Hello. As a brief introductory reminder, I'm Lauren, Kenny's fiance, and a guest blogger here when I have time (which isn't very often.) However, I am going to take some time to discuss a paper by Bas C. van Fraassen, Conditionalizing on Violated Bell's Inequalities, in which he claims that quantum mechanics creates problems for Bayesian epistemology. I have two main points to make in response, the first is that he doesn't actually need quantum mechanics for his argument, and the second is where he has failed to account for the effect of choosing which events to talk about, which changes the conclusions of his paper. I will treat these in reverse order, though.
A brief summary of van Fraassen's argument is this:
In an experimental set up involving measuring the spin of entangled photons, there are two detectors, each of which has a 50% probability of detecting something (or registering something) for each run of the experiment. (Here I'm going to be slightly sloppy and use "register something" and "detect something" interchangeably to mean "made a positive spin reading".)
However, the detectors are not uncorrelated- the probability of one detecting something is related to the cosine of the angle between the detectors squared. This is well established in quantum mechanics.
Then, van Fraasen imagines a situation where someone named Hilary is asked to predict whether or not one of the detectors registers something. She initially answers that the probability is 50%. She is then told that the other, hereafter referred to as the second, detector did register something, but she is not told what the angle is between the detectors, although she does know of the cosine squared relation. She is then asked the same question, but she now no longer knows what the probability is, because she knows it could be anything between 0% and 100% depending on the angle between the detectors.
Van Fraassen then asks, if Hilary were forced to bet, what the best thing would be for her to do. He concludes that she ought to ignore this new piece of information, even though it is relevant to the probability of the first detector registering something, and to bet the first detector registers something 50% of the time, because, he claims, she would break even doing this. Then, van Fraassen questions why Hilary is justified in ignoring the information about the second detector, since it would change her opinion. This is especially a problem for Bayesian inference, which claims that we should include all relevant evidence in our probability calculus, and as we include more relevant evidence, our probabilities become "better".
I will argue that her initial answer of 50% is actually incorrect, because of the effect of only asking her about situations where the second detector registers something, which is not a sufficiently random subset, with respect to the first detector, of all the events. Thus, her second answer is, in fact, the better answer, and Bayesian inference still stands.
Consider, for a moment, this example. (I'll explain in a moment how it relates.) I have a perfectly fair coin, which you know is fair. I then flip the coin, and ask you to guess whether or not it's heads. You win if, when I ask you, your guess matches the coin. As is well known, you should guess heads 50% of the time, to maximize your likelihood of winning. If you answer yes 49% or 51% of the time, odds are that you'll win less often than if you answered yes 50% of the time. Now, however, imagine that I flip the same fair coin, but that I look at the coin before I ask you to guess whether or not it is heads. If it's tails, then I ask you to guess whether or not it's heads. (If it's heads, I just ignore it and flip the coin again, although you don't know this.) In this case, your likelihood of winning is greatest if you never guess heads. Similarly, if I only asked you when the coin landed heads, you likelihood of winning is greatest if you always answer heads. So, when we ask someone only about a specific subset of events, the properties of that subset are relevant to rate someone should guess at. So then, if you are playing this game with someone and tell them that you're only going to ask them about a certain subset of events, but don't tell them what the subset is, they will be at a loss as to what rate they should guess, and also if they continue to guess yes 50% of the time, they will not necessary break even (depending on your subset), even though the coin lands heads 50% of the time.
Now let's look at van Fraassen's argument again, and ask whether we are, at any point, asking Hilary to guess on only a certain subset of events, and if so, whether the features of that subset was chosen would influence the probability. Recall that we do inform Hilary that the second detector did register something. Now, since the second detector will not always register something, and since we presumably are not lying to her, we are thus picking out a subset of the events, namely, the ones where the second detector goes off. Next we need to question whether this is effectually a random subset with respect to the first detector (the one we are asking Hilary about). If it is, then she will still break even guessing it detected something 50% of the time, but if it's not, then just like in the coin game above, she will no longer break when guessing yes 50% of the time. However, we know that there is a correlation between the first detector registering something and the second detector registering something (namely, that this correlation is related to the cosine of the angle squared), and so this is a NOT an effectually random subset with respect to the first detector. Hence, Hilary will not break even by guessing 50%.
But wait, you say- doesn't the first detector have to register "yes" 50% of the time? Then why doesn't she break even? Yes, the detector does register yes 50% of the time- but only when we're talking about averaging over ALL the events. Similarly, the coin lands heads up 50% of the time, over ALL the flips I make- but not over all the flips I ask you about. Similarly, we're not asking Hilary about all the events- only some of them. If you were asking Hilary about all of them, independent of what the second detector did, then she would break even guessing yes 50% of the time. But this isn't the case, since we're only asking her about ones where the second detector registered something. Thus, the error in van Fraassen's argument is when he says that "For at the right [first} side the clicks come at the 50% rate, and changes in Hilary's personal information or opinions do not affect that at all. Thus Evelyn [a hypothetical person standing at the first detector] at least would be right to advise Hilary to just ignore the ...information [about the second detector]." Evelyn would NOT be right to advise that since Hilary is being asked about a specific subset of the events Evelyn is seeing, and those events DON'T come in at the 50% rate. Evelyn should inform her of the rate for that subset of events.
Van Fraassen then goes on to investigate "Yet by what epistemic principle can one license ignoring evidence that would clearly change one's opinion if heeded?". However, this isn't necessary, because as we've seen, Hilary shouldn't ignore the evidence she has gained, because if she continues guessing 50%, she won't break even, as van Fraassen claimed. Thus, she does have a "better" probability after the evidence from the second detector than she did beforehand.
Now, you may be wondering how "I don't know anything" is "better" than 50%. The reason is that when we initially asked Hilary what the probability was of the first detector registering something and she answered 50%, she was implicitly assuming that there was no correlation between whether or not we asked here about the first detector and what the first detector registered. To be correct, she should have said "Depends- was this a randomly selected run?". As we've seen, it was not. So her answer of 50% is actually wrong- not because the second detector doesn't register something 50% of the time, but because we're asking her about a subset that she knows nothing about instead of the entire set. The extra information tells her that she was wrong in that assumption, and thus, the probability "something between 0% and 100%" is in fact better than "50%".
