on the certainty of sunrise

I have been reading Jeff Loveland’s paper on Buffon and the certainty of sunrise. (Jeff and I knew each other casually in 1983, when we both matriculated in the Michigan State honor’s college. I was a casualty of the local social scene, lasting only two years. However, we were together on the MSU Putnam team–with Frank Sottile–as sophomores, which is how I came to know him slightly better than I would have in mere virtue of our having lived on the same floor of a dorm.) I’ve been railing in various philosophy papers against the employment of finite-almost-surely random variables of unbounded expectation in quantities linear with respect to decision theoretic utility, which makes Jeff’s paper especially interesting to me.

Buffon’s inductive prescription was a bit on the strange side. If the sun has risen N times, and never failed to rise, then the probability that it will rise tomorrow is something like 1-(1/2)^(-N). Buffon’s reasoning is obscure on this point. Jeff had a hypothesis as to how he might have come by his conclusion independently (a contemporary scholar, Price, came to essentially the same conclusion) which is basically this: assume the sequence of rise opportunities is Bernoulli (i.e. independent identically distributed and two-valued) with p (the probability of a rise on a given morning) taking on one of the values 0, 1/2 or 1. We are, a priori, indifferent to the values. Buffon’s solution now falls out by standard conditionalization on having seen N consecutive rises. (Neither Price nor Loveland sees it this way, but one could.)

Loveland mentions a somewhat less arbitrary succession principle, due apparently to LaPlace. Namely, we again take the sequence to be Bernoulli, but assume a priori that the parameter p is uniformly distributed on the interval (0,1). In such a case, conditionalization on N consecutive rises yields a probability of (N+1)/(N+2) of seeing a rise on the next day. My interest is in this version of the succession principle. (Karl Popper on an unrelated note wrote a 1962 paper, published in Mind, critical of Carnap’s version of the Laplace rule of succession. Popper writes as if Carnap should condition on “at least N of these N+1 mornings witnessed a sunrise” in order to update credence in “sunrise on the (N+1)st day”…rather than the obvious “the first N of these N+1 mornings witnessed a sunrise”. That is absurd but not the last time a philosopher has made a splash being absurd in precisely this way–think Monty Hall, or, more recently, see Roger White’s A Challenge for Thirders, which defends the ill-fated Double Halfer solution to the Sleeping Beauty problem. Popper goes on to defend himself by stating that it’s owing to the fact that “order and positional properties are not expressible in Carnap’s system”, but this seems unlikely–read for example Carnap’s Statistical and Inductive Probability, in which it is clear that the underlying probability space contains atoms corresponding to each of the 2^{N+1} sunrise/no sunrise possibilities–they are in fact listed for the case N=3. It’s impossible to say for sure, but I imagine what happened here is that Popper submitted a paper to Mind with the standard protocol gaffe, a referee caught it, Popper rather than admit his error tried to point to limitations of Carnap’s language system and the then-editors of Mind didn’t have the courage to just say to him, Karl we’re very sorry, but Your Philosophy Paper Sucks.) Specifically, it interests me that it yields the unfortunate result that the number X of ensuing days of sunrise becomes an almost surely finite random variable having infinite expectation, and hence one that cannot be commensurable with utility. Whether this is fatal to the rule depends on some other factors, though certainly application of the rule will require some attenuating at least.

Here is the idea. If Laplace’s rule is adopted then the probability of at least J ensuing days of sunrise is asymptotically commensurate with 1/J. And since the probability of sunrise on the (J+1)st day, conditional on this event, is on the order of 1/J as well, the epistemic probability density function of X, the number of ensuing day of sunrise preceding the first instance of no sunrise, is something like f(x)=c/x^2, meaning that E(X)=sum of x(c/x^2) = infinity. It follows that X cannot be linear with respect to utility in any rational decision theory.

For consider the following thought experiment.

God comes to LaPlace and says: “Man can live on one of two planets…Earth or Twin Earth. The sun has risen 1,000,000,000 consecutive times, without exception, on each planet. The planets seem identical in every visible respect, though they have different suns and independent fates (the number of ensuing sunrises on Earth does not provide evidence as to the number of Twin Earth). Man currently occupies Earth, but, somewhat mysteriously, everyone on Twin Earth has just vanished. If you like I could, with a snap of my fingers, move everyone to Twin Earth. If you agree to this, in fact, I will double the number of days the sun will rise on Twin Earth.”

LaPlace has good reason to accept this trade. God (who is apparently Dutch) ensures him that, quite apart from the bit about doubling the number of sunrises, man has every cause to be indifferent between Earth and Twin Earth as to quality of life and prospects for future sunrises. (I am assuming here that once the sun fails to rise, that’s it…the planet dies.) Doubling the number of sunrises on Twin Earth sweetens the pot…clearly a choice superior to Twin Earth sans doubling and, by indifference, to Earth as well. But, having accepted the trade, God reveals the number of sunrises Twin Earth will now enjoy. Double what it would have been, but still finite, and, being finite, somewhat disappointing! After all, the expected number of sunrises on Earth is still infinite! LaPlace can’t help but think he’s made a mistake, and indeed God has some mercy, and offers to let LaPlace transport everyone back to Earth. At a price! Namely, the number of sunrises there will be cut in half. That’s pretty steep, but even with this handicap Earth is still wildly outperforming Twin Earth in expectation. So LaPlace takes the deal. Mankind is back on Earth, where they started…but they have only half the number of sunrises to enjoy than before LaPlace began his ill-fated sequence of trades!

LaPlace has plainly acted in an irrational manner. But where? Well, his rule of succession may be no good. The a priori assumption…parameter p is uniformly distributed on [0,1]…may need to be changed. One could, for example, replace the uniform distribution with a “tent shaped” density function such as f(x) = 4x when 0<x<1\2 and f(x)=4-4x when 1/2<x<1. That has the effect of making the expected number of sunrises finite, but it may seem no less ad hoc than Buffon’s solution. (Why not a tent-shaped function with non-linear sides?) Or perhaps one can prevent utility from being commensurate with number of sunrises, or stipulating that no action LaPlace can take will have an unbounded effect on utility.

The best option, perhaps, is to simply adopt a finte-expectation distribution for the age of the universe, and stipulate that all local mortality distributions are attenuated by age-of-universe considerations. (If the universe dies, everything dies.) But that, I take it, is more or less what the “certainty of sunrise” problem was about in the first place. Not that, in Buffon’s time, people thought that earth and sun were everything. But then Buffon’s thought experiment didn’t involve a contemporary of Buffon contemplating the certainty of sunrise. On the contrary, it involved a man conjured out of nothing and left to consider naught but the sun and earth. For such a man, it could be argued that the mortality of the sun’s habit just does correspond to the mortality of the universe. Or if not, it might at least be treatable in like wise.

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