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Belief Calculation : Calculating weighted opinions
We will first introduce an operator to combine opinions, and then show how we use this to calculate weigthed opinions on fact.
Lets say we have two opinions: (b1, d1, u1) and (b2, d2, u2). Then the combined opinion is defined as follows:
b = (b1 + b2)/2
d = (d1 + d2)/2
u = (u1 + u2)/2
When we want to combine two weighted opinions ((b1, d1, u1), a1) and ((b2, d2, u2), a2), having a1 and a2 reflect the credibiilty of the opinions.
Then the combined weighted opinion is:
b = (b1*a1 + b2*a2)/(a1+a2)
d = (d1*a1 + d2*a2)/(a1+a2)
u = (u1*a1 + u2*a2)/(a1+a2)
a = a1 + a2
We can define this as the + operation on weigthed opinions.
This means that when we want to calculate the weigthed opinion of a fact, and we have two users we can just combine the opinions of the two users. This generalizes to:
wo(F) = wo(F,U1) + wo(F,U2) + ... + wo(F,Un)
Where wo(F) is the notation for weigthed opinion on fact F, and wo(F,U) as notation for the weigthed opinion of user U on fact F. The users U1 until Un have a weigthed opinion on this fact.
Simple scenarios (the authority of the users is assumed to be 1.0):
##Scenario 0 (a fact without any opinions):
wo(F1) = (0.0, 0.0, 1.0)
Okay, but creating a fact should generate an opinion based on how it was created. I submit that Fr supports F and this creates Fl with opinion vector wFl = (1, 0, 0). I ask a question is F true? If F is new then that creates a fact F with opinion vector wF = (0,0,1).
##Scenario 1 (a fact with only believers):
b(U1, F1)
b(U2, F1)
wo(F1) = (1.0, 0.0, 0.0, 2.0)
And authority of 2.0 sum of the two believers authority (assumed to be 1.0 each)
##Scenario 2 (a fact with a believer and a disbeliever):
b(U1, F1)
d(U2, F1)
wo(F1) = (0.5, 0.5, 0.0, 2.0)
Authority = 2.0 again.
##Scenario 3 (a fact with a believer and an uncertain person):
b(U1, F1)
u(U2, F1)
wo(F1) = (0.5, 0.0, 0.5, 2.0)
Authority = 2.0 again.