Now on to my second point, that this doesn't actually require quantum mechanics. Hopefully by stripping away the quantum mechanics, it will become clearer where the flaw is van Fraassen's argument is. So here is an argument isomorphic to van Fraassen's, but without the quantum mechanics.
Consider this case:
Assume that it rains in Timbuktu is 50% of the time.
Also assume that due to the global air flow, ocean currents, and everything else, there is a correlation between whether or not it snows in Philadelphia and whether it rains in Timbuktu. Examples of such relations would be:
1) Whenever it snows in Philadelphia, it always rains in Timbuktu, and never rains any other time. (In this case, it'd snow 50% of the time in Philadelphia).
2) Whenever it snows in Philadelphia, it never rains in Timbuktu, and always rains when it's not snowing in Philadelphia. (In this case again, it'd snow 50% of the time in Philadelphia).
Assume that I know this exact mathematical relation, but that Kenny doesn't. He can know the form of it, but not it's exact mathematical value.
Additionally assume that Kenny and I are both aware that it's not snowing here in Philadelphia.
Finally, assume Kenny has a friend in Timbuktu.
Now, assume I tell Kenny that if it rains in Timbuktu, I will make him hot chocolate. Kenny would like to know what the odds are that I'm going to make him hot chocolate. So Kenny calls his friend in Timbuktu. We expect the conversation to goes something like this:
Kenny: "Hi. What are the odds that it is going to rain there?"
Kenny's friend: "50%."
However, the conversation really should go like this:
Kenny: "Hi. What are the odds that is going to rain there?"
Kenny's friend: "I know that it depends on whether it's snowing over there, but I don't know how."
It's wrong for Kenny's friend in Timbuktu to say 50%, because the probability of it raining in Timbuktu is actually conditional on the probability of snow in Philadelphia, and I am forcing the case where it's not snowing in Philadelphia. Essentially, I'm making a cut and ignoring the days when it snows in Philadelphia. So, the relevant probability isn't the probability that it rains in Timbuktu on any day, but the probability that, on days it isn't snowing in Philadelphia, it rains in Timbuktu. Now, if Kenny bet that in 50% of these cases he'd get hot chocolate, as van Fraassen recommends, he's not necessarily going to average out even- in the first case, he's never going to get any hot chocolate. In the second case, he'll always get hot chocolate. Thus, the "extra information" that it depends on whether it's snowing in Philadelphia is not at all irrelevant, nor should he ignore it. I hope this case is somewhat clearer than the quantum mechanical case in his paper.
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Lauren, I'm still not convinced that you have reconstructed van Fraassen's argument correctly. Firstly, you are still using objective probabilities, and not subjective degrees of rational confidence. Van Fraassen is talking about Hilary's beliefs about the (unknown) outcome of one trial. So, she's being asked, after the experiment is over, "what is your degree of confidence that the right detector registered on trial 62?" She says, "Even odds - it could've gone either way." She is then informed that the left detector registered on trial 62, and asked again. She now says "I have no opinion whatsoever as to whether the right detector registered on trial 62" (or, to use the technical term, "the probability is inscrutable"). In the last paragraph, van Fraassen gives the conclusion:
Moreover, what about stability in those parts of our opinion that mat-
ter to us? The vague areas in one’s opinion harbour possible alternatives
that are totally at odds with one’s current definite opinions. As a result,
any policy for managing opinion in response to new experience will
need to make special arrangements for ensuring the stability of opinion
that one wants to retain, or risk losing it in apparently unrelated inci-
dents. If that is correct, then such a policy will need to use more than
just sets of probability functions to store and operate on – it seems
that there would need to be ‘hidden variables’ to help maintain such
stability.
I think there may be a reply along the lines of what you've sketched here, but it's certainly not the case that van Fraassen is making a mistake by asking us only about a subset of cases. He is asking about a subset of cases, and he knows it, and that subset is a singleton. What you're doing is trying to impose frequentist objective probability ("how often can this detector be expected to register in the future, if we confine ourselves only to those cases in which the other detector registers?"), which doesn't go here.
Posted by: Kenny at November 7, 2007 06:36 PMThe problem I am pointing out is earlier in his argument. Van Fraassen concludes that one needs something beyond probability functions (the hidden variables) BECAUSE he's trying to answer this question: "Yet by what epistemic principle can one license ignoring evidence that would clearly change one’s opinion if heeded?"
He is trying to answer this question because he thinks he needs a reason to justify Hilary ignoring the information about the left detector. Why does she need to ignore the information about the left detector? Because with the information she has no clue what to guess, but without the information she would guess 50%, which he claims is the right thing to do because she would break even. Note that the claim that she would break even is a claim about an objective probability, not her confidence, and it is this claim that I am challenging.
What I'm saying is he shouldn't even be asking this question. Why? Because Hilary shouldn't be ignoring the information. Why? Because she wouldn't be breaking even at guessing 50%, so that's not what she should do. Why isn't she breaking even? Because of the subset of cases we're looking at. This is where the objective probabilities come in. I'm not imposing frequentist objective probability. I don't care about future events.
Posted by: Lauren at November 8, 2007 12:02 AMThe other thing that I'm doing is arguing why the probability is REALLY inscrutable in the first case, even when she answers "Even odds- it could've gone either way". Why is it really inscrutable and not 50%? Because we're not asking her for P(right detector). We're asking her for P(right detector given that we asked her about it). In calculating the 50%, Hilary is already ignoring facts that she knows: namely, that they asked her about the event, and that they may be asking her about every single event. If she considered these possibilities, then she'd answer that she couldn't calculate the probability then either, even before we give her any information about the other detector.
Posted by: Lauren at November 8, 2007 12:14 AMSorry, correct the line in the last comment to "She knows that they may NOT be asking her about every single event, but a special subset". It's past my bedtime and I should go to sleep.
Posted by: Lauren at November 8, 2007 12:15 AMAs far as Hilary knows, there is no connection between the outcome of the experiments and which experiment we choose to ask her about. Perhaps if she's particularly suspicious she will suppose that we're trying to trip her up somehow, but it's still got to be even odds that we'd ask her about a case where it registered and a case where it didn't. If we are going to ask her about just one event, and as far as she knows it could be any of the events, and the events are expected, on physical grounds, to split half and half, then her 50% answer is correct. We're only picking out ONE event.
