Warning: this post is technical.

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 any given arbitrary planet. If Q is false, you put a low probability on this. You have a prior probability for Q. So far you have not taken into account your observation that the planet you are on has life. How do you update on this evidence, to get a posterior probability for Q? Since your model just has a number of planets in it, with none labeled as ‘this planet’, you can’t update directly on ‘there is life on this planet’, by excluding worlds where ‘this planet’ doesn’t have life. And you can’t necessarily treat ‘this’ as an arbitrary planet, since you wouldn’t have seen it if it didn’t have life.

I have an ongoing disagreement with an associate who suggests that you should take ‘this planet has life’ into account by conditioning on ‘there exists a planet with life’. That is,

P(Q|there is life on this planet) = P(Q|there exists a planet with life).

Here I shall explain my disagreement.

Nick Bostrom argues persuasively that much science would be impossible if we treated ‘I observe X’ as ‘someone observes X’. This is basically because in a big world of scientists making measurements, at some point somebody will make most mistaken measurements. So if all you know when you measure the temperature of a solution to be 15 degrees is that you are not in a world where nobody ever measures its temperature to be 15 degrees, this doesn’t tell you much about the temperature.

You can add other apparently irrelevant observations you make at the same time – e.g. that the table is blue chipboard – in order to make your total observations less likely to arise once in a given world (at its limit, this is the suggestion of FNC). However it seems implausible that you should make different inferences from taking a measurement when you can also see a detailed but irrelevant picture at the same time than those you make with limited sensory input. Also the same problem re-emerges if the universe is supposed to be larger. Given that the universe is thought to be very, very large, this is a problem. Not to mention, it seems implausible that the size of the universe should greatly affect probabilistic judgements made about entities which are close to independent from most of the universe.

So I think Bostrom’s case is good. However I’m not completely comfortable arguing from the acceptability of something that we do (science) back to the truth of the principles that justify it. So I’d like to make another case against taking ‘this planet has life’ as equivalent evidence to ‘there exists a planet with life’.

Evidence is what excludes possibilities. Seeing the sun shining is evidence against rain, because it excludes the possible worlds where the sky is grey, which include most of those where it is raining. Seeing a picture of the sun shining is not much evidence against rain, because it excludes worlds where you don’t see such a picture, which are about as likely to be rainy or sunny as those that remain are.

Receiving the evidence ‘there exists a planet with life’ means excluding all worlds where all planets are lifeless, and not excluding any other worlds. At first glance, this must be different from ‘this planet has life’. Take any possible world where some other planet has life, and this planet has no life. ‘There exists a planet with life’ doesn’t exclude that world, while ‘this planet has life’ does. Therefore they are different evidence.

At this point however, note that the planets in the model have no distinguishing characteristics. How do we even decide which planet is ‘this planet’ in another possible world? There needs to be some kind of mapping between planets in each world, saying which planet in world A corresponds to which planet in world B, etc. As far as I can tell, any mapping will do, as long as a given planet in one possible world maps to at most one planet in another possible world. This mapping is basically a definition choice.

So suppose we use a mapping where in every possible world where at least one planet has life, ‘this planet’ corresponds to one of the planets that has life. See the below image.

Squares are possible worlds, each with two planets. Pink planets have life, blue do not. Define ‘this planet’ as the circled one in each case. Learning that there is life on this planet is equal to learning that there is life on some planet.

Now learning that there exists a planet with life is the same as learning that this planet has life. Both exclude the far righthand possible world, and none of the other possible worlds. What’s more, since we can change the probability distribution we end up with, just by redefining which planets are ‘the same planet’ across worlds, indexical evidence such as ‘this planet has life’ must be horseshit.

Actually the last paragraph was false. If in every possible world which contains life, you pick one of the planets with life to be ‘this planet’, you can no longer know whether you are on ‘this planet’. From your observations alone, you could be on the other planet, which only has life when both planets do. The one that is not circled in each of the above worlds. Whichever planet you are on, you know that there exists a planet with life. But because there’s some probability of you being on the planet which only rarely has life, you have more information than that. Redefining which planet was which didn’t change that.

Perhaps a different definition of ‘this planet’ would get what my associate wants? The problem with the last was that it no longer necessarily included the planet we are on. So what about we define ‘this planet’ to be the one you are on, plus a life-containing planet in all of the other possible worlds that contain at least one life-containing planet. A strange, half-indexical definition, but why not? One thing remains to be specified – which is ‘this’ planet when you don’t exist? Let’s say it is chosen randomly.

