If you haven't the faintest clue whether some proposition p is true or false, what subjective probability (credence) should you give it? 1/2? A common answer these days is that you shouldn't give it a precise credence at all. Instead, your credence should be spread over an interval, such as [0,1]. Greater precision than that ought to be based on real knowledge, e.g. of objective probabilities. Mere ignorance doesn't qualify one to make such claims.
Roger White, in his talk 'Evidential Symmetry and Mushy Credence', offers a neat argument for the old-fashioned answer of 1/2,* which goes roughly as follows:
Coin Game: Suppose you're given a fair coin which has 'p' plastered on one side, and '~p' on the other. Moreover, you know that whichever one is true was plastered over the Heads side. You toss the coin and it happens to land on 'p'.
(1) It's a fair coin, so you should initially give P(heads) = 1/2.
(2) This should not changed upon seeing the coin land on 'p' -- you have no idea whether p is the true one or not, so there is no new evidence for you here. So your updated P+(heads) = initial P(heads) = 1/2.
(3) Since the coin landed on 'p', this will be heads-up iff p is true. Hence P+(p) = P+(heads) = 1/2.
(4) But the coin landing on 'p' doesn't tell you anything new about the proposition's truth value. So your prior credence P(p) should also have been 1/2.
Convinced?
* Correction: the argument merely shows that your credence in p shouldn't be imprecise (for that entails, contradictorily, that it should be precisely 1/2). Maybe it should be some other precise value, though; that will depend on the details of what proposition p is.
Friday, February 15, 2008
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I think 1/2 is a legitimate assumption - the problem arises in that you might be systematically tricked - or that you might be ignoring other information.
ReplyDeleteAlso I think almost any - possibly every, assessment of probability has a known and an unknown probability component.
I'm tired, so my brain isn't quite working right, but I think there's a stronger argument working in the other direction: consider two propositions p and q. If you have no idea whether they're true, and no idea whether consistent sentences of first-order logic constructed from them are true [like (p&q) and (p&~q)] then you can't assign all the relevant propositions a subjective probability of 1/2. I think. I think assigning p&q a probability of 1/2 would require assigning p&~q a probability of 0, but my brain isn't quite in the condition to write out a formal proof of that.
ReplyDeleteIt's my impression that 50% is the standard answer these days and that this reasoning is well known. It is taken for granted that any other answer is a logical fallacy in one of the Judgment Under Uncertainty books.
ReplyDeleteThe alternative, multi-term probability theory, doesn't gain you anything on Bayesian in terms of predictive power or scope, though it is probably more psychologically realistic and may gain you some accuracy in the context of other known sources of human irrationality.
Color me unconvinced.
ReplyDeleteThe labels p and ~p are merely masking the outcome. It would be the same as if you were to flip the coin but cover it before you saw the result. You'd expect a 1/2 chance of getting heads after the flip, but you can't tell because it's covered. Your updated chance is still 1/2.
There seems no legitimate reason to update when the initial move does nothing to move one closer to useful information.
Or maybe I'm talking out of my hat.
Jason - right, that's the argument.
ReplyDeleteHallq - good point. But perhaps we could reasonably give (p&q) credence of just 1/4, if our ignorance of the atomic sentences p,q is more fundamental? (I'd be curious to hear what others think of this problem.)
Michael - You may be right, but I think the 'mushy credence' solution is more recent. White attributed the view to Jim Joyce and Elliot Sober, for example, both of whom are big names in the contemporary field. (Certainly, the principle of indifference is no longer widely accepted, thanks to Bertrand's paradox, etc.)
Also, I don't think that mushy credence is in any way opposed to Bayesianism. The way White described it, we were simply to model a mushy believer as having a "committee" of homunculi in their head, each one with precise credences subject to Bayesian updating.
The problem with your solution is knowing which sentences should be fundamental when we're dealing with real statements. Consider "the present king of Finland is bald," from the point of view of someone who has no idea whether Finland has a king (yes, I know, there are proposals for how to write that one in symbolic logic, but I suspect it would only be the beginning of our troubles).
ReplyDeleteIt's tricky, I agree, but it doesn't seem that outrageous to require that a perfectly rational agent be able to work out which sentences are fundamental.
ReplyDeleteThen again, it's an interesting question to ask what is rational for limited agents (such as real-life humans) who aren't capable of this. How ought we to make judgments under this meta-uncertainty?
I gotta say I don't get this.
ReplyDeleteHow can you argue for how much credence to give to an truly unknown proposition by giving an example where the outcome is obviously a 50/50 chance even though the name of the state is covered by tape. You are assuming a dichotomy and therefore artificially setting up the "uncertainty" by strict limits.
Why not try a simpler example without all the fancy notation?