You also talk as if we only have probabilistic beliefs if we can calculate probabilities, but Bayesians don't believe that at all: we have probabilistic beliefs about all sorts of things that we can't calculate odds on. So, for instance, when Hilary tries to assign a (vague) value to P(the detector registered on trial x | she was asked about trial x), she's going to think about the motiviations of the people asking and what they were trying to do, and realize that there is no difference to any of us between them picking a case in which the other detector registered and a case where it didn't. That is, our purposes in asking her are served just as well. So won't she then suppose that the trial chosen could be any trial from either set, and shouldn't she therefore bet at 50%? She is then informed of which set it came from, but because she doesn't know the angles she is in the same position as before: in one set, the number of trials in which the detector registered is higher, and in the other they're lower, but she still doesn't know whether she is in the higher set or the lower. From her perspective, there's even odds on which set she's in, so she should take the probability over the whole set, ignoring the new evidence.
Posted by: Kenny at November 8, 2007 07:51 AM"As far as Hilary knows, there is no connection between the outcome of the experiments and which experiment we choose to ask her about."
Correct. But what does "no connection" mean? Like everything else in the English language, there's an ambiguity in that term, which is part of the what makes this problem tricky. "No connection" can mean just that, that there isn't a connection of any type. But, to say something is uncorrelated is actually to make a strong claim about the type of connection between objects, even though we also call that "no connection". Hilary has no reason to assume this connection- instead she should treat it as one of the possible correlations of the events she might be asked about. But she makes that assumption that they're uncorrelated when she answers 50%.
"it's still got to be even odds that we'd ask her about a case where it registered and a case where it didn't"
No, it's not, because we don't know it's uncorrelated. We choose what cases we ask her about, and in this case, we're tripping her up because we're asking about cases where it registered, i.e. there is a correlation. Think about this. I buy a bag of marbles. There's 25% red, 25% green, 25% yellow, 25% blue in the bag I buy. And then I put some of them in a box, and ask you to bet on the odds that I pull a red one out. You shouldn't guess 25%, you should say "how on earth do you expect me to know?". As far as you know, there's no connection between the marbles I placed in the box and their color- but that doesn't mean you can assume that their color is completely uncorrelated with their presence in the bag. That's the case here.
As to your second part, sure, it's reasonable for her to pick 50% initially, because all those beliefs are reasonable and true, she doesn't have any reason to suspect us. Just like it's reasonable for you to pick 25% for the marbles in the case above. But you shouldn't be too confident in it, and neither should she (despite what van Fraassen says).
And similarly at the end. If, after we tell her the angle, she still wanted to bet (or we forced her to), she might as well pick 50%. But actually, she has no more justification for that than any other number. You picked the set where the ratio is 1-1 (i.e. higher or lower). But you could also divide them into the set where it's 2-1, or 1-7, or 4-5, or any other number you want. She doesn't know which of those sets she's in, and so she could take the probability of those sets, and depending on what number we pick, she might do better or worse than 1-1 odds.
There's also a big difference in this approach and what van Fraassen does in his paper. He claims Hilary must guess 50% based on whether or not that's the objective probability she should guess to get the most number right. What you're establishing is that, given what she knows, she might as well guess 50%. There's a big difference. In case case she must throw out the additional information. In your case, she keeps it, saying well I might as well guess 50% but now I'm really not certain in that at all.
Here's what van Fraassen should have done. He should have told Hilary he was picking a random subset of events, and then told her the result of the left detector (even if it was low). In this case, she would be absolutely confident at first that probability is 50%, and in the second case, she would be inscrutable.
However, the solution here is that she's epistemically justified in going back to the 50% even after she gets the new information because she knows that over the whole set that they ask her (even if it's just one event), that's going to be better than any other number she pulls out of a hat.
To go to the marble analogy, what should happen is that I say, ok, I put a random selection of marbles in my box. What's the probability I pull a red one? You can say 25% certainly now. And then I say, wait, I've put some of the marbles into baskets inside my box. There's a different percentage of red marbles in each basket, but I'm not telling you what it is. I pulled this marble from the right basket. Now what's the odds of it being red?
You should still answer 25%- in this case, that IS the best answer. This is what van Fraassen wanted to show in the paper above- that Hilary would just be justified in answering 50%, but that it IS the right answer she should give. That's it's a better answer than throwing up your hands and walking away. However, here you can reason to the fact that it's the right answer using statistics, but you can't in the above case, which is why it's a "well you might as well pick 50%" deal.
How do you reason to that? Basically you say, gee, it'd be really nice if I could know what the percentages were in the baskets- if I could know that, then I could increase my accuracy in guessing. But I don't know that. And my probability of guessing the right percentage is low enough that, even accounting for the great gain I would get if I knew what they were, my odds are still lower than if I stick with 25%.
Now hos is this different than the previous case? Because in this case you know something about the set you're actually being asked about. If you were only being asked about marbles in a certain basket in the box, then once again, you should refuse to answer, or any answer is just as good as 25%.
In short, it's reasonable for her to pick 50%, but it's not necessary, as van Fraassen wants to claim that it is. In picking 50%, the new information's influence is to make her really not certain of it, knowing in fact that any other number is just as likely to be right as 50%. van Fraassens, because he thinks she must pick 50%, wants to introduce new hidden variables that make the new information so uncertain that it's better not to use it, instead of here, where she uses it to determine she has no clue but decides that in the end she doesn't have anything better to guess anyway. Do you see the difference?
Posted by: Lauren at November 8, 2007 08:55 AMWhere does van Fraassen say that she necessarily has to keep her 50%? Are you sure that's what he's arguing? I don't think it is. All he's claiming is that if we make no adjustment to Bayesian epistemology at all, if we just follow where Bayesian inference leads us, there will be cases where a new piece of true information that is relevant evidence to the case causes us to know less about it, to be unjustified in holding any opinion whatsoever on the subject, whereas we were previously justified in holding an opinion. (It won't just lead our level of confidence to be closer to .5, but lead us to be so confused that we don't even HAVE a level of confidence.) So, he says, we need some kind of "hidden variable" (I think he may be using the phrase a bit facetiously, given the QM context) or principle of epistemic conservativism, so that we don't change our beliefs in cases like this one.
