You should expect that, on average, a test will leave your beliefs unchanged.
Emphasis mine.
When I shake the box, my belief that the coin landed heads is 50%. When I look inside, my belief changes, yes, but two one of two options of equal probability: 0% (I see it came out tails), or 100% (I see it came out heads.)
It is trivial to see that my expected posterior belief is 0% 1⁄2 + 100% 1⁄2 = 50%, or in other words, it’s exactly equal to my prior belief.
The question is whether ‘change’ signifies only a magnitude or also a direction. The average magnitude of the change in belief when doing an experiment is larger than zero. But the average of change as vector quantity, indicating the difference between belief after and before the test, is zero.
If you drive your car to work and back, then the average velocity of your trip is 0, but the average speed is positive.
You should expect that, on average, a test will leave your beliefs unchanged.
Emphasis mine.
The statement is still wrong: Opening the box always changes your beliefs, therefore, it also changes your beliefs on average.
The correct version of this statement is “your belief over the beliefs that you will have after performing a test must be equivalent to your current belief”, which seems to be a trivial claim.
The correct version of this statement is “your belief over the beliefs that you will have after performing a test must be equivalent to your current belief”, which seems to be a trivial claim.
It may seem trivial but then again so does the claim that P(A and B) ⇐ P(A), and still...
In particular, I’ve sometimes caught myself simultaneously having aliefs like ‘if she flees, then she must be a witch’, ‘if she stays, then she must be a witch’, and ‘she may or may not be a witch, and I can’t know until I see whether she flees or stays’, and until I read the post about conservation of expected evidence I never realized there was something wrong with that.
The statement, “You should expect that, on average, a test will leave your beliefs unchanged,” means that you cannot expect an unbiased test to change you beliefs in a particular direction, as is clear from the original post.
Of course you expect to hold different beliefs after the test. If you didn’t, the test would not be worth doing. But you are not more likely to end up at (100% heads, 0% tails) than (0% heads, 100% tails).
On the other hand, if you think it is more likely that you will end up at, say, (0% heads, 100% tails), then you cannot rightly claim that you currently believe the coin to be fair (your 50%, 50% estimate does not reflect your true expectations).
That said, it’s far from the most easily accessible formulation of that meaning imaginable.
I mean, sure, the future state in which half of my measure has ~1 confidence in “heads” and half my measure has ~0 confidence in “heads” is in some sense not a change from my current state where I have .5 confidence in “heads”, but that’s not the interpretation most people will adopt of “leave your beliefs unchanged.”
It seems more accessible to say that if I expect a test to update my beliefs in a particular direction, I should go ahead and update my beliefs in that direction now (and perform the test as confirmation).
Of course, this advice presumes that I won’t anchor on my new belief. Which, given that I’m human, is not a safe assumption.
I would suggest that you expect your beliefs to be changed in 100% of cases. Currently, you believe in a 50% probability. After doing the tests, we have a set of universes, some of in which you believe a 100% probability and some of in which you believe a 0% probability. Your belief changed in every single one.
X and Y can be averaged out, but belief in number X and belief in number Y don’t average out to be “belief in the average of X and Y”.
The statement, “You should expect that, on average, a test will leave your beliefs unchanged,” means that you cannot expect an unbiased test to change you beliefs in a particular direction, as is clear from the original post.
Actually you can: Shake a box with a coin you know to be biased. Before you look into the box, your belief for heads is, say, 80%. You expect that is more likely that, when you open the box, your belief will change to 100% heads rather than 0%.
I don’t think there is an useful way to patch the statement without making explicit reference to the technical definition of Bayesian belief.
I agree that the statement is not crystal clear. It makes it possible to confuse the (change in the average) with the (average of the change).
Mathematically speaking, we represent our beliefs as a probability distribution on the possible outcomes, and change it upon seeing the result of a test (possibly for every outcome). The statement is that “if we average the possible posterior probability distributions weighted by how likely they are, we will end up with our original probability distribution.”
