My biggest problem with calling a lack of evidence evidence is that it is unnecessary in the first place, which makes it confusing when it comes to discussing it.
Also, I'm not arguing for or against the existence of the fifth column. I think I was unclear about that earlier, and I think we probably got a signal or two crossed. The fifth column was a fact, it existed in Japan, and it is the reason they were afraid of a fifth column in America.
Warren also never argued their existence, only their activity, so I don't see why you have a Q and an R at all. Re-read the statement, he took the 5th column's existence as a given.
What I'm arguing is the idea that a lack of evidence of subversive activity can be strong evidence that a plan similar to Pearl Harbor is being hatched.
To that end, I went ahead and made some calculations.
These are my assumptions, and I feel they are historically reasonable (I didn't cite studies, so I can't exactly call them accurate):
1% of all subversive plots are surprise plots (a-la Pearl Harbor). I call these p(subversion).
Evidence for such plots I call p(evidence).
90% of the time when there is such a plot, there is evidence of it before the fact. I call this p(evidence|subversion).
This is the critical part of Warren's statement - he is essentially assuming the opposite of what I say here, and I assert this is not reasonable given what we know of such plots. There was even evidence of the Pearl Harbor plot before hand. An attack was expected and planned for; it was really only the location (and the lack of a prior declaration) and precise timing that was a surprise militarily. I've frankly never heard of a case of a surprise attack with absolutely no evidence that it would occur, so I believe I am being extremely generous with this number. I would not accept lowering this number much further than this.
Last, I assert that 5% of the time when evidence is found for subversion, no subversion actually occurs. Again, I think this is a reasonable number, and probably too low. I wouldn't have a problem adjusting this number down as low as 1%.
Everything else is calculated based on these three assumptions.
p(subversion) = 1% p(~subversion) = 99%
p(evidence|subversion) = 90% p(~evidence|subversion) = 10% p(evidence|~subversion) = 4.95% p(~evidence|~subversion) = 94.05%
p(evidence) = 5.85% p(~evidence) = 94.15%
p(subversion|evidence) = 15.52% p(~subversion|evidence) = 35.35% p(subversion|~evidence) = 0.11% p(~subversion|~evidence) = 99.89%
So my conclusion on the question of how likely a lack of evidence implies a plot for subversion is drawn from the last two figures. Given my assumptions, which I believe are consistent with history, 99.89% of the time when there is no evidence of a plot for a surprise attack, there is no plot for a suprise attack. This means 0.11% of the time when there is no evidence of a plot, there actually is a plot.
Thus, likelihood of a Pearl Harbor style plot when there is no evidence to that fact is 0.11%.
It looks like our views have converged. What you wrote above seems to be in agreement with what I wrote here:
...What Warren said is consistent with coherent Bayesian updating, even if he was updating on a bizarre prior. It might have been wrong to put a high prior probability on subversive activity, but the probability calculus doesn't tell you how to pick your prior. All I am saying is that the Warren quote, in and of itself, does not constitute a violation of the rules of the probability calculus.
Maybe Warren committed such a violation earlier on. Maybe
From Robyn Dawes’s Rational Choice in an Uncertain World:
Consider 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, and ¬E for the observation of no sabotage. The symbol H1 stands for the hypothesis of a Japanese-American Fifth Column, and H2 for the hypothesis that no Fifth Column exists. The conditional probability P(E | H), or “E given H,” is how confidently we’d expect to see the evidence E if we assumed the hypothesis H were true.
Whatever the likelihood that a Fifth Column would do no sabotage, the probability P(¬E | H1), it won’t be as large as the likelihood that there’s no sabotage given that there’s no Fifth Column, 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), read “A implies B,” is not equivalent to (¬A ⇒ ¬B), read “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), i.e., seeing E increases the probability of H, then P(H | ¬ E) < P(H), i.e., failure to observe E decreases the probability of H . The probability P(H) is a weighted mix of P(H | E) and P(H | ¬ E), and necessarily lies between the two.1
Under the vast majority of real-life circumstances, a cause may not reliably produce signs of itself, but the absence of the cause is even less likely to produce the signs. The absence of an observation may be strong evidence of absence or very weak evidence of absence, depending on how likely the cause is to produce the observation. The absence of an observation that is only weakly permitted (even if the alternative hypothesis does not allow it at all) is very weak evidence of absence (though it is evidence nonetheless). This is the fallacy of “gaps in the fossil record”—fossils form only rarely; it is futile to trumpet the absence of a weakly permitted observation when many strong positive observations have already been recorded. But if there are no positive observations at all, it is time to worry; hence the Fermi Paradox.
Your strength as a rationalist is your ability to be more confused by fiction than by reality; if you are equally good at explaining any outcome you have zero knowledge. The strength of a model is not what it can explain, but what it can’t, for only prohibitions constrain anticipation. If you don’t notice when your model makes the evidence unlikely, you might as well have no model, and also you might as well have no evidence; no brain and no eyes.
1 If any of this sounds at all confusing, see my discussion of Bayesian updating toward the end of The Machine in the Ghost, the third volume of Rationality: From AI to Zombies.