Posted by: Kenny at November 8, 2007 09:51 AM1) Does van Fraassen say she MUST say 50% in the second case? I think yes. This is because he is trying to justify her throwing out the information from the other detector. If she didn't need to throw out this new information, then she wouldn't need a justification for throwing it out. Hence, somewhere in his argument he concludes that she must stick with 50% and ignore the new information- it's not ok for her to do otherwise.
2) "there will be cases where a new piece of true information that is relevant evidence to the case causes us to know less about it, to be unjustified in holding any opinion whatsoever on the subject, whereas we were previously justified in holding an opinion"
He has to be claiming more than this. Consider this case. You know the weather in Philadelphia is such that there's typically a 50% chance of rain on Thursdays. So you are initially justified in believing there's a 50% chance of rain today. Then I come by and say, I looked at the weather report this morning and there's not a 50% chance of rain, but I don't remember what it is. Now, you don't have justification to hold any opinion on what the odds of rain are today, even though previously you were.
I think his argument in this area goes like this: Bayesian analysis says that she cannot hold any opinion with any degree of confidence after we tell her about the second detector. However, that's not the case because she should hold to her 50%, because that would lead her to break even. (Or at least she could hold to her 50% with some degree of confidence.) Therefore, there's something wrong with Bayesian probability and let's find a way to fix it.
It's not a problem for Bayesian analysis to say that she's not justified in holding any opinion on the subject, when it showing her that her previous assumption that justified her earlier opinion was wrong, which is the case here.
I need to run to class, but here's something to think on and I'll say more later.
Posted by: Lauren at November 8, 2007 10:20 AMBayesian epistemology says that Pf(S) = Pi(S|E), where S is a belief, E is some evidence regarding that belief, and P is the probability of that belief being true. So there would be a problem for Bayesian epistemology if, for some reason, Pf(S) didn't accurately represent the degree of confidence we should have in belief S.
Now it could be the evidence E could relate to the prior evidence one had for believing S. If E countered prior evidence for S, then P(S) decreases. If E completely contradicts S, P(S|E) is 0. Let's say S is: all desks are black. And I initially have a very high degree of confidence, because I incorrectly believe desks are defined to be black. Case 1: you come tell me that no, desks are defined by their purpose, and not by color. So P(S) decreases, although perhaps not to 0. I can still believe that all desks are black. Case 2: I find a white desk. I now have no justification for beliving S. Now P(S)=0.
So, Bayesion epistemology does not have a problem if in the final probability is 0 (i.e. you have no justification in believing it), or if it is less than it was initially- (i.e., your prior reasons for believing was countered by new evidence). It has a problem if for some reason we want to say that Pf(S) != Pi(S|E).
What are the relevant beliefs to Hilary?
S: The probability of the first detector going off is .5
Note that this a belief about a correlation.
What is Pi(S)?
According to van Haassen, 1. However, as has been discussed, it's not 1 because she should account for the fact that she knows absolutly nothing about the subset of events we are asking her about. In fact, it's substantially less than one, because her entire justification is that she cannot think of a reason that we wouldn't use a nonrandom subset. The fact that she gives it a nonzero value means that she is assuming this specific correlation (i.e. perfectly uncorrelated).
So, this is just like you don't have any solid information when I show up with the box of marbles and ask you what the probability that I pull a red one is, given that all the marbles in the box were once in a bag, with other marbles, where 25% of them were red. Maybe you think, well, I can't think of reason for her to change the density of red marbles, so I'll say 25%. I'm not so sure I'd say you are justified in saying that, but regardless, it's pretty weak justification.
So Pi(S) is small.
Now, what is Pi(S|E)?
To answer this, we need to figure out what E is. To first order, E is the fact that the other detector went off. However, to understand why this is relevant, we need to look deeper. The fact that the other detector went off tells us we're looking at a specific subset of events, and especially a non random one. This is E: The subset we are looking at is nonrandom.
What is the effect of E on Pi(S)? It decreases it. Since the assumption that they were random was your only assumption in support of Pi(S), perhaps even to 0. You now have no justification for your belief.
So Pi(S|E)=0, or is pretty near to it. (In the paper, he takes it as 0.)
However, this itself is NOT a problem for Bayesian analysis so long is Pi(S|E) is the "correct" degree of certainty for the belief. van Fraasen says no, it's not. She should continue to believe in S. He says that this is "correct" because it lines up with reality- i.e. she will break even if that is how she bets. I am arguing NO, she will not strike even, and in fact she has no better odds believing the probability is .5 than she does with .1435672. So Pi(S|E) is the "correct" degree of certainty.
---------------------------------------------
New point:
However, it initially looks like there is a way to resuscitate his above argument. What if we DO take a randomly selected subset of events from the set of events, and tell Hilary that?
Let A be the set of all events.
Let B be the subset of events we ask Hiliary about.
Then Pi(S)=1. (Assuming she believes us fully.)
Then we tell her what happened at the other detector. If it clicked, we tell her that. If it didn't click, we tell her that.
Now what effect does the information E have on her?
It has two effects.
1) What is her belief in the probability for this specific event?
2) What effect does it have on her betting on all of B?
It tells her not anything about B, but rather, that this event is in a special subset of B. However, this is a statement about this EVENT.
She then knows nothing about the probability for this specific event, because of the reasons given above. Hiliary can derive that, if she knew more information about these subsets, she would be able to guess more accurately than her 50% breaking even she's "guaranteed" with a random subset.
But- this is the important point- she still knows for sure that S is true since it's a random subset. So she knows that guessing 50% will let her break even. So P(S|E)=.5, for her betting (which is what van Fraassen cares about), even though P(this event|E)=0.
Posted by: Lauren at November 8, 2007 01:28 PMWe're not talking about the belief that the probability is .5. That's mixing in objective probabilities again. We don't care about objective probabilities. Van Fraassen uses this idea of a "representor," which is something like a range of probabilities we could assign. When we get new information, the range is supposed to shrink. So it starts out as [.45, .55], but we get more information and now its [0, 1]. The question is thrown wide open again.
Now maybe, as I think you've been trying to argue, this isn't a real problem, but I'm still not convinced that it's the same thing as what you are talking about at all.
Posted by: Kenny at November 8, 2007 01:45 PM"When we get new information, the range is supposed to shrink."
