Bayes' Theorem: "Adjustment of Subjective Confidence"

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For a fuller description of the structure of Bayes' theorem, see the page Bayes' Theorem: Conditional Probabilities
    Bayes' theorem describes the relationships that exist within an array of simple and conditional probabilities. Although its primary application is to situations where "probability" is defined according to the strict relative- frequency construction of the concept, it is sometimes also applied to situations where "probability" is constructed as an index of subjective confidence. In this latter form of application, the subjective confidence that one has in the truth of some particular hypothesis is computationally adjusted upward or downward in accordance with whether an observed outcome is confirmatory or disconfirmatory of the hypothesis. For the rest of this page, probabilities constructed in this subjective- confidence fashion will be described as "s-probabilities" and symbolized as "sP".

   For example: Suppose an investigator is 75% confident that hypothesis A is true and 25% confident that it is not true. The corresponding subjective probabilities could be constructed as
   sP(A) = .75  and
   sP(~A) = .25
Suppose also that the investigator believes event B to have a 90% chance of occurring if the hypothesis is true (B|A), but only a 50/50 chance of occurring if the hypothesis is false (B|~A). Thus:
   sP(B|A) = .9
   sP(~B|A) = .1
   sP(B|~A) = .5  and
   sP(~B|~A) = .5
   
  A = hypothesis A is true
~A = hypothesis A is false
  B = event B occurs
~B = event B does not occur
 		   
| = if
  Thus:
    (B|A) = B if A
    (B|~A) = B if not-A
    and so on
The application of Bayes' theorem to these s-probability values would lead the investigator to adjust his/her degree of subjective confidence in hypothesis A upward, from .75 to .844, if the outcome is confirmatory (event B occurs), and downward, from .75 to .375, if the outcome is disconfirmatory (event B does not occur). Similarly, the degree of subjective confidence that hypothesis A is false would be adjusted downward, from .25 to .156, if event B does occur, and upward, from .25 to .625 if event B does not occur.
To perform calculations of this type using Bayes' theorem, enter the s-probability for one or the other of the items in each of the following pairs (the remaining item in each pair will be calculated automatically). A probability value can be entered as either a decimal fraction such as .25 or a common fraction such as 1/4. Whenever possible, it is better to enter the common fraction rather than a rounded decimal fraction: 1/3 rather than .3333; 1/6 rather than .1667; and so forth.
   sP(A)  or  sP(~A)
sP(B|~A)  or  sP(~B|~A)
 sP(B|A)  or  sP(~B|A) 		    
After the three s-probability values (one from each pair) have been entered, click the cursor anywhere in the gray field of the table to complete the intermediate calculations, then click the "Calculate" button. ~~Note that no probability value can be less than 0.0 or greater than 1.0.
 		        
Initial Subjective Confidence
~A
hypothesis
false 		A
hypothesis
true
sP(~A) =
sP(A) =
B
confirmatory
outcome
sP(B|~A) =
sP(B|A) =
~B
disconfirmatory
outcome
sP(~B|~A) =
sP(~B|A) =
   
Adjusted Subjective Confidence
Adjusted
sP(~A)
Adjusted
sP(A)
B
confirmatory
outcome
sP(~A|B) =
sP(A|B) =
~B
disconfirmatory
outcome
sP(~A|~B) =
sP(A|~B) =
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