..., and this too shall pass

George Rebane

Bet you thought I was going to wax eloquent on some deep philosophical or socio-political issue, a behavior for which I have a weakness as witnessed numerous times in these pages.  Fooled you.  In another life I am currently working on a couple of very intriguing technical projects involving uncertainty and algorithmics, the kind that I often have to take off at least one shoe in order to get past the sticky parts.  One of them involves a nifty and little known method to calculate probabilities about the termination of a kind of realworld and practical processes.

Suppose you live in a metropolitan area where the atmosphere is becoming more polluted by NOX gases as measured by their concentration in the air over the city.  And all you know is the record shows that this pollution began about 37 years ago.  You need to decide whether to continue living in the city, or to move away, and an important factor in the decision is whether NOX will be brought under control within the next four years.  Since this question involves uncertainty, you want to compute the probability that the NOX trend (process) will terminate some time in these four years.  And all you know about the process is that it is 37 years old.

Well lucky you, there is a computable answer which turns out to be 0.098, or just under 10% chance that the NOX trend will be brought under control sometime in the next four years.

The solution for such problems comes through a simple formula derived from arguments first presented by physicist J Richard Gott in a quickly forgotten 1993 paper published in Nature - ‘Implications of the Copernican Principle for our Future Prospects’.  In an effort to revive the physicist’s thinking, Kierland and Monton made a somewhat cumbersome attempt to explain Gott in their ‘How to Predict Future Duration from Present Age’ (Philosophical Quarterly, 2006).  Save for a small flurry of debate, Gott’s discovery was peacefully put back to rest.

And then I came along – ta-daa!!  Plowing through the paper I was struck by the apparent unrecognized utility of Gott’s theory to the analysis of what we may call minimally known processes (MKPs).  It was immediately clear to me that in our daily round we are awash in such processes, but very few of us are able to identify them as MKPs, and fewer still have heard of Gott.  Since my professional activities continue in various areas of uncertainty, I recognized a diamond in the rough and got to work.  My humble contribution from the effort has been a clear derivation of a simple and elegant formula for Gott’s probability, that I then extended to support dealing with arbitrary future time intervals, and finally demonstrate the complete scalability of the theory.  (For those still awake, we will carry on and promise an intriguing reward to the persistent reader.  Nothing beyond clear thinking and the ability to punch numbers into a couple of simple formulas is required.)

Let’s open up the horizon with another example modified from Kierland and Monton.  If you visited a park in New Zealand’s South Island that contains various geysers, and saw from a sign near one geyser informing you that it has been spouting steadily for the last 15,000 years, you could ask for the probability that it would stop doing so within the next couple of hours that you plan to remain in the park.  Instinctively, you would conclude that the chances of that process terminating in that short interval would be very low indeed – i.e. the chance (probability) of that happening would be somewhere between slim and none (actually 0.000000015).

Walking onward you come upon another geyser encircled by recently erected yellow plastic ‘Keep Out’ tape, and a nearby sign stating that this geyser started spouting just three days ago.  If you now considered the same question, what are the chances that this MKP will terminate in the next two hours, you would naturally conclude that that probability would be much higher (actually 0.027).  Well, the Gott theory quantitatively captures those observations in a manner that is totally lucid when examined with the help of the formula we modestly here name Gott-Rebane1 or GR1.  Let’s take a look.

[To continue reading this post, please download its pdf here - Download TN1411-1-and this too shall pass]

Comments

[Account Deleted]

OK - so who's gonna win the fifth at Hialeah?
Sorry, I couldn't help it.
The proof will be in the pudding. It will be interesting to see what sort of near term predictions you might come up with and post that we can mark down and check later. The better the batting average, the more interest it will attract. I'll try a few examples myself and see how well I can do.

[George Rebane]

ScottO 647pm - OK, go for it. One of the better uses for method is as obtaining a better prior probability for Bayesian estimation. In short, if you're expecting to incorporate future evidence regarding a proposition/hypothesis that start out as a MKP, then use its Gott probability to start things off. We'll be doing more stuff with Bayes on RR that involves reader participation as the 2016 election season starts after the new year. Fun stuff.