In this post I will:
1) Show that the broadening of this range is NOT ITSELF a contradiction of Bayesian epistemology.
2) Consider again van Fraassen's argument, to look in detail at what he claims IS a contradiction of Bayesian epistemology.
First, in respons to your comment about objective probability, please keep in mind that there are two probabilities we are discussing here:
1) The actual probability of throgh the detector going off
2) Hiliary's certainty in her beliefs about the probability of the detector going off
Bayesian epistemology applies to the latter, not to the former. Thus, it IS proper to apply Bayesian epistemology to her belief that the probability of the detector is .5, and also her belief that the probability of the detector is .87453 for that matter. True, we don't care about objective probabilities. But we DO care about Hiliary's beliefs about those objective probabilities.
Now on to #1.
When we get new information, what is supposed to happen according to Bayesian epistemology? We are supposed to revise our previously held beliefs according to the condition probabilites given that eveidence. That is, when we give her the information about the second detector, Hiliary should be revising her beliefs about the probability of the detector going off.
According to Bayesian epistemology, she does NOT necessarily have to narrow the range of probabilities she believes the detector will have. Why is this? note that narrowing the range corresponds to an increase in certainty for the numbers still remaining in the range. So then, the relevant question is, does Bayesian epistemology require us to become more certain in our beliefs with more evidence?
Bayesian probability does NOT require us to become more certain with more information. Consider this quote from the Stanford Encyclopedia:
"In Bayesian Confirmation Theory, it is said that evidence confirms (or would confirm) hypothesis H (to at least some degree) just in case the prior probability of H conditional on E is greater than the prior unconditional probability of H: Pi(H/E) > Pi(H). E disconfirms (or would disconfirm) H if the prior probability of H conditional on E is less than the prior unconditional probability of H."
So we see clearly that under Bayesian epsitemology Hiliary IS allowed to become LESS certain in her beliefs about the probability of the detector going off. And what does less certain about her belief about the probability of the detector correspond to? It corresponds to the range being opened again. Thus, the mere broadening of this range is NOT a contradiciton of Bayesian epistemology.
Moving on #2.
Given that this is NOT a contradiction with Bayesian epistemology, let us look once again at van Fraassen's argument. I think we can both agree that he claims there is a contradiction with Bayesian epistemology in this experiment.
Since Bayesian epistemology claims that Pf(S) = Pi(S|E), we are looking for the point in the paper where he claims that Pf(S) != Pi(S|E). We should further be able to identify what he claims Pi(S|E) and Pf(S) are.
He claims this when he discusses her belief about the range of probility about the first detector detecting something, after she has been told about the other detector. So then, let's identify Pi(S|E) and Pf(S). These should be two beliefs Hiliary have about the probability of the first detector registering something, and secondly, they themselves should have differing degrees of certainty since they are to be unequal. Because the beliefs have differing certainties, they should be beliefs about different ranges.
What is Pi(S|E)?
As you point out, her belief after the evidence is: The probability of the second detector going off is somewhere in [0,1]. How did she arrive at this belief? By reevaluating her earlier belief that the probability of the detector is .5 using the new evidence appropriately. That is, she uses Bayesian epistemology. Therefore, her belief that the probability is in the range [0,1] is Pi(S|E).
Now, for this to be a contradiction of Bayesian epistemology, he must claim that, for some reason, Pf(S) should not be Pi(S|E). So he must make a claim about Pf(S), and justify what Hiliary should believe this. Now, justifying why someone should believe something is in general done by appealing to objective truth. Thus, here we would expect him to appeal something objective about the world to justify a certain belief that Hiliary should hold.
Consider his discussion about Evelyn and the discussion about betting. He claims that she would break even is she were betting at 50%. This is indeed a claim about an objective probability. Furthermore, the discussion regarding Evelyn makes it clear that this is what he thinks Hiliary's belief should be, albeit he is using Evelyn for his mouthpiece. So according to him, Pf(S)=.5.
This is further supported because he spends the rest of the paper explaining the rationale Hiliary could give to justify throwing out the evidence that lead to her belief that the probability is in the range [0,1]. If he thought that her belief that the probability is in the range [0,1], there would be no purpose in doing this, since this is not itself a contradiction of Bayesian epistimology, particularly since that is the conclusion Hiliary arrived at using Bayesian epistimology.
So now that we have identified Pi(S|E) and Pf(S) in his argument. Pf(S) != Pi(S|E), consistent with what one must claim if one is claiming Bayesian epistemology is wrong. Furthermore, his claims about Bayesian epistemology are consistant with the description of Bayesian epsitemology given in the Stanford Encyclopedia.
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Now where does my response come in? I'm basically saying that Pf(S) isn't .5. Because he justified Pf(S)=.5 with an appeal to an objective probability, I show that the objective probability he is discussing is not, in fact, .5.
I think you are dealing with a more simplistic version of Bayesian epistemology. Van Fraassen is interested in a more sophisticated Bayesian epistemology in which there is vagueness in degrees of confidence, since we can rarely assign numbers to our degrees of confidence. So, initially, Hilary has a vague degree of confidence in the range [.45, .55], and after the new evidence she has a vague degree of confidence in the range [0, 1]. It's not that she believes that the objective probability that the detector went off is in this range, but rather that her vague subjective confidence in the proposition that the detector went off is in this range. As such, the problem is not that her uncertainty increased, but rather that the vagueness of her degree of confidence increased.
Van Fraassen is worried that this may apply to our other beliefs as well: you may have degree of confidence .5 that it will rain today, then get some new information and suddenly be unable to assign any subjective degree of confidence to the proposition that it will rain (i.e. be completely unopinionated about whether it will rain). It's not that you come to believe that the objective probability that it could rain might be anywhere between 0 and 1, but rather that your vague degree of confidence spans the whole range from 0 to 1. The subjective probability becomes inscrutable. Furthermore, you may be epistemically required no longer to have any opinion. But that's not supposed to happen. Your degrees of confidence are supposed to get adjusted, and you could have greater or lesser certainty, yes, but you aren't supposed to lose your ability to have a degree of confidence!
Posted by: Kenny at November 10, 2007 10:07 AMFine. For the sake of argument, I will run with your interpretation, and show why what I said above fixes this too. Please reread the marble example; I'm too lazy to rewrite it here but I will refer to it below.
