You’ve used “evidence” to refer to the probability P(D | M). We’re talking about the colloquial use of “evidence for the hypothesis” meaning an observation that increases the probability of the hypothesis. This is the sense in which we’ve been using “evidence” in the OP.
If you draw 5 balls from an urn, and they’re all red, that’s evidence for the hypothesis that the next ball will be red, and so you conclude that the next one could be red, with a bit more certainty than you had before. If you draw 5 balls from an urn, and they’re blue, that’s evidence against the hypothesis that the next one will be red, so you conclude that the next one is less likely to be red than you thought before.
Your thought processes are wrong by the bayesian proof, however, if every sequence of 5 balls leads you to increase your belief that the next one will be red.
This is essentially what Warren did. If he observed sabotage he would have increased his belief in the existence of a fifth column, and yet, observing no sabotage he also increased his belief in the existence of a fifth column. Clearly, somewhere he’s made a mistake.
I see your point and I think we mostly agree about everything. My only slight extra point is to suggest that perhaps Warren was trying to use his prior beliefs to predict an explanation for absence of sabotage, rather than trying to use absence of sabotage to intensify his prior beliefs. In retrospect, it’s likely that you’re right about Warren and the quote makes it seem that he did, in fact, think that absence of sabotage increased likelihood of Fifth Column. But in general, though, I think a lot of people make a mistake that has more to do with starting out with an unreasonable prior, or making assumptions that their prior belief is independent of observations, than it has to do with a logical fallacy about letting conditioning on both A and ~A increase the probability of B.
Ah!
You’ve used “evidence” to refer to the probability P(D | M). We’re talking about the colloquial use of “evidence for the hypothesis” meaning an observation that increases the probability of the hypothesis. This is the sense in which we’ve been using “evidence” in the OP.
If you draw 5 balls from an urn, and they’re all red, that’s evidence for the hypothesis that the next ball will be red, and so you conclude that the next one could be red, with a bit more certainty than you had before. If you draw 5 balls from an urn, and they’re blue, that’s evidence against the hypothesis that the next one will be red, so you conclude that the next one is less likely to be red than you thought before.
Your thought processes are wrong by the bayesian proof, however, if every sequence of 5 balls leads you to increase your belief that the next one will be red.
This is essentially what Warren did. If he observed sabotage he would have increased his belief in the existence of a fifth column, and yet, observing no sabotage he also increased his belief in the existence of a fifth column. Clearly, somewhere he’s made a mistake.
I see your point and I think we mostly agree about everything. My only slight extra point is to suggest that perhaps Warren was trying to use his prior beliefs to predict an explanation for absence of sabotage, rather than trying to use absence of sabotage to intensify his prior beliefs. In retrospect, it’s likely that you’re right about Warren and the quote makes it seem that he did, in fact, think that absence of sabotage increased likelihood of Fifth Column. But in general, though, I think a lot of people make a mistake that has more to do with starting out with an unreasonable prior, or making assumptions that their prior belief is independent of observations, than it has to do with a logical fallacy about letting conditioning on both A and ~A increase the probability of B.