  360 days out of the year.
• P( B A1 ) 0.9 When it rains, the weatherman
  predicts rain 90 of the time.
• P( B A2 ) 0.1 When it does not rain, the
  weatherman predicts rain 10 of the time.

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Applying the Bayes theorem

• P(AkB) P(Ak). P(BAk)
  
• P(A1). P(BA1)P(A2). P(BA2) P(An).
  P(BAn)
• P(A1B) P(A1). P(BA1)
  
• P(A1). P(BA1)P(A2).
  P(BA2)
• P(A1B) 0.111