I think you’re wrong about “backwards probability”.
Probabilities describe your state of knowledge (or someone else’s, or some hypothetical idealized observer’s, etc.). It is perfectly true that “your” probability for some past event known to you will be 1 (or rather something very close to 1 but allowing for the various errors you might be making), but that isn’t because there’s something wrong with probabilities of past events.
Now, it often happens that you need to consider probabilities that ignore bits of knowledge you now have. Here’s a simple example.
I have a 6-sided die. I am going to roll the die, flip a number of coins equal to the number that comes up, and tell you how many heads I get. Let’s say the number is 2. Now I ask you: how likely is it that I rolled each possible number on the die?
To answer that question (beyond the trivial observation that clearly I didn’t roll a 1) one part of the calculation you need to do is: how likely was it, given a particular die roll but not the further information you’ve gained since then, that I would get 2 heads? You will get completely wrong answers if you answer all those questions with “the probability is 1 because I know it was 2 heads”.
(Here’s how the actual calculation goes. If the result of the die roll was k, then Pr(exactly 2 heads) was (k choose 2) / 2^k, which for k=1..6 goes 0, 1⁄4, 3⁄8, 6⁄16, 10⁄32, 15⁄64; since all six die rolls were equiprobable to start with, your odds after learning how many heads are proportional to these or (taking a common denominator) to 0 : 16 : 24 : 24 : 20 : 15, so e.g. Pr(roll was 6 | two heads) is 15⁄99 = 5⁄33. Assuming I didn’t make any mistakes in the calculations, anyway.)
The SSA-based calculations work in a similar way.
Consider the possible different numbers of humans there could ever have been (like considering all the possible die rolls).
For each, see how probable it is that you’d have been human # 70 billion, or whatever the figure is (like considering how probable it was that you’d get two heads).
Your posterior odds are obtained from these probabilities, together with the probabilities of different numbers of human beings a priori.
I am not claiming that you should agree with SSA. But the mere fact that it employs these backward-looking probabilities is not an argument against it; if you disagree, you should either explain why computations using “backward probabilities” correctly solve the die+coins problem (feel free to run a simulation to verify the odds I gave) despite the invalidity of “backward probabilities”, or else explain why the b.p.’s used in the doomsday argument are fundamentally different from the ones used in the die+coins problem.
I think you’re wrong about “backwards probability”.
Probabilities describe your state of knowledge (or someone else’s, or some hypothetical idealized observer’s, etc.). It is perfectly true that “your” probability for some past event known to you will be 1 (or rather something very close to 1 but allowing for the various errors you might be making), but that isn’t because there’s something wrong with probabilities of past events.
Now, it often happens that you need to consider probabilities that ignore bits of knowledge you now have. Here’s a simple example.
I have a 6-sided die. I am going to roll the die, flip a number of coins equal to the number that comes up, and tell you how many heads I get. Let’s say the number is 2. Now I ask you: how likely is it that I rolled each possible number on the die?
To answer that question (beyond the trivial observation that clearly I didn’t roll a 1) one part of the calculation you need to do is: how likely was it, given a particular die roll but not the further information you’ve gained since then, that I would get 2 heads? You will get completely wrong answers if you answer all those questions with “the probability is 1 because I know it was 2 heads”.
(Here’s how the actual calculation goes. If the result of the die roll was k, then Pr(exactly 2 heads) was (k choose 2) / 2^k, which for k=1..6 goes 0, 1⁄4, 3⁄8, 6⁄16, 10⁄32, 15⁄64; since all six die rolls were equiprobable to start with, your odds after learning how many heads are proportional to these or (taking a common denominator) to 0 : 16 : 24 : 24 : 20 : 15, so e.g. Pr(roll was 6 | two heads) is 15⁄99 = 5⁄33. Assuming I didn’t make any mistakes in the calculations, anyway.)
The SSA-based calculations work in a similar way.
Consider the possible different numbers of humans there could ever have been (like considering all the possible die rolls).
For each, see how probable it is that you’d have been human # 70 billion, or whatever the figure is (like considering how probable it was that you’d get two heads).
Your posterior odds are obtained from these probabilities, together with the probabilities of different numbers of human beings a priori.
I am not claiming that you should agree with SSA. But the mere fact that it employs these backward-looking probabilities is not an argument against it; if you disagree, you should either explain why computations using “backward probabilities” correctly solve the die+coins problem (feel free to run a simulation to verify the odds I gave) despite the invalidity of “backward probabilities”, or else explain why the b.p.’s used in the doomsday argument are fundamentally different from the ones used in the die+coins problem.