If that were not the case, it would imply that we were failing to make use of all of the prior information we have in our original distribution.
A misunderstood reading of the statement is that “the average of the absolute change in the probability distribution on measurement is zero.” This is not the case, as you rightly point out. It would imply that we expect the test to yield no information.
Emphasis mine.
When I shake the box, my belief that the coin landed heads is 50%. When I look inside, my belief changes, yes, but two one of two options of equal probability: 0% (I see it came out tails), or 100% (I see it came out heads.)
It is trivial to see that my expected posterior belief is 0% 1⁄2 + 100% 1⁄2 = 50%, or in other words, it’s exactly equal to my prior belief.
The question is whether ‘change’ signifies only a magnitude or also a direction. The average magnitude of the change in belief when doing an experiment is larger than zero. But the average of change as vector quantity, indicating the difference between belief after and before the test, is zero.
If you drive your car to work and back, then the average velocity of your trip is 0, but the average speed is positive.
The statement is still wrong:
Opening the box always changes your beliefs, therefore, it also changes your beliefs on average.
The correct version of this statement is “your belief over the beliefs that you will have after performing a test must be equivalent to your current belief”, which seems to be a trivial claim.
It may seem trivial but then again so does the claim that P(A and B) ⇐ P(A), and still...
In particular, I’ve sometimes caught myself simultaneously having aliefs like ‘if she flees, then she must be a witch’, ‘if she stays, then she must be a witch’, and ‘she may or may not be a witch, and I can’t know until I see whether she flees or stays’, and until I read the post about conservation of expected evidence I never realized there was something wrong with that.
The statement, “You should expect that, on average, a test will leave your beliefs unchanged,” means that you cannot expect an unbiased test to change you beliefs in a particular direction, as is clear from the original post.
Of course you expect to hold different beliefs after the test. If you didn’t, the test would not be worth doing. But you are not more likely to end up at (100% heads, 0% tails) than (0% heads, 100% tails).
On the other hand, if you think it is more likely that you will end up at, say, (0% heads, 100% tails), then you cannot rightly claim that you currently believe the coin to be fair (your 50%, 50% estimate does not reflect your true expectations).
That said, it’s far from the most easily accessible formulation of that meaning imaginable.
I mean, sure, the future state in which half of my measure has ~1 confidence in “heads” and half my measure has ~0 confidence in “heads” is in some sense not a change from my current state where I have .5 confidence in “heads”, but that’s not the interpretation most people will adopt of “leave your beliefs unchanged.”
It seems more accessible to say that if I expect a test to update my beliefs in a particular direction, I should go ahead and update my beliefs in that direction now (and perform the test as confirmation).
Of course, this advice presumes that I won’t anchor on my new belief. Which, given that I’m human, is not a safe assumption.
I would suggest that you expect your beliefs to be changed in 100% of cases. Currently, you believe in a 50% probability. After doing the tests, we have a set of universes, some of in which you believe a 100% probability and some of in which you believe a 0% probability. Your belief changed in every single one.
X and Y can be averaged out, but belief in number X and belief in number Y don’t average out to be “belief in the average of X and Y”.
Actually you can:
Shake a box with a coin you know to be biased. Before you look into the box, your belief for heads is, say, 80%. You expect that is more likely that, when you open the box, your belief will change to 100% heads rather than 0%.
I don’t think there is an useful way to patch the statement without making explicit reference to the technical definition of Bayesian belief.
I agree that the statement is not crystal clear. It makes it possible to confuse the (change in the average) with the (average of the change).
Mathematically speaking, we represent our beliefs as a probability distribution on the possible outcomes, and change it upon seeing the result of a test (possibly for every outcome). The statement is that “if we average the possible posterior probability distributions weighted by how likely they are, we will end up with our original probability distribution.”
If that were not the case, it would imply that we were failing to make use of all of the prior information we have in our original distribution.
A misunderstood reading of the statement is that “the average of the absolute change in the probability distribution on measurement is zero.” This is not the case, as you rightly point out. It would imply that we expect the test to yield no information.