Re #38 (and #34), what seems to me to be going on (and I’d welcome correction if I’m wrong, ‘cause it’s been bugging me) is that applied to E3, Gott’s method throws away any info on ‘x’, and implies joint distribution assumptions (i.e., dependence or perfect correlation) re A and B that aren’t absurd as such but which are facially inconsistent with the supposed lack of information on the processes.

As for trying to make hay from the direction of the method’s probabilities for E1 and E2, to decide on how to make use of Pr(E1 at z) and Pr(E2 at z), it’s necessary to make some assumption on the joint distribution, and a “reasonable” assumption probably won’t lead to the Gott’s Pr(E3 at z).

It should be possible to work backward from Gott’s conclusions to see what assumption he makes about the distribution of N. I haven’t done that.

Choose that has a distinct beginning and a distinct end. Note the beginning. Then at some random time between the beginning and end, note that it has not ended yet. Since we assume the time was random, we can suppose that it’s a sample from a uniform distribution. There’s 1/2 chance it is in the first half and 1/2 chance it’s in the back half, so that’s a 50% chance the end will come in less time than has already passed, and a 50% chance it will last longer than it’s already lasted. By the same reasoning there’s only a 5% chance that it will end in less than 1/19 the time that’s already passed, and a 5% chance its total life will be 20 times or more what it’s already survived.

This bothers me somehow. If it’s right I should be able to get the same result with a different method. So—If the interval is some unknown N, and I assume nothing about N except that it’s larger than my observation A, what is the maximum entropy distribution? N is in the interval (A, infinity) and I’ve assumed nothing else about it, so I must suppose that every value in that interval is equally likely. So any value I choose for the mean is too small. I get a different result.

Let me try again. In a finite universe there can be a lot more material things that last a short time than that last a long time. The longer something lasts the longer it takes up mass and space that something else could use. There’s only room for so many long-lived stars. There’s room for a lot of short-lived tritium atoms. So maybe I shouldn’t assume that every lifetime is just as likely.

If we make the assumption that there is some unknown mean duration M, and nothing else, then the maximum entropy distribution is an exponential. And if we remove all the cases where x

I don’t know why I am taking it upon myself to defend this principle. I am in no way committed to it, but neither seems to me to be so crazy. We don’t derive anything more than we put in, that is the assumption of the a priori distribution (I don’t know why some call it uniform distribution; the assumption is that the a priori probability that the total duration is T is proportional to 1/T ). In most cases the assunption is inappropriate, because most events we want to consider are very complex and arise from thousands of different factors, so even if the assumption holds for each of the factors, it will be very far off for the total event. One would be better off using a distribution that is more stable, like a Gaussian; but then one needs to put in more knowledge, namely the variance of the distribution. If one is unwilling to put in any information whatsoever, then the 1/T law is the simplest guess, and it may even be the unique possibility. It may not be very useful, I totally agree, but its application shouldn’t lead to contradictions.

we always know that two events collectively entail, and individually are entailed by, their conjunction.

What does this mean? In your example, the relationship between E1 and E3 is by no means symmetrical. E3 implies E1, the other way around is not true. So it is not a contradiction that you could use the formula for E1 but not for E3.

Gott’s method is to make a frequentist statement about tfuture based on the long run properties of a random sample.

What was the first thing any of us were taught about frequentist statistics? You determine what hypothesis you’re going to test, then you look at your data. The example supposes we’ve looked at the data, seen that both plays are still running, then decided to do the calculations for E1, E2 and E3.

This means the results for E1 & E2 are wrong. We wouldn’t be worrying about this example had Play B not yet started, or the play’s not overlapped, or had Play A ended. So the probability statements about E1 & E2 can’t be justified based on the long run properties of random samples that Gott uses to contruct his formula. Our running these tests have been influenced by our looking at the data.

I think E3 is the only circumstance in your example where it’s sound to apply the principle. You can’t apply it to E1 or E2.

I don’t remember ever learning about this principle, but what it sounds like is simply this: if something we don’t know much about has been going on for a while, chances are it’s going to go on for a while longer.

And if that’s what it’s saying, then perhaps the phrase I quoted needs to be modified to say “the probability … is at least 1/2”. Which means that the guy who suggested this principle likes all the same odds you do.

