A theorem: Absence of evidence is evidence of absence. 1) Let X be any arbitrary event, or combination of arbitrary events. 2) Let ~X denote the probability of X occuring 3) Let E represent the existence of a set of positive evidences that would indicate the terms/reality of X 4) Let ~E denote, or the total absence of positive evidence 5) Let P(X) denote the probability of event X. 6) Let P(X|E) denote the probability of event X given E. By definition, this is the joint probability of the logical conjunction X ∧ E divided by the probability of E. P(X|E) = P(X∧E)/P(E) Logical Assumptions: 1) If any event like X were to genuinely occur, then it very likely left some evidence of itself. P(E|X) > P(~E|X) Establishing the point, the probability of E, given X, is greater than the probability of ~E, given X. P(E|X) > P(~E|X) 1 - P(~E|X) > P(~E|X) P(~E|X) < ½ The prior assumption is logically deemed fair under the confines of definitive evidence. Mostly anything of meaningful significance tends to leave empirical data. Stercus accidit. P(~E|X) < ½ Invoking Bayes Theorem: P(~E|X) = P(X|~E)P(~E)/P(X) ½ > P(X|~E) P(~E)/P(X) P(X|~E) < 1/2 P(X)/P(~E) Given the sequence, is X a likely or unlikely event? And that varies, giving logical impetus for assumption 2. The event X is extraordinary. P(X) 1/2 Thus, we finally arrive at: P(~X|~E) > P(X|~E) Given an absence of any evidence for X, the more likely event X did NOT occur. This is a textbook example of a common epistemic principle called “inference to the best explanation”. Quod Erat Demonstrandum. Many things can never be known with absolute certainty, but we can show which explanations are more probable and plausible.
Posted on: Sun, 04 Aug 2013 18:36:58 +0000
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