Bayes' Theorem Application QuestionI'm reading Networks by Mark Newman and I'm confused about one of his applications of Bayes' theorem. He begins with the following equation: P ( A , x , y ∣ data ) = P ( data ∣ A , x , y ) P ( A ) P ( x ) P ( y ) P ( data ) He then assumes the prior probabilities P(x) and P(y) are uniform and therefore constants. He then states that we cannot assume the same for P(A) and introduces the following as its prior probability P ( A | p )(where p is another probability on which A depends)From this he updates the formula above as follows: P ( A , x , y , p ∣ data ) = P ( data ∣ A , x , y ) P ( A ∣ p ) P ( p ) P ( x ) P ( y ) P ( data ) Why doesn't P(data|A,x,y) include p (and why did he introduce p into the posterior probability)?

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Bayes&#039; Theorem Application QuestionI&#039;m reading Networks by Mark Newman and I&#039;m confused about one of his applications of Bayes&#039; theorem. He begins with the following equation:  P  (  A  ,  x  ,  y  ∣  data  )  =            P      (      data      ∣      A      ,      x      ,      y      )      P      (      A      )      P      (      x      )      P      (      y      )              P      (      data      )      He then assumes the prior probabilities P(x) and P(y) are uniform and therefore constants. He then states that we cannot assume the same for P(A) and introduces the following as its prior probability   P  (  A      |    p  )(where p is another probability on which A depends)From this he updates the formula above as follows:  P  (  A  ,  x  ,  y  ,  p  ∣  data  )  =            P      (      data      ∣      A      ,      x      ,      y      )      P      (      A      ∣      p      )      P      (      p      )      P      (      x      )      P      (      y      )              P      (      data      )      Why doesn&#039;t P(data|A,x,y) include p (and why did he introduce p into the posterior probability)?

Camron Herrera · Accepted Answer

Note that P(data) depends on x,y,A. So, data is independent of p when A is given, hence,   P  (  d  a  t  a  ∣  A  ,  p  ,  x  ,  y  )  =  P  (  d  a  t  a  ∣  A  ,  x  ,  y  ).For the second question, you need p in the posterior to get the Bayes update with   A  ∣  p, which you know the distribution.

Bayes' Theorem Application Question I'm reading Networks by Mark Newman and I'm confused about one

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