Absence of Evidence is still evidence even with possibility of hallucination

http://rationalwiki.org/wiki/Absence_of_evidence

lemma 1: by definition, an event E is evidence for a hypothesis H if and only if P(H|E) > P(H), therefore if P(H|E) < P(H) then E is evidence for ~H. (intuitively this seems correct since the derivative of P(H) with respect to P(~H) is negative, d(P(H)/d(P(~H)<0)

proof:

if P(H|E) < P(H)

< 1-P(~H)

→  -P(H|E) > P(~H)-1        (negating both sides and inverting the inequality)

→   1-P(H|E) > P(~H)           (adding 1 to both sides)

the left hand side can be simplified to this expression

1-P(H|E) =   1- P(H∩E)/P(E)

=  (P(E)-P(H∩E))/P(E)

=  (P((~H∩E)∪(H∩E))-P(H∩E))/P(E)

=  (P(~H∩E)+P(H∩E)-P(H∩E))/P(E)

=   P(~H∩E)/P(E)                 since E

=   P(~H|E)

∴  P(~H|E)>P(~H)

and therefore E is evidence for ~H when P(H|E) < P(H)

and we now can also say that E is not evidence for or against H when P(H|E) = P(H), in other words an event is not evidence if and only if the event E and the hypothesis H are INDEPENDENT of eachother.

 

definitions :

H = the hypothesis that X exists (the conceptual set X has instances in reality)

E = the event that evidence for H was present during perception

~E = the event that no evidence for H was present during perception

T= what is perceived is a true/accurate representation of reality ( when X = aliens, H= “aliens exist” and E=”i perceive evidence that alien exists” then  E∩T = “i perceive evidence that aliens exist and that evidence does exist”)

F=~T= what is perceived is false/inaccurate (due to being a hallucination/illusion) = ∼T ( when H= “aliens exist”and E=”i perceive evidence that alien exists” then  E∩F = “i perceive evidence that aliens exist and that evidence does not exist”. )

 

claim 1: if E∩T is evidence for an uncertain hypothesis H (P(H|E∩T)> P(H)) and there is a non-certain possibility of hallucination (1>P(E∩F)>0 and E∩F is not evidence for or against H (P(H|E∩F)=P(H) by lemma 1) , then E is evidence for H

assumptions:

1.the probability function P(X) represents the subjective probability of a “perfectly rational” (mathematical) perceiving subject.

2.we are unsure about H (1>P(H)>0)

3.we have some faith in our experiences (1>P(E∩T)>0)

4. P(H|E∩T)>P(H)               E∩T is evidence supporting H

5.P(H|E∩F)=P(H)               can a hallucination of perceiving X be evidence against H? here i assume no. having a false experience is equivalent to no evidence for or against H since hallucination is now equivalent to no attempt at perception. (in claim 2 i go by the strict definition of E∩T and E∩F)

proof:

P(H|E) = P(H∩E)/P(E)

= P(H∩((E∩T)∪(E∩F))) / P(E)

=P((H∩E∩T)∪(H∩E∩F)) / P(E)

=(P(H∩E∩T) +P(H∩E∩F))) / P(E)        by mutual exclusivity

=(P(H|E∩T)P(E∩T) +P(H|E∩F)P(E∩F))) / P(E)

=(P(H|E∩T)*P(E∩T) +P(H)P(E∩F))) / P(E)    by 5.

and by 3. and 4. the following strict inequality holds

>(P(H)*P(E∩T) +P(H)P(E∩F))) / P(E)

=(P(H)(P(E∩T) +P(E∩F))) / P(E)

=(P(H)P(E)) / P(E)

=P(H)

if 1>P(E∩F)>0 then the following equality holds

∴P(H|E) > P(H)

∴P(~H|~E) > P(~H)                  by the absence of evidence theorem.

Q.E.D.

claim 2: if E∩T is evidence for an uncertain hypothesis H (P(H|E∩T)> P(H)) and there is a non-certain probability of hallucination (1>P(E∩F)>0) , then E is evidence for H  iff P(E∩T)/P(E) > P(H)/P(H|E∩T) or equivalently P(T|E)> P(H)/P(H|E∩T).

assumptions:

1.the probability function P(X) used here represents the subjective probability of a “perfectly rational” (mathematical) subject.

2.we are unsure about H (1>P(H)>0)

3.we have some faith in our experiences (1>P(E∩T)>0)

4.P(H|E∩T)> P(H)       E∩T is evidence supporting H

 

proof:

by definition:

P(H|(E∩F) <P(H)         E∩F is evidence supporting ~H (by lemma 1), or alternatively evidence against H

also

P(H|E) = P(H∩E)/P(E)

= P(H∩((E∩T)∪(E∩F))) / P(E)

=P((H∩E∩T)∪(H∩E∩F)) / P(E)

=(P(H∩E∩T) +P(H∩E∩F))) / P(E)        by mutual exclusivity

=(P(H|E∩T)P(E∩T) +P(H|E∩F)P(E∩F))) / P(E)

>(P(H|E∩T)*P(E∩T) +0*P(E∩F)) / P(E)           (the strong inequality becomes an equality when the strength of E∩F is sufficient to prove ~H)

=P(H|E∩T)P(E∩T)  / P(E)

=P(H|E∩T)P(T|E)

therefore

P(H|E)> P(H) if

P(T|E)> P(H)/P(H|E∩T)

Q.E.D.

we can call the probability, P(T|E), the trust function since it is a measure of how much we trust our perceptions.

as the evidence gets stronger, P(H|E∩T) → 1, (as in the case of directly seeing an instance of X) and if our trust is greater than the unconditional  probability of H then we can necessarily say E is evidence for H and by the absence of evidence theorem ~E is evidence for ~H. notice in this case, the higher the unconditional probability, the greater trust we need in order for our perceptions to be evidence. in the case of no information and a prior uniform distribution for H and ~H (50/50), then all it takes for any direct perception of X to be evidence is for us to have more trust in our senses than distrust.

also notice, if we had certainty our senses were deceiving us, P(F)=1 and P(T) =0, then we could be certain that E is evidence against H since P(H|E∩F)<P(H) and by lemma 1  .

 

 

 

 

 

 

 

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