[connoredel]

The key insight is that you are unlikely to be experiencing the thing at a special time in its life. This is the Copernican principle (which J. Richard Gott uses in his version of this that Wikipedia mentions), which was basically "we (on Earth) are unlikely to occupy a special place in the solar system -- it's much more likely that some other object is the center."

Gott says you can be 95% confident that you're experiencing the thing in the middle 95% of its life. Let's say x is its life so far. If x is 2.5% of its eventual life (one extreme of the middle 95%), then the thing still has 39x to go. If you're at 97.5% (the other extreme), then the thing only has x/39 left. So the 95% confidence interval is between x/39 and 39x

Of course, 5% of the time you actually are experiencing something at the very beginning or very end of its life (outside the middle 95%), which is a unique thing. But that's why it's a confidence interval < 100% :)

I prefer this form of the principle a lot more than "the expected life is equal to 2X, always."

Side note: I took J. Richard Gott's class in college called The Universe. Maybe not the best use of a credit in hindsight, but we studied some really interesting things like this.

[tgb]

And the real fun is when you apply this to humanity itself: https://en.m.wikipedia.org/wiki/Doomsday_argument

[simonh]

Lots of interesting stuff in there. The problem I have with naive versions of this is that they assume as random people we don't live in a special time in human history, but if you look at human history so far the current era is both extremely short yet also spectacularly atypical in almost every conceivably way. It is also a period of still very rapid change. It's hard to get my head around what that means for estimating future trends or outcomes.

[indigochill]

Funny thing with exponential curves, no matter where on the curve you are, everyone behind you seems mind-numbingly slow and everyone ahead seems mind-bogglingly fast.

[grkvlt]

This always confused me - people talk about an exponential explosion, but the rate of change of e^x is e^x, so there is no actual 'knee' in the curve with a huge speedup afterwards...

[kenbellows]

or, if you prefer, every point is a 'knee' in the curve with a huge speedup afterward

[simonh]

Except in reality, when samples are infrequent and there is significant 'noise' in the data, it can be far from clear what shape the graph is. This is usually particularly true in the early stages of the development of a trend. How do you know you're in the early stages? You don't, but it's a big mistake to think that trends which eventually turn out to be exponential must therefore be obviously so at all times.

[simonh]

You're betraying an extremely. And I mean extraordinarily, laughably modernistic world view. We were making accheulian hand axes for well over one million years. Plus. Technological development during that period really didn't look very exponential. How do you measure change when literally nothing new is developed for tens of thousands of generations?

If you only look at development since agriculture then yes, but that's less than one percentof the time since the taming of fire.