You definitions do not match mine, which come from here :
The key data-dependent term Pr(D | M) is a likelihood, and is sometimes called the evidence for model or hypothesis, M; evaluating it correctly is the key to Bayesian model comparison. The evidence is usually the normalizing constant or partition function of another inference, namely the inference of the parameters of model M given the data D.
The evidence for the hypothesis M is Pr(D | M), regardless of whether or not Pr(D) > Pr(D | M), at least according to that page and this statistics book sitting here at my desk (pages 184-186), and perhaps other sources.
If it’s just a war over definitions, then it’s not worth arguing. My point is that it’s misleading to act like that attribute you call ‘consistency’ doesn’t play a role in what could fuel reasoning like Warren’s above. It’s not about independence assumptions or mistakes about what can be evidence (do you really think Warren cared about the technical, Bayesian definition of evidence in his thinking?). It’s about understanding a person’s formation of prior probabilities in addition to the method by which they convert them to posteriors.
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.
You definitions do not match mine, which come from here :
The evidence for the hypothesis M is Pr(D | M), regardless of whether or not Pr(D) > Pr(D | M), at least according to that page and this statistics book sitting here at my desk (pages 184-186), and perhaps other sources.
If it’s just a war over definitions, then it’s not worth arguing. My point is that it’s misleading to act like that attribute you call ‘consistency’ doesn’t play a role in what could fuel reasoning like Warren’s above. It’s not about independence assumptions or mistakes about what can be evidence (do you really think Warren cared about the technical, Bayesian definition of evidence in his thinking?). It’s about understanding a person’s formation of prior probabilities in addition to the method by which they convert them to posteriors.
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.