There is some extremely small probability that the theory of evolution is false, and the evidence of this has been withheld from us by some kind of plot. This hypothesis is supported by the absence of time cameras (since time cameras would resolve the matter), and so the absence of time cameras must increase the probability that evolution is false… even if only by 1/3^^^^^3, or something like that.
You can’t imagine anything that improbable. Unless we adopt Robin’s anthropic penalty, in which case “I am in a unique position to affect 3^^^^^3 other people” is that improbable.
Actually, the beauty of mathematics is that it enables us to imagine such things—just as surely as it tells us that there ain’t nothin’ we’re talkin’ about that’s anywhere near that.
1/3^^^^^3 is a probability. A stupid probability, but a probability nonetheless. And if you declare 1/3^^^^^3 to be not a probability because of it’s unimaginable uselessness then by the same standard I expect you to consider 3^^^^^3 ‘Not a Number’. I know you routinely use arbitrarily large numbers like 3^^^3 for decision theoretic purposes (on Halloween costumes!) and that is a number that is more or less chosen because it is already unimaginable.
Ah, so you meant: No physically possible series of Bayesian updates can promote a hypothesis to prominence if its prior probability is that low. And Peter meant: It is decision-theoretically useless to include a subroutine for tracking probability increments of 1/3^^^^^3 in your algorithm.
But the non-Bayesian source of your Bayesian prior might output 1/3^^^^^3 as the prior probability of an event—as surely for the coin flip example as for Robin Hanson’s anthropic one.
To be precise, it’s impossible to describe any sense event with a prior probability that low. You can describe hypotheses conditional on which a macro-event has a probability that low. For example, conditional on the hypothesis that a coin is fixed to have a 1/3^^^3 probability of coming up heads, the probability of seeing heads is 1/3^^^3. But barring the specific and single case of Hanson’s hypothesized anthropic penalty being rational, I know of no way to describe, in words, any hypothesis which could justly be assigned so low a prior probability as 1/3^^^3. Including the hypothesis that purple is falling upstairs, that my socks are white and not white, or that 2 + 2 = 5 is a consistent theorem of Peano arithmetic.
If this were the case, then what is to stop me from thinking of N>3^^^^^3 impossible methods of gaining evidence (aliens from Mars, or Planet X, or the past, or from Cygnus XJ45, or another dimension...), and then claiming that since these probabilities are mutually independent, summing up the positive probabilities, and claiming evolution (or any theory) to be unlikely to be true? I mean, aside from the thing about probability theory being invalid, which I haven’t seen before.
Also, thank you Eliezer, for explaining why the argument about cryonics is logically rude. I’ve been banging my head on this exact topic with a friend for the past week and have been unable to get past that road block with her.
First, you can’t think of 3^^^^^3 ways of gaining evidence, possible or impossible, because there are not that many possible distinct states of a human brain (or of the physical universe, for that matter.)
Second, the more complex your hypothesis, the less probable it will be, so some hypotheses might only change the probability by 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3, or even less, and so it is perfectly possible to sum them all and still only move the probability by a very small amount.
You and Eliezer make good points, thank you. I just started reading about negative probabilities. I don’t believe I’ve heard of them before. Just to be clear, I never claimed that the sum of possibilities would diverge, though I don’t think I gave proper attention to the prior probability distribution summing to 1.
I did not mean to imply that I would individually think up every single impossible possibility. I figured it would be enough to hook into some countably infinite set and show that it is just a subset of all the possible impossibilities we could generate. One could simply tap into the Infinite Earths of DC Comics to construct an argument that resembles the original lack of time camera argument:
The absence of any Superman from Earths 1 through 3^^^^^^^^^^^^3 confirming evolution must elevate the probability that evolution is false.
In my obviously absurd example, I am not sure why Superman from Earth 5000 would be any more complex than Superman from Earth 500. Though I suppose the numbering system would indicate an elevated degree of difficulty crossing over, perhaps. It is true, I didn’t account for all the parallel Earths where Superman is evil, or disinterested in our Earth, or unable to get here. My mind boggles at the possibilities.
