The probability of both, in that case, plummets, and you should start looking at other explanations. Like, say, that the victim was shot with a rifle at close range, which only leaves a bullet in the body 1% of the time (or whatever).
It might be true that, between two hypotheses one is now more likely to be true than the other, but the probability for both still dropped, and your confidence in your pet hypothesis should still drop right along with its probability of being correct.
So say you have hypothesis X at 60% confidence and hypotheses Y at 40% New evidence comes along that shifts your confidence of X down to 20%, and Y down to 35%. Y didn’t just “win”. Y is now even more likely to be wrong than it was before the new evidence came it. The only substantive difference is that now X is probably wrong too. If you notice, there’s 45% probability there we haven’t accounted for. If this is all bound up in a single hypothesis Z, then Z is the one that is the most likely to be correct.
Contradictory evidence shouldn’t make you more confident in your hypothesis.
That’s just not so, since the total of the two probabilities equals one. If the probability of murder with a rifle drops, the probability of murder with a handgun necessarily rises. I’m not sure how to make this point any clearer . . . . perhaps a couple equations will help:
Let’s suppose that X and Y are mutually exclusive and collectively exhaustive hypotheses.
If either X or Y has to be true, you cannot have 20% for X and 35% for Y. The remaining 45% would be a contradiction (Neither X nor Y, but “X or Y”).
While you can work with those numbers (20 and 35), they are not probabilities any more—they are relative probabilities.
It is very unlikely that the murderer won in the lottery. However, if a suspect did win in the lottery, this does not reduce the probability that he is guilty—he has the same (low) probability as all others.
I’m talking about probability estimates. The actual probability of what happened is 1, because it is what happened. However, we don’t know what happened, that’s why we make a probability estimate in the first place!
Forcing yourself to commit to only one of two possibilities in the real world (which is what all of these analogies are supposed to tie back to), when there are a lot of initially low probability possibilities that are initially ignored (and rightly so), seems incredibly foolish.
Also, your analogy doesn’t fit brazil84′s murder example. What evidence does the lottery win give that allows us to adjust our probability estimate for how the gun was fired? I’m not sure where you’re going with that, at all.
The real probability of however the bullet was fired is 100%. All we’ve been talking about are our probability estimates based on the limited evidence we have. They are necessarily incomplete. If new evidence makes both of our hypotheses less likely, then it’s probably smart to check and see if a third hypotheses is now feasible, where it wasn’t before.
brazil84 stated that there are just two options, so let’s stick to that example first.
“[rifle] no bullet will be find in or around the person’s body 0.01% of the time” is contradictory evidence against the rifle (and for the handgun). But “[handgun] no bullet will be find in or around the person’s body 0.001% of the time” is even stronger evidence against the handgun (and for the rifle). In total, we have some evidence for the rifle.
Now let’s add a .001%-probability that it was not a gunshot wound—in this case, the probability to find no bullet is (close to) 100%. Rifle gets an initial probability of 60% and handgun gets 40% (+ rounding error).
So let’s update:
No gunshot: 0.001 → 0.001
Rifle: 60 → 0.006
Handgun: 40 → 0.0004
Of course, the probability that one of those 3 happened has to be 1 (counting all guns as “handgun” or “rifle”), so let’s convert that back to probabilities:
0.001+0.006+0.0004 = 0.0074
No gunshot: 0.001/0.0074=13.5%
Rifle: 0.006/0.0074=81.1%
Handgun: 0.0004/0.0074=5.4%
The rifle and handgun numbers increased the probability of a rifle shot, as the probability for “no gunshot” was very small. All numbers are our estimates, of course.
The probability of both, in that case, plummets, and you should start looking at other explanations. Like, say, that the victim was shot with a rifle at close range, which only leaves a bullet in the body 1% of the time (or whatever).
It might be true that, between two hypotheses one is now more likely to be true than the other, but the probability for both still dropped, and your confidence in your pet hypothesis should still drop right along with its probability of being correct.
So say you have hypothesis X at 60% confidence and hypotheses Y at 40% New evidence comes along that shifts your confidence of X down to 20%, and Y down to 35%. Y didn’t just “win”. Y is now even more likely to be wrong than it was before the new evidence came it. The only substantive difference is that now X is probably wrong too. If you notice, there’s 45% probability there we haven’t accounted for. If this is all bound up in a single hypothesis Z, then Z is the one that is the most likely to be correct.
Contradictory evidence shouldn’t make you more confident in your hypothesis.
That’s just not so, since the total of the two probabilities equals one. If the probability of murder with a rifle drops, the probability of murder with a handgun necessarily rises. I’m not sure how to make this point any clearer . . . . perhaps a couple equations will help:
Let’s suppose that X and Y are mutually exclusive and collectively exhaustive hypotheses.
In that case, do you agree that P(X) + P(Y) = 1?
Also, do you agree that P(X|E) + P(Y|E) = 1 ?
If either X or Y has to be true, you cannot have 20% for X and 35% for Y. The remaining 45% would be a contradiction (Neither X nor Y, but “X or Y”). While you can work with those numbers (20 and 35), they are not probabilities any more—they are relative probabilities.
It is very unlikely that the murderer won in the lottery. However, if a suspect did win in the lottery, this does not reduce the probability that he is guilty—he has the same (low) probability as all others.
I’m talking about probability estimates. The actual probability of what happened is 1, because it is what happened. However, we don’t know what happened, that’s why we make a probability estimate in the first place!
Forcing yourself to commit to only one of two possibilities in the real world (which is what all of these analogies are supposed to tie back to), when there are a lot of initially low probability possibilities that are initially ignored (and rightly so), seems incredibly foolish.
Also, your analogy doesn’t fit brazil84′s murder example. What evidence does the lottery win give that allows us to adjust our probability estimate for how the gun was fired? I’m not sure where you’re going with that, at all.
The real probability of however the bullet was fired is 100%. All we’ve been talking about are our probability estimates based on the limited evidence we have. They are necessarily incomplete. If new evidence makes both of our hypotheses less likely, then it’s probably smart to check and see if a third hypotheses is now feasible, where it wasn’t before.
brazil84 stated that there are just two options, so let’s stick to that example first.
“[rifle] no bullet will be find in or around the person’s body 0.01% of the time” is contradictory evidence against the rifle (and for the handgun). But “[handgun] no bullet will be find in or around the person’s body 0.001% of the time” is even stronger evidence against the handgun (and for the rifle). In total, we have some evidence for the rifle.
Now let’s add a .001%-probability that it was not a gunshot wound—in this case, the probability to find no bullet is (close to) 100%. Rifle gets an initial probability of 60% and handgun gets 40% (+ rounding error).
So let’s update: No gunshot: 0.001 → 0.001 Rifle: 60 → 0.006 Handgun: 40 → 0.0004
Of course, the probability that one of those 3 happened has to be 1 (counting all guns as “handgun” or “rifle”), so let’s convert that back to probabilities: 0.001+0.006+0.0004 = 0.0074 No gunshot: 0.001/0.0074=13.5% Rifle: 0.006/0.0074=81.1% Handgun: 0.0004/0.0074=5.4%
The rifle and handgun numbers increased the probability of a rifle shot, as the probability for “no gunshot” was very small. All numbers are our estimates, of course.