This is a very good argument/summary of arguments/questions
I would analyse this in sequence (like, taking quotes in order) and then recursively go back to relook at the initial state of understanding to see if it’s at least consistent. If it isn’t, serious updates to worldview might have to occur.
These can be deferred and interleaved concurrently with other either more-interesting or higher-priority updates. Deferral can work by “virtualising” the argument as a “suppose (this: … )” question.
From 2: (now 2 layers of indirection to avoid updating on my own argument until later):
Suppose you know that there are a certain number of planets, N. You are unsure about the truth of a statement Q. If Q is true, you put a high probability on life forming on a given arbitrary planet. If Q is false, you put a low probability on this. You have a prior probability for Q.
Here I stop and summarise:
Suppose (there are) N planets (called “N” in totality). Q can be true or not-true. If Q, observe life. If not-Q, observe no-life. <-: already falsified by evidence. :-> if not-Q, “small finite number compared to N” of planets with life. :: Q cannot be false. Question: Can Q be P=1? Yes, as P=1 is just a logical, technical criterion and not necessarily relevant to a real world except in theory. Can Q be logically true? No, as that excludes the nuance of “how many planets out of N have life” which is the entire interesting part of the question. → using “there is actually a difference between ‘logical certainty’ and ‘P=1’.”.
So the question so far is to construct a prior CDF based on the previously quoted text. Since N is a finite, specific number of planets, this could be done by exhaustively checking each case, in each case for each N. Suppose N=1. Done.
Suppose N=2. Then n (number of planets) = either 1 or 2. Is it 1? yes. Is it 2? life has been observed on comets. Therefore likely to be 2, if N were to be “much larger” than 1. If N=2 then either the comets came from the 1 other planet or from the 1 planet already with life. A priori much more likely that N=1 in this case, given that life is observed to be a surprisingly rare phenomenon, however there must be some probability mass assigned to the idea that n=N=2, given that our previously described reasoning has some relevance to the question we are actually interested in, which is “N = some very large number roughly about the size of the number of planets we observe to be probably there in the universe or something”.
Suppose N is some large number compared to 1, 2, etc. Either N is prime or it can be divided by some factors up to (sqrt N). Either way, it can be “added up to” by using only 1, 2, 3, and 4, or some other small subset of numbers less than 10 like 6, 5, 2. ::<- implicitly defines subtraction.:: <->. If division is also allowed as an operation, then N either has a prime factorisation which can be calculated fairly straightforwardly or is prime.
If N is prime, then we should only use linear operations to obtain our probability distribution. If it is not prime, we may use nonlinear methods in addition. Either way, we can use both, and concurrently run the calculation to see whether some specific N is prime. Or we may choose a large N directly which has been already shown to be prime or not prime. Suppose N is 10,000,000,000,000,000,000,000,000. This is known to be not prime, and would likely be considered large compared to 1, 2, etc. We may also choose N as “some prime number close to about that value” and then apply only the linear part of the logic, and this would give us a close estimate for that N, which we can then apply some form of insertion sort/expansion/contraction/interpolation using all the tools available to us in accordance with the rules of probability theory to obtain a “best estimate” for the prime N which doesn’t require much extra calculation, and is likely good enough for cosmological estimates. See https://xkcd.com/2205/. Remember that after obtaining this prior we can “just multiply” to update it based on further observations. This is probably why it’s a good idea to get the prior from a very small number of observations if possible.
...
Now that we have worked out how we would calculate an example, it is not necessary to do so yet as this can be done after (and indeed, should be) writing down the full response, because it may turn out not to be necessary to answer the actual, quoted question which this article is about.
So what is “the rough shape” of our prior, given the response I myself have written so far?
Well, if the stated observations:
N is large compared to 1, 2 etc. (at least on the order of 10^25)
n is not 0
“You have a prior probability for Q”
‘Comets aren’t all generated from our planet originally, and (at least precursors to) life have been observed on comets’
The question we are interested in answering isn’t about cosmology but whether “Your existence is informative”
are taken as a starting point, then we can make a rough prior for Q, which is roughly that “n is small compared to N.” This is equivalent to saying that “life is unlikely” as there are much more big numbers than small numbers, and on a uniform distribution n would likely be not small (ie, within a few orders of magnitude) compared to N. “What does the evidence say about whether life is unlikely?” is now a relevant question for our larger question of the informativeness of the original question about Q.
Separately, N may not be finite, and we are interested in the actual question of the article in this case too. So we’re not actually that interested in the previous stuff, but as a prior we have that “life is unlikely even for infinite N” but that would still mean that for infinite N there would be an infinity of life.
