Edit: the argument is presented more clearly in a subsequent post.

Many thanks to Paul Almond for developing the initial form of this argument.

It is well known in these circles that the self-sampling assumption (SSA) leads to the doomsday argument. The self-indication assumption (SIA) was developed to counter the doomsday argument. This is a old debate; but what is interesting is that SIA has its own doomsday argument - of a rather interesting and different form.

To see this, let's model the population of a planet somewhat like Earth. From century to century, the planet's population can increase, decrease or stay the same with equal probability. If it increase, it will increase by one billion two thirds of the time, and by two billion one third of the time - and the same for decreases (if it overshoots zero, it stops at zero). Hence, each year, the probability of population change is:

During the century of the Three Lice, there were 3 billion people on the planet. Two centuries later, during the century of the Anchovy, there will still be 3 billion people on the planet. If you were alive on this planet during the intermediate century (the century of the Fruitbat), and knew those two facts, what would your estimate be for the current population?

From the outside, this is easy. The most likely answer if that there is still 3 billion in the intermediate century, which happens with probability 9/19 (= (3/9)*(3/9), renormalised). But there can also be 4 or 2 billion, with probabilities 4/19 each, or 5 or 1 billion, with probabilities 1/19 each. The expected population is 3 billion, as expected.

Now let's hit this with SIA. This weighs the populations by their sizes, changing the probabilities to 5/57, 16/57, 27/57, 8/57and 1/57, for populations of five, four, three, two and one billion respectively. Larger populations are hence more likely; the expected population is about 3.28 billion.

(For those of you curious about what SSA says, that depends on the reference class. For the reference class of people alive during the century of the Fruitbat, it gives the same answer as the outside answer. As the reference class increases, it moves closer to SIA.)

SIA doomsday

So SIA tells us that we should expect a spike during the current century - and hence a likely decline into the next century. The exact numbers are not important: if we know the population before our current time and the population after, then SIA implies that the current population should be above the trendline. Hence (it seems) that SIA predicts a decline from our current population (or a least a decline from the current trendline) - a doomsday argument.

Those who enjoy anthropic reasoning can take a moment to see what is misleading about that statement. Go on, do it.

Go on.

Still there? Then you've certainly already solved the problem, and are just reading to check that I got it too, and compare stylistic notes. Then for you (and for those lazy ones who've peaked ahead), here is the answer:

It's one of those strange things that happen when you combine probabilities and expectations, like the fact that E(X/Y)>1 does not imply that E(X)>E(Y). Here, the issue is that:

• Expectations of decline from a trendline for any given future populations, does not translate in expectations of decline from a trendline when we don't know the future population.

Confused? Let's go back to our previous problem, still fix the past population at 3 billion, and let the future population vary. As we've seen, if the future population was 3 billion, SIA would boost the probability of an above-trendline present population; so, for instance, 3-5-3 is more likely among the 3-?-3 than it would be without SIA.

But now consider what would happen if the future population was 7 billion - if we were in 3-?-7. In this case, there is only one possibility in our model, namely 3-5-7. So large present populations are relatively more likely if the future population is large - and SIA makes large present populations more likely. And this removes the effects of the SIA doomsday.

To sumarise:

1. For known future population, SIA increase the probability of the present population being above trend (hence increasing the chance of a trend downswing in future).
2. The current population is more likely to be high if the future population is high (hence increasing the chance of a trend upswing in future).
3. These two effects exactly compensate. SIA has no impact on the future, once the present is known.