Now is learning that ‘this planet’ has life any different from learning that some planet has life? Yes. Now again there are cases where some planet has life, but it’s not the one you are on. This is because the definition only picks out planets with life across other possible worlds, not this one. In this one, ‘this planet’ refers to the one you are on. If you don’t exist, this planet may not have life. Even if there are other planets that do. So again, ‘this planet has life’ gives more information than ‘there exists a planet with life’.

You either have to accept that someone else might exist when you do not, or you have to define ‘yourself’ as something that always exists, in which case you no longer know whether you are ‘yourself’. Either way, changing definitions doesn’t change the evidence. Observing that you are alive tells you more than learning that ‘someone is alive’.

GD Star Rating

Hmm, I am confused.  So I can’t say anything very helpful except clarify the confusion.  Let me go through point by point, perhaps someone can help.

A) I assume you are saying that we have some prior distribution for Q (call it a gut instinct) and you want to update that distribution for the fact that earth has life.

B) Our ability to measure as being roughly 15 degrees depends on us having priors for what kinds of temperatures are likely; how our thermometer behaves; and how likely the 15-degreeness of the solution is to influence our decision whether or not to make the measurement.

The equavelent priors for life-on-earth are much harder to establish.    I don’t actually know what effect that has on the argument, but it just makes me  pause.

C) I disagreed with the first paragraph after the diagram; but I didn’t understand the subsequent paragraph that refutes it.  Perhaps they cancel out then.

More generally I think the game of trying to define what planet “this planet” is among an imaginary ensemble of Universes is bogus.  But then I guess that’s why there is a controversy in the first place.

Fundamentally, it seems the fact of our existence merely prooves that life is not impossible.  To go beyond that we need additional prior assumptions, and I am suspicious of arguments that do not involve estimating those priors explicitly.

I changed the post a little to make some bits clearer. Thanks.

Yes, you have priors for everything.

What’s wrong with defining entities across possible worlds? I agree that it’s bogus to think there is a correct answer about it, but as far as I know there are few constraints on defining stuff. Part of my point is that this is all irrelevant to what evidence you receive when you observe your own existence.

The argument would go through regardless of your choice of priors, so why estimate priors as part of it?

• V V

I can’t really parse the first paragraph.

Please consider rewriting it using explicit variables and explicit quantifiers.

• V V

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.

Do you mean

“There exists a planet X such that Pr[there is life on X | Q is true] > 0.9 and Pr[there is life on X | Q is false] 0.9 and Pr[there exists a planet X such that there is life on X | Q is false] < 0.1"

Since you don’t know which is ‘this’ planet, with respect to the model,
you can’t update directly on ‘there is life on this planet’, by
excluding worlds where this planet doesn’t have life.

I don’t understand that sentence. Why is “this” between quotes? What does it mean “with respect to the model”?

• Tyrrell McAllister

I suspect that the difficulty you are running into when you try to parse is precisely the difficulty with indexical talk that Katja is pointing out.

We want to do Bayesian updating on propositions that include indexicals such as “this planet that I am pointing at beneath my feet.”  But, at a certain point, it becomes impossible to pick out the referent of “I” and “my feet” using “explicit variables and explicit quantifiers”.

The naive first-try would be to replace “I” and “my feet” with “someone” and “that person’s feet”, respectively.  But it quickly becomes apparent that this gives wrong — or at least controversial — answers when you try to do Bayesian updating with the resulting “de-indexified” propositions.

So we are left wondering just how to make sense of the indexicals, where “make sense” means “use to arrive at rational probabilistic credences”.  Thus the problem you had parsing is precisely the point.

• Orphan

Am I correct in assessing that the point of this post is that our existence proves that -our- existence is possible, which is somewhat more information than that our existence proves that -life- is possible?
Our existence has more information than the existence of some arbitrary lifeform.  I don’t think this information gives us any additional information about the probability of life, or the probability of its priors, however, because our existence must be a given in any evaluation of such probabilities.

Or, to put it another way, our existence verifies only that the probability of -us- is nonzero.  The existence of “other life” would only verify the probability of -life- is nonzero.  So there is additional information, it just has nothing to say about the probability of life in general, except to exclude a particular value, zero.

• V V

Makes sense.

