Suppose you’re observing something that changes over time — say the Dow Jones average, or the temperature in Barrow, Alaska, or the number of people who have been shot by terrorists so far this year. Suppose you have absolutely no prior information about how this thing behaves — in particular, you might have no way of knowing whether it changes continuously (like the temperature) or whether it’s subject to sudden changes (like the number of terrorist victims). You have no formula for it; you don’t even know whether there is a formula. It could be absolutely anything.
(For those who want more precision: This thing you’re observing is a real-valued function defined on the positive real numbers — which we can think of as a function of time. It can be any function whatsoever.)
Now suddenly, at some randomly chosen moment, your ability to observe comes to a halt. If that randomly chosen moment is, say, 6:23 AM, then your observations go right up to , but do not include, 6:23 AM.
Your task is to accurately guess the value of the variable at that first moment that you can’t observe. In fact, let me make your job a little harder. If your observations stop at 6:23AM, then I want you to accurately guess the value at 6:23 AM and for a little while thereafter.
Now here’s the theorem: There is a strategy that will allow you to succeed at this task with probability 100%.
If that sounds even remotely plausible to you, then you haven’t understood it.
If you could implement this strategy, you could certainly beat the stock market, unless you preferred to spend all your time winning bar bets about the paths of raindrops along windowpanes. Unfortunately, the strategy is (provably) too complex to implement on any computer. But there’s a sense in which it’s easy to describe, though not to implement. Sometime soon I’ll blog the description and a sketch of why it works. If you just can’t wait, here is the paper I learned this from.