It seems more important, by the numbers, to consider first the case of infinite N, which I will do in a reply.
Now, to restate the original “thing” we were trying to honestly say we had a prior for:
Suppose you know that there are a certain number of planets, N. You are unsure about the truth of a statement Q. If Q is true, you put a high probability on life forming on a given arbitrary planet. If Q is false, you put a low probability on this. You have a prior probability for Q.
Does this work, given this and our response?
We do not actually have a prior for Q, but we have a rough prior for a highly related question Q’, which can be transformed likely fairly easily into a prior for Q using mechanical methods. So let’s do that “non-mechanically” by saying:
If we successfully generate a prior for Q, that part is OK.
If Q is false, (::<- previously transformed into the more interesting and still consistent with a possible meaning for this part question “if Q is not-true”) :: use “If Q is not-true” as this proposition, then it is OK. But also consider the original meaning of “false” meaning “not true at all” meaning logically has 0 mass assigned to it. If we do both, this part is OK.
If Q is true, you put a high probability on life forming on a given arbitrary planet. This was the evidential question which we said the article was not mainly about, so we would continue reasoning from this point by reading the rest of the article until the next point at which we expect an update to occur to our prior; however if we do the rest of the steps (1-end here) then these updates can be relatively short and quick (as it’s just, in a technical sense, a multiplication. This can definitely be done by simultaneous (not just concurrent) algorithms). ::-> (predicted continuation point)
You are unsure about the truth of a statement Q. OK.
Suppose you know that there are a certain number of planets, N. This directly implies that we are only interested in the finite case and not the infinite case for N. However, we may have to keep the infinite case in mind. OK.
-> [Now to be able to say “we have a prior,” we have to write the continuation from 3. until the meaning of both
“what the article is about” becomes clear (so we can disambiguate the intended meanings of 1-5 and re-check)
We recognise that the prior can be constructed, and can roughly describe what it would look like.
From our previous response, our prior for finite N and a small amount of evidence was that “life is unlikely” (because there were two separate observationally consistent ways which resulted in ‘the same’ answer in some equivalence or similarity sense). For infinite N, it looks like “n is of lower dimension in some way” (dimension here meaning “bigness”) than N.
Now we have a prior for both, so we can try to convert back to a prior for the original proposition Q, which was:
You are unsure about the truth of a statement Q. If Q is true, you put a high probability on life forming on a given arbitrary planet. If Q is false, you put a low probability on this.
Our prior is that Q is false.]
In retrospect, the preceding (now in square brackets, which were edited in) could be considered a continuation of 3. So we are OK in all 5 ways, and we have a prior, so we can continue responding to the article.
My response to this would be:
This is a very good argument/summary of arguments/questions
I would analyse this in sequence (like, taking quotes in order) and then recursively go back to relook at the initial state of understanding to see if it’s at least consistent. If it isn’t, serious updates to worldview might have to occur.
These can be deferred and interleaved concurrently with other either more-interesting or higher-priority updates. Deferral can work by “virtualising” the argument as a “suppose (this: … )” question.
From 2: (now 2 layers of indirection to avoid updating on my own argument until later):
Here I stop and summarise:
Suppose (there are) N planets (called “N” in totality). Q can be true or not-true. If Q, observe life. If not-Q, observe no-life. <-: already falsified by evidence. :-> if not-Q, “small finite number compared to N” of planets with life. :: Q cannot be false. Question: Can Q be P=1? Yes, as P=1 is just a logical, technical criterion and not necessarily relevant to a real world except in theory. Can Q be logically true? No, as that excludes the nuance of “how many planets out of N have life” which is the entire interesting part of the question. → using “there is actually a difference between ‘logical certainty’ and ‘P=1’.”.
So the question so far is to construct a prior CDF based on the previously quoted text. Since N is a finite, specific number of planets, this could be done by exhaustively checking each case, in each case for each N. Suppose N=1. Done.
Suppose N=2. Then n (number of planets) = either 1 or 2. Is it 1? yes. Is it 2? life has been observed on comets. Therefore likely to be 2, if N were to be “much larger” than 1. If N=2 then either the comets came from the 1 other planet or from the 1 planet already with life. A priori much more likely that N=1 in this case, given that life is observed to be a surprisingly rare phenomenon, however there must be some probability mass assigned to the idea that n=N=2, given that our previously described reasoning has some relevance to the question we are actually interested in, which is “N = some very large number roughly about the size of the number of planets we observe to be probably there in the universe or something”.