What is the representator?
"One way to think about my representor is that it contains ALL THE POSSIBLE OPINIONS that I could have by becoming precise: that is, opinions I would have if I filled in blanks in my present opinion." [emphasis mine].
Thus, we need to ask if Hiliary's initial representator contained, so far as she could reason, ALL the possible opinions she could have had if she got new information. You claim here inital representator is [.45,.55], van Fraassen claims it's .5 exactly. I claim that both of those are wrong, and it should be [0,1], if she includes the fact that they might be asking her about a nonrandom subset.
Back to the marble example, if I show up with my box of marbles containing the marbles drawn from the bag where red marbles were 25% of the total, and ask you what the probability of me pulling a red marble is, you may have justification in guessing 25%, because you have no reasons to assume I picked a nonrandom subset. BUT you do NOT have justification in making your representator .25 exactly- in fact, you must say it's [0,1].
Why? Because you know that if I told your more information- by the way, I pulled only the red marbles from the bag, or by the way, I pulled only the blue marbles from the bag- you would have a different opinion. Since your representator includes ALL these opinions you MIGHT have if I gave you more information, you MUST include these in your representator, even though you might have justification in better 25%. Your representator is NOT the value you should bet at.
Hence, if Hiliary had thought just a little bit harder, she would have realized that the subset could have been anything. And therefore her representator initially would have been [0,1], although, being forced to bet and being unable to think of any reason we'd want a nonrandom subset, she could be justified in saying 50%. But making her representator is .5 is NOT allowed- it MUST be .5+/-.5 to account for the fact that she should be able to reason that if we gave her information about the subset her opinion would no longer be .5.
Posted by: Lauren at November 10, 2007 11:53 AMNo, that's not the way the representor works. Van Fraassen talks about the probability being .5 for simplicity, but, in fact, Hilary's subjective degree of confidence is vague, and falls somewhere within a range - it doesn't have a specific value. The representor is the set of opinions Hilary could have with her current information, if she got rid of the vagueness. It is not the range of opinions she could have by getting new information. That should always be [0, 1].
Posted by: Kenny at November 12, 2007 10:33 AMThere IS a vagueness in her current information- the vagueness in what subset she is being asked about. She's ignoring that vagueness inappropriately. Secondarily, van Fraassen doesn't talk about the probability being .5 for simplicity, he thinks Hiliary can be absolutely certain that it's .5. So with her current information, she could have any opinion in the range [0,1].
Posted by: Lauren at November 12, 2007 10:37 AMJust to clarify:
Representator: "ALL THE POSSIBLE OPINIONS that I could have by becoming precise: that is, opinions I would have if I filled in blanks in my present opinion."
What vagueness does Hiliary have in her current opinion? What subset she is being asked about.
If that vagueness were to become precise, what opinions could shes have? Anything between [0,1].
Van Fraassen ignores the fact that Hiliary is making an unjustified opinion about what subset she is being asked about. Now, you could talk about after she makes her unjustified assumption, as discussed above, this isn't a problem because Bayesian epistemology allows for the case where you find out your assumption was wrong and your representator becomes larger again.
Posted by: Lauren at November 12, 2007 10:46 AMNo. Vagueness and uncertainty are two totally different things. A question with vagueness is one like "how many hairs do you have to pull out of a man's head before he becomes bald?" or, "how many grains of sand do you need to make a dune?" The point being made is that in real life we never have precise opinions, we have vague ones. Van Fraassen probably holds a linguistic theory of vagueness (I don't know of any philosopher who doesn't except Peter van Inwagen), so the idea is that we simply haven't bothered to define the terms "bald" and "dune" fully, and there are good reasons why we wouldn't. So, to resolve the vagueness doesn't mean to get enough information to determine the objective probability: it means to pick one precise number to represent her opinion, which is not something we normally do. The idea of the representor is that it's like mapping the one vague notion of "baldness" to a class of precise notions, e.g. "having no hairs," "having no more than 1 hair," "having no more than two hairs," etc.
It's not a matter of being certain that it's .5 - we're talking about her subjective opinion; she doesn't have to discover it, she just has to decide. She doesn't need any additional information to resolve the vagueness.
Posted by: Kenny at November 12, 2007 04:02 PMFor the record, I disagree with your interpretation. When van Fraassen discussing the representator, this is what he says:
"Suppose a given person Hilary has subjective probability function P with domain D. We can think of D as part of a larger domain D¢, and P as part of each of a whole range of probability functions P¢ defined on D¢. Presumably Hilary has no opinion at all about any propositions in D¢ that are logically independent of those in D. But there are many mixed cases. For example if P(A) = 0.5 then whatever B is, P¢(A&B) £ 0.5. The standard amendment, as elaborated in the literature by Isaac Levi, Richard Jeffrey and Patrick Suppes, among others, is to represent Hilary’s opinion here by the entire class of probability functions P¢ rather than by a single one.1 Call that class her representor. Her opinion about A&B is then represented by the class of numbers assigned to it by the members of her representor. That class will be an interval."
This most certainly includes vagueness as you have defined it, but it also includes uncertainty. Note that if Hiliary had uncertainty about B, as opposed to mere vagueness, she would still have a range of values.
Regardless, if it is as you claim, and Hiliary just has uncertainty at the end, and not vagueness. She is uncertain about the angle. She is just as justified in choosing an angle at the end as she is in choosing a subset in the beginning. If there's no "vagueness" in her opinion because she should be uncertain about the subset she's being asked about, then there's no "vagueness" in her opinion because she's uncertain about the angle she's being asked about. They are completely isomorphic situations. She is just as justified in deciding on an angle in the end as she is on deciding on a subset at the beginning.
Posted by: Lauren at November 12, 2007 05:18 PMCorrection:
Regardless, if it is as you claim, and Hiliary just has uncertainty at the BEGINNING, and not vagueness. I'm arguing she just has uncertainty at the end also.
I found the quote I was looking for. I would like to point out that van Fraassen uses "vagueness" to refer to her lack knowledge about the angle, NOT anything about an undefined term, which similar to how I would call her lack of knowledge about the subset.
