Clearly this “prediction” is wrong. Just model the population growth based on dice throw. Each pair of parents has on average 3.5 children, so the population grows exponentially. The real question is, where did the logic go so wrong?
First, what are the aposteriori probabilities of each outcome of the dice roll by your parents, according to the SIA? How do you calculate that? Forget 6-sided die, let’s try something simpler: a random choice of 0 or 1 child (furthermore, only one parent is required for procreation). SIA, SSA or any other model implies apriory probabilities of 0.5 for each outcome but aposteriori probabilities 0 and 1. Thus the “prediction” is that the growth rate was zero in the past, but will be negative in the future.
Now the issue is much clearer: if you trace your reasoning into the past, you see that only those parents who had children got accounted for. Those who had zero children dropped out from the calculations, since there is no one to remember them. And the number of people who had exactly one child (culminating with you) has always been constant (namely 1).
Extrapolating this to your case, your “prediction” under-samples those parents who had fewer children.
Imagine, that you have a 3 sided dice. This way the population is stable.
Yet, your parents have probably more children than you will have. You will have 1, 2 or 3 (p=1/3 for each case) − 2 on average.
But since you are alive, it is only 1⁄6 that your parents have 1 child, 1⁄3 that they have 2 and 1⁄2 that they have 3. It looks like you will have smaller number of children than your parents. Most probably.
A small modification of the initial post and there is no population growth or decline.
Clearly this “prediction” is wrong. Just model the population growth based on dice throw. Each pair of parents has on average 3.5 children, so the population grows exponentially. The real question is, where did the logic go so wrong?
First, what are the aposteriori probabilities of each outcome of the dice roll by your parents, according to the SIA? How do you calculate that? Forget 6-sided die, let’s try something simpler: a random choice of 0 or 1 child (furthermore, only one parent is required for procreation). SIA, SSA or any other model implies apriory probabilities of 0.5 for each outcome but aposteriori probabilities 0 and 1. Thus the “prediction” is that the growth rate was zero in the past, but will be negative in the future.
Now the issue is much clearer: if you trace your reasoning into the past, you see that only those parents who had children got accounted for. Those who had zero children dropped out from the calculations, since there is no one to remember them. And the number of people who had exactly one child (culminating with you) has always been constant (namely 1).
Extrapolating this to your case, your “prediction” under-samples those parents who had fewer children.
Imagine, that you have a 3 sided dice. This way the population is stable.
Yet, your parents have probably more children than you will have. You will have 1, 2 or 3 (p=1/3 for each case) − 2 on average.
But since you are alive, it is only 1⁄6 that your parents have 1 child, 1⁄3 that they have 2 and 1⁄2 that they have 3. It looks like you will have smaller number of children than your parents. Most probably.
A small modification of the initial post and there is no population growth or decline.
Yes, this was an example I considered, too, but it does not seem to highlight the problem with under-sampling of the low-sibling families as much.
A side note. In the real world, on average, one HAS less children than his parents. Parents can’t have zero children, a child can.
Both are true, but the former doesn’t follow from the latter. In particular, I suspect the former was false a few decades ago.
Indeed, this self-selection is the reason that the SIA Doomsday is a bad prediction.