Degrees of Belief, and Evidence of Ignorance vs. Ignorance of Evidence
Suppose I know that coin A is fair, and have no information whatsoever about whether coin B is fair. To what degree should I believe that coin A will land heads if tossed, and that coin B will land heads if tossed?
In both cases, my intuitions say “1/2″. But it is sometimes pointed out that there’s an important difference between the two cases that one’s degrees of belief should reflect. In the first case, one has evidence about whether coin A will land heads—one knows that the chance of the coin landing heads is 1/2. In the second case, however, one has no evidence whatsoever about whether coin B will land heads—one has no idea what the chance of the coin landing heads is.
I’m not sure, however, why the difference between the two cases should be reflected in one’s degrees of belief. Set aside degrees of belief for the moment, and let’s just focus on good (?) old all-or-nothing beliefs. In the first case, I suspend belief on whether the coin will land heads. In the second case, I suspend belief too. But suppose someone points out—correctly—that the two cases are different in the way described above. It doesn’t seem reasonable to conclude that this difference should somehow be reflected in your suspension of belief. But then, why might it be reasonable to make such a demand when it comes to degrees of belief?
May 21st, 2008 at 6:09 pm
Suppose that (to modify the case slightly) you’ve already had a series of coin tosses for both coin A and B, and every toss in the series came up heads. It seems to me that your credence that coin A will come up heads next time you flip should still be .5 no matter how many times it has come up heads already. It is far less clear that the same holds for coin B. The fact that coin B has come up only heads in the past gives you evidence for thinking that B is not a fair coin, but the fact that coin A has come up only heads in the past does not give you evidence for thinking that A is not a fair coin. You know that coin A is fair, but you don’t know that about coin B. So I agree with you that before having gone through the initial series of tosses, your credence with respect to whether A and B will land heads should be equal, but not after you have seen them both land only (or mostly) on heads. Also, for this to occur, there must have been an initial difference in your epistemic situation with regards to the two coins, but whether that difference needs to be reflected in your credences about the initial tosses themselves is a separate question. You may give some credence to coin B being unfair, but you’d give equal credence to it being biased towards heads as to it being biased towards tails. So either way, for the very first toss, you ought to give ‘B will land heads’ .5 credence.
May 23rd, 2008 at 6:19 pm
Hey Leon,
Thanks for the comment—I agree with everything you said! The modified case is interesting, because it suggests that although your initial credences about whether each coin will land heads if tossed need not reflect the initial difference in your epistemic situation, the difference may be reflected in the way you update your credences after repeated tosses. If you’re rational, you’ll still assign a credence of 0.5 to coin A landing heads on the next toss, but you’ll assign a greater credence to coin B landing heads on the next toss. (If I remember correctly, Wolfgang made a similar point in a reading group a few months ago…)
June 17th, 2008 at 9:34 am
This seems to suggest that we cannot adequately describe the constraints rationality sets on the epistemic stance we should take towards propositions given evidence using just the degrees of belief we should adopt. We also need a variable for how easily evidence should impact on our credence, how ‘robust’ the degree of belief should be to new evidence. (I’d like to know whether having a credence n held with robustness m in proposition p can be redescribed without loss as having a different credence simpliciter in some other proposition.)
Let’s say you have observed a series of a 15 tosses landing heads for both coins. Knowing as you do that A is fair, this does not change your credence in it landing heads next time; however your credence that B will land heads may have gone up by a bit. Now suppose someone who has been standing quietly in the corner comes over and tells you – to all appearances sincerely – that coin B is indeed unfair. Presumably your credence that B will land heads next time now increases further.
But now suppose that your coin tossing experiments occur on an island, on which you are shipwrecked. On your arrival you found a letter in handwriting you recognise as that of the world’s foremost anthropologist. Also shipwrecked, she had discovered that people there suffer from a religiously induced paranoia vis-à-vis coins, leading them to quickly conclude that perfectly fair coins are unfair (all of this leading to much wasted energy spent on melting down and carefully remaking perfectly ordinary coins). In this case, although the observation of 15 heads in a row might have led you to increase your credence that coin B will land heads next time, the testimony will not increase it further. So the robustness with which you should hold a certain credence in a proposition has to describe how different types of evidence should impact on your credence in various circumstances.
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