"Let us write ‘Right’ for ‘There was a click on the right hand side’, and similarly for ‘Left’. So her representor is simply the set of all probability functions p such that:
• p(Right | Left, and angle = x) = cos2(x)
• p(Left | Right, and angle = x) = cos2(x)
• p(Right) = p(Left) = 0.5.
Conditionalizing any one such function p on the information ‘Left’ yields a function q such that
q(Right, and angle x) = 2 cos2(x)p(Left, and angle x). This can be any number in [0, 1], because p(Left and angle x) can have any value in [0, 0.5]. The value of q(Right) should presumably be this very range for q(Right and angle x), since it is a given that the angle must have some magnitude or other. Therefore Hilary’s posterior probability
for ‘Right’ equals [0, 1], which is the vaguest it can be at all."
He is not talking about the angle being a "vague notion"- instead, he seems to take the angle has having an exact, if unknown, value throughout the paper. From this we see that he isn't basing his argument on vagueness in the notion of angle, but rather in its value- that is to say, an uncertainty.
Thus her opinion is vague because she knows nothing about the angle, and it could have a cosine values anywhere between 0 and 1. Similarly, I am claiming her opinion beforehand SHOULD have been vague in the same sense, because the probability could have been anything between 0 and 1 depending on the subset. We can quibble about whether or not that's properly "vague" or "uncertain", but I think it's clear that her lack of knowledge about the subset is identical to the lack of knowledge she has about the angle, which is what van Fraassen bases his argument on.
Posted by: Lauren at November 12, 2007 10:22 PMThe angle is certainly not a vague concept. The problem is that you are still confusing the SUBJECTIVE probabilities of Bayesian epistemology with the OBJECTIVE probabilities of quantum mechanics. Hilary has the greatest degree of UNCERTAINTY when she assigns a probability of .5 - that means that she thinks the two outcomes are equally likely, and therefore has no idea which one actually happened. She has the greatest degree of VAGUENESS of opinion when the probability is the range [0, 1] - that means she has no opinion whatsoever as to how likely each outcome is. It is not the REALITY or OBJECTIVE PROBABILITY that is vague (as I said, most philosophers don't think it is possible for such things to be vague), it is only HER OPINION that is vague. In principle (van Fraassen thinks), the more evidence you have, the less vague your opinion should be. On the other hand, if new evidence increases your uncertainty, that's not a problem at all (in fact, that had better be possible or Bayesian inference is in trouble).
If you can show that Hilary's opinion at the end is less vague than her opinion at the beginning, THEN you will have shown that van Fraassen's argument fails, even if her uncertainty is greater at the end, but, in order to do this, you MUST stop confusing subjective and objective probabilities, and stop confusing vagueness with uncertainty.
Now, your substantive claim is that she should have realized that there might be selection bias, and therefore should have had only vague opinion before. This can be answered quite simply by saying that Hilary is not asked to report P(Right | we asked), but simply P(Right). That is, suppose we ask her to report what her opinion was before she knew we were going to ask her about this trial. Then, even if you are right that the fact that we asked her about it is a new piece of evidence in light of which she should revise her opinion, she will nevertheless be justified in assigning a definite (i.e. non-vague) probability (before she knew we would ask her) to the detectore going off on that particular trial. Now, because you've made her suspicious of selection bias, she will have definite opinion BEFORE we ask her, but maximally vague opinion AFTER we ask her, and this may make the problem even worse than before.
Posted by: Kenny at November 12, 2007 10:41 PMTo your first point-
As I have been saying all along, at the beginning she SHOULD have said the probability is in the range [0,1], and nothing more, because she can say nothing about the subsets. She SHOULD have no opinion whatsoever about the outcome. You say: "She has the greatest degree of VAGUENESS of opinion when the probability is the range [0, 1]" Thus this IS a vagueness, as I initially claimed, or else you should clarify this more. I have been claiming that van Fraassen is WRONG when he says .5, that it should have been [0,1]. I am NOT the one claiming in should be .5- I am not claiming she should have maximum uncertainty- but maximum vagueness (going by what you're saying vagueness is now).
To your second point-
P(right) is DEFINED as part of the SET. You can't have the probability of drawing a red ball unless you know what set it's from. Similarly, the probability of it going off .5% with respect to the set of all experiments. She is justified in thinking that the probability with respect to all experiments in 50% beforehand. But then as soon as she gets more information, she has to reconsider what the relevant subset is.
Consider the marble example above: I come over and tell you that I'm pulling marbles from a bag where the have a .25 chance of being red. You think they have a .25 chance of being red. Then I tell you that I'm really pulling them from a box that has marbles from the bag in it. You assume the probability of the box is .25. I then tell you that I didn't necessarily choose a random subset. Your opinion is [0,1]. Do you consider this to be a problem for Bayesian epistemology, and if not, where do you think the difference is between this and van Fraassen's argument?
I think this example makes it clear that your belief about probability has to be defined in terms of a set. You are ALWAYS at least implicitly defining a set when you talk about probability, which is what van Fraassen forgot, and what you are forgetting in your second case. There is no P(right)- there is P(right our of all of them), P(right out of the ones I ask you about), P(right out of whatever set you want).
Posted by: Lauren at November 12, 2007 11:08 PMSecondarily, which I forgot to mention above, I have NEVER claimed the OBJECTIVE probability is vague. The set/angle are well defined the entire time- they are vague to Hiliary because she knows nothing about them. And the probability over the entire set is well defined from quantum mech. I have no idea why you seem to think that I have been claiming this; if you point me to where I have been unclear I will most certainly clarify myself.
Also, I'm betting you are going to say that I haven't established that she should have said the probability is in the range [0,1], so I'm going to resummarize that argument here.
Consider why van Fraassen says that her belief should be [0,1] at the end. This is because she knows about the correlation with the angle, and she has enough information that if she knew the angle, she could calculate it out, and she knows it would fall in the range [0,1], but cannot narrow it down further.
Consider her initial case, if she accounts for sampling bias. If she thought, she would realize that they could be asking her about the set where the detector never went off, or they could be asking her about the set where it always went off, or anywhere in between. By the exact same reasoning (I can make it more mathematically rigorous if you like), she knows that the probability is in the range [0,1], but nothing else.