Suppose N is some large number compared to 1, 2, etc. Either N is prime or it can be divided by some factors up to (sqrt N). Either way, it can be “added up to” by using only 1, 2, 3, and 4, or some other small subset of numbers less than 10 like 6, 5, 2. ::<- implicitly defines subtraction.:: <->. If division is also allowed as an operation, then N either has a prime factorisation which can be calculated fairly straightforwardly or is prime.
If N is prime, then we should only use linear operations to obtain our probability distribution. If it is not prime, we may use nonlinear methods in addition. Either way, we can use both, and concurrently run the calculation to see whether some specific N is prime. Or we may choose a large N directly which has been already shown to be prime or not prime. Suppose N is 10,000,000,000,000,000,000,000,000. This is known to be not prime, and would likely be considered large compared to 1, 2, etc. We may also choose N as “some prime number close to about that value” and then apply only the linear part of the logic, and this would give us a close estimate for that N, which we can then apply some form of insertion sort/expansion/contraction/interpolation using all the tools available to us in accordance with the rules of probability theory to obtain a “best estimate” for the prime N which doesn’t require much extra calculation, and is likely good enough for cosmological estimates. See https://xkcd.com/2205/. Remember that after obtaining this prior we can “just multiply” to update it based on further observations. This is probably why it’s a good idea to get the prior from a very small number of observations if possible.
...
Now that we have worked out how we would calculate an example, it is not necessary to do so yet as this can be done after (and indeed, should be) writing down the full response, because it may turn out not to be necessary to answer the actual, quoted question which this article is about.
So what is “the rough shape” of our prior, given the response I myself have written so far?
Well, if the stated observations:
N is large compared to 1, 2 etc. (at least on the order of 10^25)
n is not 0
“You have a prior probability for Q”
‘Comets aren’t all generated from our planet originally, and (at least precursors to) life have been observed on comets’
The question we are interested in answering isn’t about cosmology but whether “Your existence is informative”
are taken as a starting point, then we can make a rough prior for Q, which is roughly that “n is small compared to N.” This is equivalent to saying that “life is unlikely” as there are much more big numbers than small numbers, and on a uniform distribution n would likely be not small (ie, within a few orders of magnitude) compared to N. “What does the evidence say about whether life is unlikely?” is now a relevant question for our larger question of the informativeness of the original question about Q.
Separately, N may not be finite, and we are interested in the actual question of the article in this case too. So we’re not actually that interested in the previous stuff, but as a prior we have that “life is unlikely even for infinite N” but that would still mean that for infinite N there would be an infinity of life.
It seems more important, by the numbers, to consider first the case of infinite N, which I will do in a reply.
Now, to restate the original “thing” we were trying to honestly say we had a prior for:
Does this work, given this and our response?
We do not actually have a prior for Q, but we have a rough prior for a highly related question Q’, which can be transformed likely fairly easily into a prior for Q using mechanical methods. So let’s do that “non-mechanically” by saying:
If we successfully generate a prior for Q, that part is OK.
If Q is false, (::<- previously transformed into the more interesting and still consistent with a possible meaning for this part question “if Q is not-true”) :: use “If Q is not-true” as this proposition, then it is OK. But also consider the original meaning of “false” meaning “not true at all” meaning logically has 0 mass assigned to it. If we do both, this part is OK.
If Q is true, you put a high probability on life forming on a given arbitrary planet. This was the evidential question which we said the article was not mainly about, so we would continue reasoning from this point by reading the rest of the article until the next point at which we expect an update to occur to our prior; however if we do the rest of the steps (1-end here) then these updates can be relatively short and quick (as it’s just, in a technical sense, a multiplication. This can definitely be done by simultaneous (not just concurrent) algorithms). ::-> (predicted continuation point)
You are unsure about the truth of a statement Q. OK.
Suppose you know that there are a certain number of planets, N. This directly implies that we are only interested in the finite case and not the infinite case for N. However, we may have to keep the infinite case in mind. OK.
-> [Now to be able to say “we have a prior,” we have to write the continuation from 3. until the meaning of both
“what the article is about” becomes clear (so we can disambiguate the intended meanings of 1-5 and re-check)
We recognise that the prior can be constructed, and can roughly describe what it would look like.
From our previous response, our prior for finite N and a small amount of evidence was that “life is unlikely” (because there were two separate observationally consistent ways which resulted in ‘the same’ answer in some equivalence or similarity sense). For infinite N, it looks like “n is of lower dimension in some way” (dimension here meaning “bigness”) than N.
Now we have a prior for both, so we can try to convert back to a prior for the original proposition Q, which was:
Our prior is that Q is false.]
In retrospect, the preceding (now in square brackets, which were edited in) could be considered a continuation of 3. So we are OK in all 5 ways, and we have a prior, so we can continue responding to the article.
(To be continued in reply)