the sleeping Brad DeLong problem

Brad DeLong was sound asleep for several months, but after he woke up he found an old blog post and quickly calculated the probability that Lubos is an April fool; he wrote a blog post about it, because he had no memory of all the earlier debates about this.
Of course, Lubos disagreed with this calculation, the probabilities are very different according to his argument. So who is right?

I wrote about the Sleeping B. problem almost ten years ago, when it was mostly discussed in academic papers. Meanwhile it has become an internet standard and turned into a political debate: Lubos and the reactionaries vs. Sean and the liberals. Unfortunately, all the nuances of the problem have been lost, as it happens in tribal conflicts, such as the existence of solutions which are neither 1/2 nor 1/3.

In one of the many comments somebody wrote something like "if I would be really interested in this problem, I would code a simulation and see what it does". Unfortunately, if she would actually sit down to implement a simulation, she would quickly find that it does not really answer anything; this is one of the problems which cannot be settled with an experiment or simulation, because it is about the question what we actually mean with "probability". Both physicists and economists are not well trained at thinking through what it is exactly they are talking about, therefore I predict that this debate will not go away anytime soon.

So what is really the issue with this? Well, if the "a priori probability" is 1/2 then S.B. has to bet "as if" it was 1/3 due to the setup of the problem. So if you think probabilities are defined as betting odds (as many Bayesians do) then you will prefer 1/3. If you think probabilities are objective properties of physical systems then you will probably (!) prefer 1/2. (Btw, notice that S.B. has to (implicitly) use the 1/2 "a priori" to calculate the 1/3.)

I actually prefer the 3/8 solution, because it is not so clear how one should understand it. But I realized that I am a tiny minority of one a long time ago ...


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Perhaps it is useful to repeat the 3/8 solution here:
If the outcome was Head, she will only wake up on Tuesday; but if it was Tail, she will either wake up Tuesday or Wednesday with "a priori probability" 1/2 for each, since she cannot distinguish the two days.
Therefore, the probability for the just awoken S.B. that today is Tuesday is
p(Tue) = (1/2) + (1/2)(1/2) = 3/4
where the 1st term corresponds to Head and the 2nd to Tail.

Now we can calculate the probability for Head as p(Head) = (1/2)*p(Tue) + 0*p(Wed) = (1/2)*(3/4) = 3/8,
knowing that the "a priori probability" for Head on Tuesday is 1/2.

the chaos computer club

A recent paper suggests a fundamental limit on the chaos in physical systems.
Lubos wrote an easy to read introduction to the main idea.

I think this might be interesting for Scott and everybody else interested in (quantum)computing. If one considers a Turing machine sensitive to initial conditions (i.e. the input string) or if one considers a (quantum)computer simulating chaotic systems, the conjecture seems to imply a limit on computability.

Or think about a device which measures the position x of the butterfly wings, sends the result to a computer, which calculates a function f(x) to determine its output. The conjecture seems to suggest a limit on the functions the computer can calculate in a finite amount of time.

Is it correct to read the result as "the number N of different internal states any computer can reach after a time T is bounded by e^wT where w is a fundamental constant"?

life expectancy

Recently I stumbled upon this math problem:
The positive integer N has a finite value but is unknown to us.
We are looking for a function f(X) which minimizes the sum
E = |f(0) - N| + |f(1) - N| + |f(2) - N| + ... + |f(N-1) - N|
for almost all N.

Notice that we do not know the value of N, so f(X) cannot depend on it, which eliminates the trivial solution f(X) = N among others.
In order to illustrate what I am looking for, consider as first example
1] f(X) = 0, which results in E = N².
However, there is a better solution
2] f(X) = X, which results in E = (N+1)*N/2, which is less for almost all N (except N=1).

Unfortunately, I do not know the best solution f(X) and this is where you are invited to leave a comment to help me out.
But I do have strong evidence (i.e. a numerical test up to large values of N) that
3] f(X) = X + sqrt(X) is an even better solution.

So what is the motivation for this problem and why the title for this blog post?
Consider a process or phenomenon which exists already for X years and we try to estimate the total lifetime N without any further information. So our estimate can only depend on X and we try to find a function which minimizes the total estimation error E as described above; every year we make an estimate f(X) which is wrong by the amount |f(X) - N| and we try to minimize the sum of those errors. In some sense this is a variation of the infamous 'doomsday argument'.

It is now obvious why 1] is a bad solution and 2] is much better. Btw the function f(X) = 2*X would give the same total error E minus a small constant, so whether we assume that the process ends immediately or estimate that it will last twice as long (as it already did) makes no significant difference.

Btw the solution 3] creates a paradox: The best estimate for the life expectancy seems to depend on what units one uses: We get a different result if we calculate with days rather than years.


added later: If I assume f(X) = c*X, perhaps motivated by that paradox, then I can show that c=sqrt(2) minimizes E for large enough N. However, the function
4] f(X) = sqrt(2)*X + sqrt(X) seems to be an even better candidate, but I do not know how to determine a,b to minimize E for f(X) = a*X + b*sqrt(X) or determine the general solution for f(X).


added later: There are better ways to illustrate the problem; e.g. a box contains an unknown number N of candies. You take out one after another and at each step X you have to guess N. At the end there is a penalty proportional to the sum of your wrong guesses.
Perhaps a "deeper" example considers a Turing machine, which performed already X steps and we try to guess after how many steps N it halts.
The reason these examples are "better" is that there is no change of physical units (e.g. from years to days) that would affect N.

no black holes?

Laura Mersini-Houghton and Harald Pfeiffer published a paper with numerical results suggesting that black holes may not really exist (see also this earlier result). As one would expect, several pop. sci. webpages have already picked this story up.

The paper is of course not a general proof, but describes a particular model using certain assumptions; it considers the spherically symmetric collapse of pressure-less dust and it makes simplifying assumptions about the Hawking radiation: The energy tensor for the Hawking radiation is taken from earlier calculations for (static) black holes, proportional to 1/R^2, and I don't think this is justified if one wants to prove that black holes do not exist. Further it is assumed that most of the radiation is generated by the collapsing body itself (*) and finally assumptions are made about the heat transfer function C which I cannot follow (yet).

The resulting differential equations are numerically integrated until a shell-crossing singularity appears, in other words a naked singularity (presumably an artifact of the model assumptions, i.e. perfect spherical symmetry, so it is only slightly embarrassing in a paper which wants to remove black hole singularities).
The behavior of the dust suggests a rebound near the horizon, but it is too bad the full evolution is unknown, because it raises interesting questions.
What happens to the pressure-less dust in the long run? Will it collapse again after the rebound, perhaps infinitely often?
What does the final state (including Hawking radiation and the "influx" of negative energy) actually look like?

I am sure this paper will generate several responses and eventually more realistic calculations will follow.
Until then I remain skeptical that this result will actually hold in general.


(*) I admit that I do not understand this passage in the earlier paper: "Hawking radiation is produced by the changing gravitational field of the collapsing star, i.e. prior to the black hole formation [..]. Otherwise the surface gravity of the black hole κ, and the temperature of Hawking radiation would increase with time..."
I thought the standard picture is that the "influx" is at the event horizon (not the collapsing body) and the temperature does indeed increase with time...


added later: Supposedly William Unruh was more direct and he thinks that the paper is nonsense.