I think my original complaint remains, though. Why would the absence of evidence from something that is admittedly impossible increase the probability of something being false? I suppose I am complaining too much for such a tiny marginal increase in probability, since a random person on the street shouting “evolution is false!” is probably going to sway your opinion to a far larger degree than some 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3 event.
However, it strikes me as strange that someone should feel obligated to disbelieve (even a tiny bit) evolution on the grounds that Superman 3^^3 didn’t tell him/her that it was true.
The point was that you can easily have a sum of an infinite series that adds to some small finite amount. Assuming “Earth 5000” is defined differently from “Earth 500″ (which it must be in order to have new hypothesis), your different hypotheses will different complexities depending on the complexity of the number. Overall (but not in every single instance) the higher the number, the more complex the hypothesis, so the less the probability will be changed. There is no reason for this infinite sum not to converge to an extremely small quantity.
In any case (and this may be Peter de Blanc’s point), these probabilities are smaller than the sensitivity of the human judgement: so in fact, subjectively you don’t need to feel obliged to change your opinion at all based on them.
That makes sense. I guess as long as the sum of the infinitely many absurdly contrived possibilities remains less than rounding error and/or sensitivity of human judgment, I have no qualms with your original point.
There are counterbalancing negative possibilities, you can’t sum over just the positive ones. And since a prior probability distribution sums to 1, the contribution of even solely the positive possibilities must converge to a finite sum rather than diverging.
It happens that I’ve also been banging my head on this exact topic with a friend about a week ago and failed to get past that road block with him. Upvotes are in order.
There is some extremely small probability that the theory of evolution is false, and the evidence of this has been withheld from us by some kind of plot. This hypothesis is supported by the absence of time cameras (since time cameras would resolve the matter), and so the absence of time cameras must increase the probability that evolution is false… even if only by 1/3^^^^^3, or something like that.
Silly Unknowns. 0, 1, and 1/3^^^^^3 are not probabilities.
Why not 1/3^^^^^3?
You can’t imagine anything that improbable. Unless we adopt Robin’s anthropic penalty, in which case “I am in a unique position to affect 3^^^^^3 other people” is that improbable.
Actually, the beauty of mathematics is that it enables us to imagine such things—just as surely as it tells us that there ain’t nothin’ we’re talkin’ about that’s anywhere near that.
I can’t imagine quarks either.
1/3^^^^^3 is a probability. A stupid probability, but a probability nonetheless. And if you declare 1/3^^^^^3 to be not a probability because of it’s unimaginable uselessness then by the same standard I expect you to consider 3^^^^^3 ‘Not a Number’. I know you routinely use arbitrarily large numbers like 3^^^3 for decision theoretic purposes (on Halloween costumes!) and that is a number that is more or less chosen because it is already unimaginable.
log_2(3^^^^^3) heads in a row?
Coin’s fixed.
Ah, so you meant: No physically possible series of Bayesian updates can promote a hypothesis to prominence if its prior probability is that low. And Peter meant: It is decision-theoretically useless to include a subroutine for tracking probability increments of 1/3^^^^^3 in your algorithm.
But the non-Bayesian source of your Bayesian prior might output 1/3^^^^^3 as the prior probability of an event—as surely for the coin flip example as for Robin Hanson’s anthropic one.
To be precise, it’s impossible to describe any sense event with a prior probability that low. You can describe hypotheses conditional on which a macro-event has a probability that low. For example, conditional on the hypothesis that a coin is fixed to have a 1/3^^^3 probability of coming up heads, the probability of seeing heads is 1/3^^^3. But barring the specific and single case of Hanson’s hypothesized anthropic penalty being rational, I know of no way to describe, in words, any hypothesis which could justly be assigned so low a prior probability as 1/3^^^3. Including the hypothesis that purple is falling upstairs, that my socks are white and not white, or that 2 + 2 = 5 is a consistent theorem of Peano arithmetic.