No. My point is that your being alive (ignoring what kind of creature you are, and all other info) gives more information than ‘there exists someone who is alive’, about Q (and so about the probability of life).  I don’t know what you mean by ‘the probability of its priors’. Why must your existence be a given in any evaluation of the probability of life? I didn’t even know you existed until this day, and yet I have assessed the probability of life before. What you say after ‘to put it another way’ is what I am criticizing.

• Orphan

Because your observing your own existence is contingent on your own existence.  Regardless of the probability of life, your existence is necessary to calculate that probability, and therefore your existence is certain entirely independent of that probability for any nonzero value.

If you didn’t exist, what would you calculate the likelihood of your own existence to be?

The null hypothesis doesn’t even make sense.  You’re attempting to assert a likelihood based on a relationship with no logical negation.  You can’t reject your own existence.

For an analog case, assume you have a display that reads “9”, and a button that can be pushed to display a number; the only information you have is that the button was pushed at least once, pushing the button is the only way to change the display, and that the current display is “9”.  What is the likelihood that, if pressed, it will display “9”?

You can assume 9 is possible.  That’s it.  The problem you are solving asserts that 9 is currently the case; you don’t know if it was pressed once or a billion times or a billion to the power of a billion times.  Because 9 is a given, it provides absolutely -no- information about the probability of 9 actually displaying, -except- that 9 is currently the case; for all you know, somebody who lived forever and really liked the number 9 pressed that button until it came up.  Every time you press the button yourself provides meaningful information about the display; the value it displayed when you came upon the display is meaningless.

• Silent Cal

You can make an a priori calculation about the likelihood of a universe producing life without updating on your own existence. If you’re interested in an accurate estimate of that probability, you *should* update on your own existence, but you can make the calculation just fine without doing so.

• Orphan

Silent – except that using your own existence is like using the original value of the display in a problem scenario.  The prior probability, in our problem definition, of the display showing “9” is 100%, because the problem definition requires that “9” be displayed.  It adds no information about the odds of “9” displaying on any other button press.  Indeed, the problem definition even permits that the posterior likelihood of “9” showing up is 0%, if, for example, the display will only ever display a given value once.

Similarly, our own existence is part of the problem definition in trying to ascertain the likelihood of life; the probability of our existence is 100%, because we exist, as part of the problem definition.  This adds no information about the likelihood of the existence of life, except to rule out 0%.  We don’t know how many times the button was pushed to result in the scenario of the problem definition.

• V V

@82d99766d5d827acb8915675200c7906:disqus

You can make an a priori calculation about the likelihood of a universe producing life without updating on your own existence.

Formally, you can, if you know the priors.

However, how do you esitmate the priors given that you can’t observe any universe where you don’t exist?

• Silent Cal

Orphan:

If you just mean that seeing 9 is meaningless because it’s already in the problem definition that a 9 is currently displayed, then you still have to update on the fact that a 9 is currently displayed. Any textbook Bayesian reasoning problem has all of the evidence contained in the problem definition, but you still have to update on it.

The other thing I think you might mean is that we don’t even have a prior about who pushed the button how many times, i.e. someone who lived forever and really liked a number is no less likely than any other possibility. This seems odd to me; I’m pretty sure a Bayesian always has a prior, even if it’s not a very good one. Also, if this is what you meant, the same point stands if the current number is an observation instead of a part of the problem definition.

V V: I’m not sure I fully understand your objection, but couldn’t you use the universal prior? http://wiki.lesswrong.com/wiki/Solomonoff_induction

Such an a priori estimate is clearly not a practical thing to do, but the fact that it’s possible shows how updating on your own existence makes sense.

• V V

@82d99766d5d827acb8915675200c7906:disqus

I’m not sure I fully understand your objection, but couldn’t you use the universal prior?

Long answer: the “universal prior” is not so much universal after all:

1) It embedds implicit assumptions on what it is physically computable in the universe.

2) If you exists inside the universe, then you can’t use the universal prior, because it is uncomputable within the universe that it is supposed to model.

3) It is defined in terms of strings. Mapping strings to universes, assuming that it is possible, requires an encoding, which adds a layer of arbitrarity.

4) It is defined up to the specification of an universal Turing machine. While asymptotic equivalence theorems do hold, the choice of the specific Turing machine might be relevant for finite strings.

• Silent Cal

V V: This sounds like a big epistemological problem. Cool.