In any case, I certainly wasn’t assuming that the events were independent, or that either distribution was uniform. I was just applying the principle given; a principle I think is utterly crazy. The principle says that in the absence of information to the contrary, we can assume we are dealing with uniform distributions here.

Now either knowing the logical relations between E1, E2 and E3 counts as information to the contrary, or it doesn’t. If it does, then we can never apply the principle, because we always know that two events collectively entail, and individually are entailed by, their conjunction. So the principle is vacuous. If it doesn’t, then the principle leads to an absurd result. So we get the conclusion that it seems to me we always get with these principles saying we can derive something from ignorance, either vacuity or absurdity.

Your mockeries are similarly bad. WWW is a communication protocol. If it started when 2 users used it, and it will end when the last 2 users quit, that isn’t nearly uniform. When there are 4 million users, the chance they’ll all quit in a short period is much less than when there are 4 users. Similarly with australians and iPods.

But I agree with your criticism of the method. When you don’t have much information to work with, you get predictions that aren’t very useful. The main thing they might possibly be good for is as a baseline for additional work. “This is our Bayesian prior. Now when we add this information, how does that change it?”

Very often the assumption of uniformity is a bad assumption. If you assume there’s a single interval and you’re trying to find the length of that interval in the lack of other informtion then it makes sense. But if you’re sampling from a distribution of intervals that’s exponentially distributed, then you’re more likely to sample from close to the beginning.

Say you have 1000 japanese cars and each of them have a lifetime of precisely 150,000 miles, but their odometers are broken. You have reason to think that their mileage is all the same within 3000 miles, and they’re uniformly distributed within that interval. Then you run them all the same distance. How long will they last? You can get a pretty good estimate when the first one quits. You can expect the last one to quit within 3000 miles or so. Contrived? Yes. To make it useful you need a sample size larger than 1. But you need a fixed interval or the distribution of interval sizes will matter.

When you argue from ignorance you have to be careful about your assumptions about just what it is you’re ignorant of.

So we’ve no business using the formula to calculate Pr(E1 at z). The formula does not apply to E1.

I also thought at first that the time of observation could not be random for both A and B, but this is not so as long as the time was not chosen to be in the lifetime of A and B, it just happened to be so. Otherwise, whenever I observed an event E, someone else could argue that the time was not random, simply by observing another event that started after E. Imagine that the observations of E1 and E2 were done by two different persons, each one not knowing anything about the other play. Then the first person, observing E1, would apply the Copernican formula; whether he can apply it or not cannot depend on whether some other person is observing another event.

Besides, even if you leave out E1, you are still left with the problem that Pr(E2 at z)=Pr(E3 at z).

Similarly, there are cases when you can be confident that the partition to which you are applying the principle of insufficient reason is symmetric and hence the principle is valid. The problem is to specify the criterion.

The problem more fundamental. If we observe Plays A and B are both running – and we know Play A started before Play B – then we know our observation isn’t a random sample from within the life span of Play A. Because we can’t have observed the duration of time when Play A was running but Play B wasn’t. So we’ve no business using the formula to calculate Pr(E1 at z). The formula does not apply to E1.

“The principle is meant to give us probabilities of the duration of things of unknown length. If we only apply it when we know how long the event is, well then it will give plausible results, but won’t tell us anything we don’t already know.”

I don’t want to sound like I’m being too picky about something you took three seconds to type into a comment box, but I think that phrasing misses an important distinction. Your claim is different to Gott’s. There’s a difference between: (A) knowing how long an event is, and (B) knowing that you’ve a random sample from within the lifespan of the event.

B’s is significantly different to A. You can know you’ve chosen a random raffle ticket without knowing how many tickets are in the raffle. In this different context you can use Gott’s maths to infer the number of tickets. All we need to know is that we’ve randomly sampled them.

Obviously, that kicks the debate back to arguing for or against the validity of the random sampling assumption. I think John’s wrong that Gott’s suggestion is just the principle of insufficient reason. I think Gott would argue that in some cases you can have positive reason to believe that your observation is a random sample – from knowledge of the nature of what your observing and the circumstance of the observation – and this justifies applying the formula.

Likewise, the Copernican principle applies to measurements of the event, not to the fact of its existence. To assume that the existence of everything is a stochastic process that flips from true to false with some fixed probability is, as you demonstrate ridiculous. It also throws out a lot of information about the event in question.