Absence of Evidence Is Evidence of Absence.
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You are probably familiar with the claim that absence of proof is not proof of absence.
Logically, this is correct. If P implies Q, and I have proof of (not) P, I don't have a logical proof of (not Q)
But if I am collecting evidence for a propositon, then this result in formal logic doesn't hold.
Absence of proof is proof of absence.
For a complete explanation see the article Absence of Evidence Is Evidence of Absence in Robyn Dawes's Rational Choice in an Uncertain World:
Post-hoc fitting of evidence to hypothesis was involved in a most grievous chapter in United States history: the internment of Japanese-Americans at the beginning of the Second World War.'
When California governor Earl Warren testified before a congressional hearing in San Francisco on February 21, 1942, a questioner pointed out that there had been no sabotage or any other type of espionage by the Japanese-Americans up to that time.'
Warren responded, 'I take the view that this lack [of subversive activity] is the most ominous sign in our whole situation.
It convinces me more than perhaps any other factor that the sabotage we are to get, the Fifth Column activities are to get, are timed just like Pearl Harbor was timed... I believe we are just being lulled into a false sense of security.'
This sounds convincing, but rethink Warren's argument from a Bayesian perspective.'
When we see evidence, hypotheses that assigned a higher likelihood to that evidence, gain probability at the expense of hypotheses that assigned a lower likelihood to the evidence.'
This is a phenomenon of relative likelihoods and relative probabilities.'
You can assign a high likelihood to the evidence and still lose probability mass to some other hypothesis, if that other hypothesis assigns a likelihood that is even higher.
Warren seems to be arguing that, given that we see no sabotage, this confirms that a Fifth Column exists.'
You could argue that a Fifth Column might delay its sabotage.' But the likelihood is still higher that the absence of a Fifth Column would perform an absence of sabotage.
Let E stand for the observation of sabotage, H1 for the hypothesis of a Japanese-American Fifth Column, and H2 for the hypothesis that no Fifth Column exists.'
Whatever the likelihood that a Fifth Column would do no sabotage, the probability P(E|H1), it cannot be as large as the likelihood that no Fifth Column does no sabotage, the probability P(E|H2).' So observing a lack of sabotage increases the probability that no Fifth Column exists.
A lack of sabotage doesn't prove that no Fifth Column exists.' Absence of proof is not proof of absence.' In logic, A->B, 'A implies B', is not equivalent to ~A->~B, 'not-A implies not-B'.
But in probability theory, absence of evidence is always evidence of absence.'
'If E is a binary event and P(H|E) > P(H), 'seeing E increases the probability of H'; then P(H|~E) < P(H), 'failure to observe E decreases the probability of H'.'
P(H) is a weighted mix of P(H|E) and P(H|~E), and necessarily lies between the two.'
See An Intuitive Explanation of Bayesian Reasoning.
You might want to read this article and comments, at the Overcoming Bias website, Overcoming Bias
The point can be summarized. When you are doing your due diligence, you are not a logician.
You have to decided whether you believe the evidence is stronger or weaker for a particular hypothesis -you do not reside the Platonic heavens of formal logicians.
You are going to have to act, so make sure that the evidence appears to be on your side.
