PRIOR PROBABILITY

PROPOSITION

 SYMBOL PRIOR PROBABILITY (p)
Peter will catch tanigue d 0.4
there will be no dynamite fishing s1 0.8
there will be fresh bait available s2 0.5
there will be good sea conditions s3 0.3

tanigue or narrow-barred Spanish mackerel (scomberomorus commerson)
 

Prior probability refers to the probability of the propositions prior to any evidence [e.g. without knowing anything else about the probability that Peter will catch tanigue tomorrow is 0.4].

CONDITIONAL PROBABILITIES

p(d|s1) denotes the probability of d assuming s1 (on the condition that s1) [e.g. the probability that Peter will catch tanigue tomorrow when it is already known that there will be no dynamite fishing].

p(d|s1) = p(d&s1)/p(s1) = p(d&s1)/0.8 = p(d&s1)*1.25  

p(d&s1) = p(s1|d)*p(d) = p(s1|d)*0.4

p(d|s1)  = p(d&s1)*1.25  = p(s1|d)*0.4*1.25 = p(s1|d)*0.5

p(d|s1&s2&s3) = p(s1&s2&s3|d)*p(d)/p(s1&s2&s3|d)

= p(s1&s2&s3|d)*p(d)/(p(s1|s2&s3)*p(s2|s3)*p(s3))

= p(s1&s2&s3|d)*0.4/(0.65*0.6*0.3) [the tree factors in the divisor are assumed estimates]

=p(s1&s2&s3|d)*3.42

if all sk are independent, then 

p(d|s1&s2&s3) = p(s1&s2&s3|d)*p(d)/(p(s1)*p(s2)*p(s3)) = p(s1&s2&s3|d)*0.4/(0.8*0.5*0.3) = p(s1&s2&s3|d)*3.33

CERTAINTY FACTORS

MB(d,s1) stands for the measure of the increase in belief in d given s1.given that p(d|s1) > p(d). MD(d,s1) stands for the measure of the increase in disbelief in d given s1.given that p(d|s1) < p(d). 

MB(d,s1) = (p(d|s1)-p(d))/(1-p(d))

if p(d|s1) is 0.45 (i.e. > p(d)) then MB(d,s1) = (0.45-0.4)/(1-0.4) = 0.083  [increase in belief in d because of s1]  

MD(d,s1) = (p(d)-p(d|s1))/p(d)

 if p(d|s1) is 0.35 (i.e. < p(d)) then MD(d,s1) = (0.4-0.35)/0.4 = 0.125  [increase in disbelief in d because of s1]  

Increment and Decrement Factors in Pro/3

The degree of increase in belief (or disbelief) in a proposition d given another proposition s are specified by a increment factor (and decrement factor). The first comes into play when the "post-evidence" probability of s is higher than the prior probability, while the latter comes into play when post-evidence probability is lower (ref. Bayesian-type certainty rules).  

The following table shows the certainty factors (CF) (computed by Pro/3) for d and the implied conditional probabilities p(d|s1) under four assumptions for p(s1) and four increment/decrement values:

  INC=1.1 DEC=0.9 INC=1.3 DEC=0.7 INC=1.9 DEC=0.1 INC=3.8 DEC=0.05
p(s1)=0.0 CF=-0.0.62 p(d|s1)=0.42 CF=-0.204 p(d|s1)=0.48 CF=-0.842 p(d|s1)=0.74 CF=-0.918 p(d|s1)=0.77
p(s1)=0.25 CF=-0.042 CF=-0.134 CF=-0.512 CF=-0.53
p(s1)=0.75 CF=-0.002 CF=-0.010 CF=-0.034 CF=-0.0.36
p(s1)=1.0 CF=0.038 p(d|s1)=0.42 CF=0.106 p(d|s1)=0.46 CF=0.264 p(d|s1)=0.56 CF=0.528 p(d|s1)=0.72

The next table shows the certainty factors for d and the implied conditional probabilities p(d|s2) under the same assumptions as above:

  INC=1.1 DEC=0.9 INC=1.3 DEC=0.7 INC=1.9 DEC=0.1 INC=3.8 DEC=0.05
p(s2)=0.0

same as for s1
p(s2)=0.25 CF=-0.0.3 CF=-0.157 CF=-0.547 CF=-0.35
p(s2)=0.75 CF=0.018 CF=0.056 CF=0.152 CF=0.358
p(s2)=1.0

same as for s1
And finally, the certainty factors for d and the implied conditional probabilities p(d|s3) under the same assumptions:
  INC=1.1 DEC=0.9 INC=1.3 DEC=0.7 INC=1.9 DEC=0.1 INC=3.8 DEC=0.05
p(s3)=0.0

same as for s1
p(s3)=0.25 CF=-0.01 CF=-0.030 CF=-0.094 CF=-0.100
p(s3)=0.75 CF=0.024 CF=0.070 CF=0.186 CF=0.418
p(s3)=1.0

same as for s1
Then some combinations of assumptions:
  All calls with INC=1.1 and DEC=0.9 s1:INC=1.1 DEC=0.9 s2:INC=1.3 DEC=0.7 s3:INC=1.9 DEC=0.1 s1:INC=1.9 DEC=0.1 s2:INC=1.3 DEC=0.7 s3:INC=1.1 DEC=0.9 All calls with INC=3.8 and DEC=0.05
p(s1)=0.0 p(s2)=0.0 p(s3)=0.0 CF=-0.182 p(d|s1&s2&s3)=0.47 CF=-0.898            p(d|s1&s2&s3)=0.76 CF=-0.898 p(d|s1&s2&s3)=0.76 CF=-0.998 p(d|s1&s2&s3)=0.8
p(s1)=0.0 p(s2)=0.5 p(s3)=1.0 CF=-0.006 CF=0.220 CF=-0.828 CF=-0.718
p(s1)=0.25 p(s2)=0.5 p(s3)=0.75 CF=-0.004 CF=0.158 CF=-0.466 CF=-0.016
p(s1)=0.75 p(s2)=0.5 p(s3)=0.25 CF=-0.012 CF=-0.098 CF=-0.044 CF=-0.136
p(s1)=1.0 p(s2)=1.0 p(s3)=1.0 CF=0.116 p(d|s1&s2&s3)=0.47 CF=0.406     p(d|s1&s2&s3)=0.64 CF=4.06 p(d|s1&s2&s3)=0.76 CF=0.954 p(d|s1&s2&s3)=0.98

 

Using the combination-type rule (in place of the bayesian-type rule), gives the following results (the certainties in the center column has been computed via the prior probability):
p(s1)=0.0 p(s2)=0.0 p(s3)=0.0 cf(s1)=-1.0 cf(s2)=-1.0 cf(s3)=-1.0 CF=-1.000
p(s1)=0.0 p(s2)=0.5 p(s3)=1.0 cf(s1)=-1.0 cf(s2)=0.0 cf(s3)=1.0 CF=0.000
p(s1)=0.25 p(s2)=0.5 p(s3)=0.75 cf(s1)=-0.689 cf(s2)=0.000 cf(s3)=0.643 CF=-0.120
p(s1)=0.75 p(s2)=0.5 p(s3)=0.25 cf(s1)=-0.063 cf(s2)=0.000 cf(s3)=-0.170 CF=-0.22
p(s1)=1.0 p(s2)=1.0 p(s3)=1.0 cf(s1)=1.0 cf(s2)=1.0 cf(s3)=1.0 CF=1.000