Friday, September 14, 2018

the probability of the unknown

John. C. Wright (incidentally one of the finest science fiction and fantasy writers of our time) discusses a problem called the Carter Catastrophe. Mr. Wright argues that the problem statement is nonsense. I argue in opposition that the reasoning of the problem is correct as far as it goes, but that it offers almost no information. Mr. Wright comes to his conclusion from the position that probability is about repeatable events, and if he were right about that, his conclusion would be correct, but although he has a lot of brilliant scientists and mathematicians who agree with him, I do not.

I am one of a minority who believe that probability is, in fact, a form of logic. Traditional logic is based on two possibilities called truth values: True and False. Although this is by far the most common kind of logic, there are others. Some logics use three truth values: True, False, and Unknown. Others use four: True, False, Unknown, and Contradictory. Probability is just another form of logic that has a range of values from True (probability 1) to False (probability 0) and all shades of unknown in between.

Mr. Wright actually wrote a great science fiction book involving a non-two-valued logic. The book is The Null-A Continuum, and "Null-A" stands for non-Aristotlean logic. Aristotle's logic was the usual True/False variety, and the name was no doubt intended for similarity to non-Euclidean geometry.

So, given the view of probability as a form of logic, what is the probability of a statement x that you have absolutely no information about? Clearly you have no reason to prefer one value over another, so the probability is 1/2, which means completely unknown. Now let me give you a little more information about x:
x is the statement: when I throw this 6-sided die, I will roll a 1
Now you have the additional information that there are six mutually exclusive possibilities. Which one should you favor? Well, you have no  reason to prefer one possibility over the other, so the probability is 1/6. If you are a frequentist like Mr. Wright, you can't actually give a number until someone tells you that the die is a fair one, but as a matter of logic, if all you know is that one of six possibilities is true, and you have no other information at all, then the probability of any of the possibilities is 1/6.

Notice, however, that once we have the additional information, we changed the probability from 1/2 to 1/6. The 1/2 is no longer relevant because we have more information (this is one of the ways that probability is less nice than traditional logic; traditional logic is monotonic in the sense that adding more information cannot change your answer).

With this background, let's get back to the Carter Catastrophe, which makes sense if logic is probability. Suppose you have no more information than this:
Event e has not happened in the last hour but will happen eventually.
Let's say that the time from one hour ago to the next time e happens is t. We have no information at all about where now sits in the time range from 0 to t, so any time in that range is equally likely. This means that, for example, the chance that were are in the top half (or the bottom half) of the range is 1/2. In fact, the chance that we are in any 1/n of the range is 1/n.

This is the sort of reasoning that the Carter Catastrophe is based on, and it is logically sound, but the issue is that it is logically sound given that we have almost no information. Any other probability based on any more information at all will be a better estimate. You can't argue that something is likely based on almost no information when there is actual information to look at.

Frequentists like Mr. Wright object to assigning probabilities like this for non-repeatable events, but there is a semi-frequentist interpretation: if someone were to pick a bunch of random problems of this low-information type and offer you bets based on the problem, then reasoning with logical probability is the best strategy you can use to maximize your wins (or minimize your loses), assuming the questions were not deliberately chosen to defeat probability logic.

1 comment:

  1. "Event e has not happened in the last hour but will happen eventually."

    FALSE! Zeno's paradox proves that nothing ever happens. You can't argue with logic. (Tee hee!)