"Yes or No: is 1781 the last four digits of President Bush's phone number"
You could calculate that the odds of a random number being the correct 4 digit number would be 10 to the 4th, but that assumes the number is random. Let's posit a situation where I may know the correct answer, therefore the odds are <10 to the 4th. You have no idea of the odds. Are you still going to claim that the answer to the "yes no question" is 50/50?
Now, admittedly I may completely mis understand the intricacies of the question. On the other hand, you may have over thought it.
Gerard - I'm not sure, but that still sounds like a situation where I have more reason to think the numbers aren't correct. (Maybe. It's hard to know what to say there. But that's the advantage of the argument I discuss in the post: it forces us to one clear conclusion.)
ReplyDeleteI'm not sure I follow your objection to the earlier argument. Which of the four premises do you think is false? Or do you think the conclusion does not follow from the premises? (It looks pretty water-tight to me, but I may be missing something.)
Hi, I think my objection is that there is already a defined probability in the example given, and thus is not at all analogous to the question it proposes to explore, which is "If you haven't the faintest clue whether some proposition p is true or false."[emphasis added] In your example, you do, in fact, have more than the faintest clue. You know the odds are 50/50.
ReplyDeleteYour conclusions do not follow your premiss. You've only proven that for situations where there are known 50/50 odds, the odds remain 50/50
Since the identity of the outcome is masked, multiple permutations do not add more certainty--nor do they subtract certainty--thus they are irrelevant.
So, in the case of unknown probability, or as you say, where you "haven't the faintest clue whether some proposition p is true or false" you cannot ascribe probability of 50/50 since the probability is unknown. Just because the answer is bimodal does not mean that the probability can be rationally assumed to be 50/50. Unknown certainty == unknown odds, not 50/50 odds.
Hold on, the example stipulates an objective probability of 50/50 for the coin flip (landing heads), but it doesn't stipulate anything about p. No objective probability for p is discussed at all, and that our subjective probability (credence) for p ought to be 50% is the conclusion of the argument, not anything assumed in the set-up.
ReplyDeleteNote that the conclusion is that one ought to have 50% credence in p even before the coin flip. So, even if the coin flip never takes place, one's credence should still be 50%. It doesn't depend on being in the peculiar situation with the coin. The coin game simply serves to demonstrate why this is so: if you started with any other credence -- including no credence at all -- then you would end up with the wrong result in this case. Unless, perhaps, you deny premise (4) and think that the flipping of the coin may change your credence in p from undefined to 0.5?
I think we are talking cross-purposes.
ReplyDeleteYour premise was "If you haven't the faintest clue whether some proposition p is true or false, what subjective probability (credence) should you give it? 1/2? "
Which you then follow up with an example with a stipulative probability of 50/50, artificially giving credence with your proposition. You can't stipulate a specific situation that contradicts the premiss, which is unknown probability. In effect, you've merely made an argument by assertion by stipulating a known prior probability in your analogy without regard for your original premiss.
Why not propose a situation where the stipulative prior probability is unknown, as I tried to do in my earlier post. When you start from such a position then you don't wind up with a conclusion of "50/50" but unknown, where you can't assign a fixed level of credence based on any prior probability because there is none.
The probability of unknown is unknown, regardless of whether the answer is expressed in a bimodal fashion.
Yes, we're talking at cross-purposes. One last attempt at clarification, then I think I'll have to call it a day...
ReplyDeleteYou need to distinguish between objective and subjective probabilities, and also between the propositions 'p' and 'heads'. Your objections all seem to stem from a failure to carefully distinguish these. (Whenever you speak of a 'probability', I don't know which of the four possible combinations you are talking about.)
In particular, note that there is no contradiction in the following set of claims, all of which are true in the scenario I discuss:
(A) The objective probability of p is unknown. Also, you have no reason to think either p or not-p more likely true. (This is what I mean by the 'haven't the faintest clue' line.)
(B) The objective probability of heads is 1/2
(C) One's subjective credence in heads (both before and after the flip) should be 1/2
(D) One's subjective credence in p (both before and after the flip) should be 1/2.
In particular, note that the stipulation in (B) is compatible with (A). Writing a sentence on a coin doesn't affect the objective probability that the sentence is true (nor does it give you any new evidence for or against its truth). So I haven't a clue what you're talking about when you accuse me [or Roger White] of "stipulating a known prior probability in your analogy without regard for your original premiss."
"In particular, note that there is no contradiction in the following set of claims, all of which are true in the scenario I discuss:"
ReplyDeleteYes, I would agree. I think the question you asked and the one you answered in you OP are not the same. The imprecision you accuse me of has its orgins in your original premise: "If you haven't the faintest clue whether some proposition p is true or false, what subjective probability (credence) should you give it? 1/2?"
Perhaps if you re-phrased the question to match the answer you have carefully given your conclusions would follow your premises?
I will stipulate a certain amount of ignorance on the subject and concede that I may need to review my own arguments but I do think the distinctions you make to me are post hoc even though you may have meant them to be part of your original premise.