Posted by: Lauren at November 12, 2007 11:27 PMPerhaps I've been a bit sloppy when I talk about the probability of pulling a red marble from the box in some of my earlier comments- I should clarify that I have been speaking from your perspective. I know what subset I put in the box, so I know what the probability of pulling a red marble is (because the subset determines the probability). You don't know what subset I put in the box, so as far as you know, the probability is anywhere between 0 and 1 TO YOU. The actual probability is determined exactly (say, .645645 for example), but you don't know anything about that and the best you can say is [0,1]. So when I say the probability is in the range [0,1], I mean from your perspective that's the best you can say. The later comments were clearer on that point, but in case you missed that, I hope this clarifies how I have been talking about subjective probabilities and not objective, since you seem to think I'm talking about objective probabilities more than I think I am.
Posted by: Lauren at November 13, 2007 08:16 AMBut we're NOT talking about probabilities over a set. Because the expected outcome (in terms of objective probability) is that the right detector will register in 1/2 of all trials, Hilary's subjective degree of confidence that the detector went off in the case of EACH INDIVIDUAL TRIAL was initially .5. I don't understand how anything you have said undermines this in the least. Even if she has to change her mind once we ask her, that doesn't change this fact. Before we asked, her belief with respect to trial 62 had subjective degree of confidence .5, with no vagueness at all. Why wouldn't it?
Posted by: Kenny at November 13, 2007 01:50 PMWhat is the probability I pull a red marble?
You're hopefully staring at the computer screen blankly, wondering whether I'm pulling the red marble from the box, the bag, or out of thin air. The probability is different depending on the set, and you HAVE to assume a set to get a probability. (In QM, the default set is a hypothetical entity called an ensemble, because I know you're itching to ask me about that.)
So now you're asking about individual trials.
Let's go back to my marble analogy for a minute. To make it more rigorous for the case of individual trials, imagine that I take my box of marbles and put it in the bag, with the remaining marbles in the bag but not in the box. I then tell you that I'm pulling the marbles out of the bag and that .25 of the marbles in the bag are red. I pull marbles out of the bag, but if I didn't pull it out of the box, I ignore that case and move on. So, I'm really only asking you about the ones pulled out of the box that is inside the bag of marbles.
Now, you are confident- for each and every trial- that the probability of me pulling a red marble is .25, because you know the probability of pulling a red marble out of the bag is .25, and I'm pulling the marbles out of the bag. That is objectively true, also.
Now imagine that you're getting upset because the numbers aren't lining up. So then I tell you that I'm only actually only asking about the marbles from a box inside the bag, and that there's a nonrandom mixture of marbles in the box. Now- for every trial- you know what the probability I pull a red marble is. BUT when I ask you about the probability I pull a red marble, because you know then that it's in a new subset, and you have no answer. (Why that is will be explained in detail below.)
Now the complexity here is, am I pulling marbles from the bag or from the box, and is that relevant? I hope it's obvious that it's relevant, since the objective probabilities will be different.
Also, recall the box is in the bag, so I'm not *technically* lying to you when I tell you I'm pulling marbles from the bag where the probability is .25- although it does seem like cheating on my part if I only tell you this, and not about the box. Why is this? It's because of how probabilities work.
Specifically, it's because of the ASSUMPTIONS that go into calculating the probability of an individual event. You have to make an assumption when you apply that .25 to the individual event- namely, that is was a randomly selected invidual out of all the possible occurances. Here, the "occurrances" are me pulling each of the marbles in the bag out.
Consider what you had to do to calculate the .5 for the xth time I pulled a marble. Assuming I replace the marbles when I draw them out, you calculated the probability for THAT SPECIFIC EVENT by taking the number of red marbles in the bag divided by the number of total marbles in the bag You at least IMPLICITLY assumed a set (the bag) to calculate that. You're assuming it is possible for me to draw every marble in the bag when you put that in the denominator.
Now when I ask you about the marble, you know that it came from a box. Thus, the numerator and denominator you use to calculate the probability are no longer all the red or total marbles in the bag, but the ones in the box- which you know nothing about. This is why you have no opinion- you know you're talking about a new set, but you don't know it's properties.
This imagine what happens for trial 42. I pull a marble from the bag.
t1: You know the probability of me pulling it is .25.
Now I ask you about it.
t2: There are now two subcases:
1) I didn't tell you about the box. You may still think .25, and you are wrong objectively wrong. If you're smart, as I have shown, you will consider the fact that I may be cheating you and have no idea.
2) I did tell you about the box. You know this is part of a special subset where the probability isn't necessarily .25. You have no idea what it is.
Van Fraassen's argument centers around going from case 1 to case 2, which I have shown is a nonissue. You are now asking about going from t1 to t2, which is NOT the issue addressed in his paper, but I will speculate on it anyway.
So there are several different directions we can take with this. The first is to challenge whether Bayesian epistemology means you must always become less vague. By your own admission, vagueness is correlated to the range Hiliary thinks the probability of the detector can be in. So does Bayesian epistemlogy require Hiliary to always narrow the range she thinks the probability of the detector can be in? I don't think so. Note that under your definition of vagueness as the range, and under the claim that Bayesian epistemlogy requires vagueness to decrease, if Hiliary was thought it [.6,.10] because someone told her so, and then she was corrected and told it was between [.49,.51], this would be a contradiction. If this is a problem with Bayesian epistemology, then we should throw it out.
However, I would contend that it's not a problem Why? If she had a narrow range based on something, and then she received "disconfirming" evidence, then she would be justified in broadening her range again. Now it's clear in the immediately above case that the disconfirming evidence is whoever corrected her. But what is "disconfirming" evidence for probabilities? And is there anything in going from t1 to t2 that is disconfirming evidence? I would contend that disconfirming evidence for probabilities is anything that tells you that the relevant set to the issue at hand isn't the one you used to calculate the probability. So when she finds out about the box, that is disconfirming evidence for her probability- even though that may seem counterintuitive to us, it's clear from the mathematics. We're not used to working with probabilities and moving from one set to another, how something can be true with respect to one set and false with respect to another. However, you can point to what in the calculation is wrong now, and so I would claim it's perfectly acceptable for her probability range to become broader, and her vagueness to increase, without difficulty in Bayesian epistemology.
Posted by: Lauren at November 13, 2007 04:20 PMfo103.txt;3;6
Posted by: mCVUfCWYYOITiBbx at July 24, 2008 10:43 PM