The log_2(3^^^^^3) consecutive binary digits of pi starting from number 3^^^^^3 are 0?
The simulators are messing with you.
Then our minds are “fixed” too, just like the coin.
How many dustspecks in the eye are you willing to bet on that?
The log_2(3^^^^^3) consecutive binary digits of pi starting from number 3^^^^^3 are 0?
If this were the case, then what is to stop me from thinking of N>3^^^^^3 impossible methods of gaining evidence (aliens from Mars, or Planet X, or the past, or from Cygnus XJ45, or another dimension...), and then claiming that since these probabilities are mutually independent, summing up the positive probabilities, and claiming evolution (or any theory) to be unlikely to be true? I mean, aside from the thing about probability theory being invalid, which I haven’t seen before.
Also, thank you Eliezer, for explaining why the argument about cryonics is logically rude. I’ve been banging my head on this exact topic with a friend for the past week and have been unable to get past that road block with her.
First, you can’t think of 3^^^^^3 ways of gaining evidence, possible or impossible, because there are not that many possible distinct states of a human brain (or of the physical universe, for that matter.)
Second, the more complex your hypothesis, the less probable it will be, so some hypotheses might only change the probability by 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3, or even less, and so it is perfectly possible to sum them all and still only move the probability by a very small amount.
You and Eliezer make good points, thank you. I just started reading about negative probabilities. I don’t believe I’ve heard of them before. Just to be clear, I never claimed that the sum of possibilities would diverge, though I don’t think I gave proper attention to the prior probability distribution summing to 1.
I did not mean to imply that I would individually think up every single impossible possibility. I figured it would be enough to hook into some countably infinite set and show that it is just a subset of all the possible impossibilities we could generate. One could simply tap into the Infinite Earths of DC Comics to construct an argument that resembles the original lack of time camera argument:
The absence of any Superman from Earths 1 through 3^^^^^^^^^^^^3 confirming evolution must elevate the probability that evolution is false.
In my obviously absurd example, I am not sure why Superman from Earth 5000 would be any more complex than Superman from Earth 500. Though I suppose the numbering system would indicate an elevated degree of difficulty crossing over, perhaps. It is true, I didn’t account for all the parallel Earths where Superman is evil, or disinterested in our Earth, or unable to get here. My mind boggles at the possibilities.
I think my original complaint remains, though. Why would the absence of evidence from something that is admittedly impossible increase the probability of something being false? I suppose I am complaining too much for such a tiny marginal increase in probability, since a random person on the street shouting “evolution is false!” is probably going to sway your opinion to a far larger degree than some 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3 event.
However, it strikes me as strange that someone should feel obligated to disbelieve (even a tiny bit) evolution on the grounds that Superman 3^^3 didn’t tell him/her that it was true.
(edit: I got the Superman 500 mixed up with 5000)
The point was that you can easily have a sum of an infinite series that adds to some small finite amount. Assuming “Earth 5000” is defined differently from “Earth 500″ (which it must be in order to have new hypothesis), your different hypotheses will different complexities depending on the complexity of the number. Overall (but not in every single instance) the higher the number, the more complex the hypothesis, so the less the probability will be changed. There is no reason for this infinite sum not to converge to an extremely small quantity.
In any case (and this may be Peter de Blanc’s point), these probabilities are smaller than the sensitivity of the human judgement: so in fact, subjectively you don’t need to feel obliged to change your opinion at all based on them.
That makes sense. I guess as long as the sum of the infinitely many absurdly contrived possibilities remains less than rounding error and/or sensitivity of human judgment, I have no qualms with your original point.
There are counterbalancing negative possibilities, you can’t sum over just the positive ones. And since a prior probability distribution sums to 1, the contribution of even solely the positive possibilities must converge to a finite sum rather than diverging.
It happens that I’ve also been banging my head on this exact topic with a friend about a week ago and failed to get past that road block with him. Upvotes are in order.