To better understand the suggested model of small anthropic update I imagined the following thought experiment: my copies are created in 4 boxes: 1 copy in first box, 10 in second, 100 in third and 1000 in forth. Before the update, I have 0.25 chances to be in 4th box. After the update I have 0.89 chances to be in 4th box, so the chances increased only around 3.5 times. Is it a correct model?
Nope, that’s not the model. Your initial expected population is 1111/4≈278. After the anthropic update, your probabilities of being in the boxes are 1/1111, 10/1111, 100/1111 and 1000/1111 (roughly 0.09%, 0.9%, 9% and 90%). The expected population, however is (11+102+1002+10002)/1111≈909. That’s an expected population update of 3.27 times.
Note that, in this instance, the expected population update and the probability update are roughly equivalent, but that need not be the case. Eg if your prior odds are 1:1:10−36 about the population being 1, 1000, or 1012, then the expected population is roughly 500.5, the anthropic-updated odds are 1:1000:10−24, and the updated expected population is roughly 1000. So the probability boost to the larger population is roughly (10−24/1000)/10−36=1021, but the boost to the expected population is roughly 2.
To better understand the suggested model of small anthropic update I imagined the following thought experiment: my copies are created in 4 boxes: 1 copy in first box, 10 in second, 100 in third and 1000 in forth. Before the update, I have 0.25 chances to be in 4th box. After the update I have 0.89 chances to be in 4th box, so the chances increased only around 3.5 times. Is it a correct model?
Nope, that’s not the model. Your initial expected population is 1111/4≈278. After the anthropic update, your probabilities of being in the boxes are 1/1111, 10/1111, 100/1111 and 1000/1111 (roughly 0.09%, 0.9%, 9% and 90%). The expected population, however is (11+102+1002+10002)/1111≈909. That’s an expected population update of 3.27 times.
Note that, in this instance, the expected population update and the probability update are roughly equivalent, but that need not be the case. Eg if your prior odds are 1:1:10−36 about the population being 1, 1000, or 1012, then the expected population is roughly 500.5, the anthropic-updated odds are 1:1000:10−24, and the updated expected population is roughly 1000. So the probability boost to the larger population is roughly (10−24/1000)/10−36=1021, but the boost to the expected population is roughly 2.