Let me propose a charitable interpretation of what brazil84 is saying (he can correct me if I am wrong). Here is an example:
We are discussing who committed a crime. There are three and only three suspects: Peter, Paul and Mary. Mary has an excellent alibi, so she’s basically out of the running. There is some evidence both for Peter’s and for Paul’s guilt. Let’s say we agree that the probabilities of each being guilty are: Mary 2%, Peter 49%, Paul 49%.
Then a witness comes up who saw someone wearing a dress in the scene of the crime. Since men are a priori unlikely to wear dresses, this lowers the probability of Peter or Paul doing it.. Let’s say however that for whatever reason, we agree that it slightly less unlikely a priori that Peter would wear a dress than that Paul would wear it. Mary’s alibi is so good that the new evidence only raises very slightly her probability of being guilty. The posterior probabilities are: Mary: 6%, Peter: 48%, Paul 46%.
This seems like a situation which might be described with brazil84′s quote
“so situations can arise where evidence comes out which contradicts a hypothesis but still makes that hypothesis more likely to be correct”
in the sense that Peter’s guilt, even though in the absolute sense less likely (the evidence “contradicted” it) should now be our top hypothesis; it is “more likely to be correct” compared to the only plausible alternative.
I agree that brazil84′s way of putting it was a bit confusing, if this is what he meant.
I certainly agree that the situation you describe can occur. (I could quibble about whether the probability-shift for Mary actually depends on the quality of her alibi here, as that seems like double-counting evidence, but either way it’s entirely possible for the posterior probabilities to come out the way you describe.)
And, OK, sure, if “more likely to be correct” is understood as “more likely [than some other hypothesis] to be correct”, rather than “more likely [than it was before that evidence arrived] to be correct”, I agree that the phrase describes the situation. That is, as you say, a bit confusing, but not false.
So, OK. Provisionally adopting that interpretation and returning to the original comment… their initial comment was “situations can arise where evidence comes out which contradicts a hypothesis but still makes that hypothesis more likely to be correct”. Which, sure, if I understand that to mean “more likely [than some other hypothesis] to be correct” is absolutely true.
All of which was meant, I think, to refute bigjeff5′s comment about what sort of evidence should increase confidence in the belief that there is no bias. Which I understood to refer to increasing confidence relative to earlier confidence.
I think that’s pretty close. If I am arguing that Paul committed the murder (and you are arguing that Peter committed the murder) it doesn’t really help your argument to point out that there is evidence the murderer was wearing a dress since it undermines your own position just as much as it undermines the position you have taken.
Getting back to the original discussion, another poster pointed out that my “contested cases later” hypothesis is undermined by the fact is undermined by the observation that for some judges there is a zero percent approval rate for later cases. The problem with this argument is that it undermines the “hunger” hypothesis even more than the “contested cases later” hypothesis.
Let me propose a charitable interpretation of what brazil84 is saying (he can correct me if I am wrong). Here is an example:
We are discussing who committed a crime. There are three and only three suspects: Peter, Paul and Mary. Mary has an excellent alibi, so she’s basically out of the running. There is some evidence both for Peter’s and for Paul’s guilt. Let’s say we agree that the probabilities of each being guilty are: Mary 2%, Peter 49%, Paul 49%.
Then a witness comes up who saw someone wearing a dress in the scene of the crime. Since men are a priori unlikely to wear dresses, this lowers the probability of Peter or Paul doing it.. Let’s say however that for whatever reason, we agree that it slightly less unlikely a priori that Peter would wear a dress than that Paul would wear it. Mary’s alibi is so good that the new evidence only raises very slightly her probability of being guilty. The posterior probabilities are: Mary: 6%, Peter: 48%, Paul 46%.
This seems like a situation which might be described with brazil84′s quote
in the sense that Peter’s guilt, even though in the absolute sense less likely (the evidence “contradicted” it) should now be our top hypothesis; it is “more likely to be correct” compared to the only plausible alternative.
I agree that brazil84′s way of putting it was a bit confusing, if this is what he meant.
I certainly agree that the situation you describe can occur. (I could quibble about whether the probability-shift for Mary actually depends on the quality of her alibi here, as that seems like double-counting evidence, but either way it’s entirely possible for the posterior probabilities to come out the way you describe.)
And, OK, sure, if “more likely to be correct” is understood as “more likely [than some other hypothesis] to be correct”, rather than “more likely [than it was before that evidence arrived] to be correct”, I agree that the phrase describes the situation. That is, as you say, a bit confusing, but not false.
So, OK. Provisionally adopting that interpretation and returning to the original comment… their initial comment was “situations can arise where evidence comes out which contradicts a hypothesis but still makes that hypothesis more likely to be correct”. Which, sure, if I understand that to mean “more likely [than some other hypothesis] to be correct” is absolutely true.
All of which was meant, I think, to refute bigjeff5′s comment about what sort of evidence should increase confidence in the belief that there is no bias. Which I understood to refer to increasing confidence relative to earlier confidence.
I think that’s pretty close. If I am arguing that Paul committed the murder (and you are arguing that Peter committed the murder) it doesn’t really help your argument to point out that there is evidence the murderer was wearing a dress since it undermines your own position just as much as it undermines the position you have taken.
Getting back to the original discussion, another poster pointed out that my “contested cases later” hypothesis is undermined by the fact is undermined by the observation that for some judges there is a zero percent approval rate for later cases. The problem with this argument is that it undermines the “hunger” hypothesis even more than the “contested cases later” hypothesis.