Does it ameliorate it at all if you can use your own existence as evidence? Granted, you then know that the universe has definitely produced a nonzero amount of life, but if you’re trying to learn the probability of Q (as defined in the OP) and you have no sensible way of getting a prior, it seems like knowing that you exist wouldn’t help.

• Mark M

I don’t know how well I can respond to the whole article because, like others who have posted, I’m having a hard time parsing it.  The term ‘world’ does not seem to be used consistently.  Are you talking about possible realities that could have come about?  Different planets in our universe?  Different universes in the multiverse?  There is also ambiguity of “you” as a living breathing person on “this planet,” and “you” as a detached observer contemplating from afar, possibly across existential boundaries.  I believe the article uses both these viewpoints, but the context shift is not readily apparent.

In any case, perhaps I can address the original question of whether “you should take ‘this planet has life’ into account by conditioning on ‘there exists a planet with life’.”

My answer is whether it makes sense to predicate ‘this marble is blue,’ on ‘there exists a marble that is blue,’ while observing a blue marble.  You certainly can do this, but since it doesn’t change anything the extra effort doesn’t make much sense.

Unless, of course, you are contemplating a set of worlds (from across existential boundaries) where blue, marbles, or blue marbles may not exist.

You’re right there were a few places where I meant ‘planet’ but wrote ‘world’. Fixed that, so ‘world’ = ‘possible world’ = possible state of affairs. Thanks.

‘You’ are an entity who can observe things and reason about them, and also who exists in the world, and has other characteristics, e.g. a tendency to breathe.

• Silent Cal

I have an example that I think might clarify things. At least, it did for me.

We know that all worlds have ten planets, and they’re numbered one through ten (by number of moons, perhaps). Half of worlds are X worlds, which have life on all planets, and the other half are Y worlds, which have life on exactly one planet. A tenth of Y worlds have life on planet one, a tenth have life on planet two, and so on.

You observe that you’re on planet three, so there’s life on planet three. If you ignore the index and condition on “some planet has life”, you learn nothing, as all worlds have life on some planet. Alternatively, you could condition on “planet three has life”, which will give evidence for being in an X world. To see this, call the observation “planet three has life” E. P(X | E) = P(E | X) * P(X) / P(E) = 1 * .5 / (.5 + .5 / 10) = .90909…  The evidence is in fact quite strong.

Note that there’s nothing special about planet three; in every world, X or Y, with life on any planet, that planet’s inhabitants will follow the above reasoning to conclude that they’re in. This sounds wrong at first (at least, it did to me). But the fact is, given our specification of the set of worlds, most planets with life are in X worlds. This fact also doesn’t change if you don’t know your world’s number, so we can discard the assumption of a numbering existing (maybe).

The more I think about this, the more it sounds like SIA vs. SSA, with this post arguing for SIA.

• Orphan

The facts do change if you discard your specification for the set of worlds, and are trying to figure out which set of worlds you exist in.

You’re positing additional prior knowledge in order to make this information meaningful; this information is only meaningful in relation to that prior knowledge.

• Silent Cal

Are you objecting to the fact that I’ve made a specification of the possible worlds, or to the particular set of worlds I’ve specified?

It sounds to me like your objection is the former. The thing is, to get any answer at all, you have to have a prior, which is a probability distribution over possible worlds. Now, I spoke in terms of proportions of a set of worlds that all exist instead of in terms of probabilities,  but the results will be the same. I find it helps my intuition to imagine that all of the worlds exist, but the conclusions are the same if you change it to “We know that the world has to have ten planets; there is a 50% prior probability that it is an X world, and a 50% prior probability that it is a Y world” and so on.

If you’re objecting to my particular specification, it clearly was cooked up to get the result I wanted, but it does demonstrate that observing that one’s planet has life can make a difference for at least some priors. Moreover, my intuition says that this would have an effect for most real cosmological priors. I don’t have a formal argument for this now, but at the very least, this would mean that the relevant debate should be over what cosmological priors are like.

You have taken E to mean “planet 3 has life”, and get P(E) == 0.55.  Opposing philosophers would say that E means “I am on planet 3”, in which case P(E) = 0.1 and the update gives you no information.

I am not saying that one philosophy or the other is right, only that the specification you gave doesn’t lay the groundwork for doing Bayesian statistics.  That’s interesting in itself, in my response to Katja I talked vaguely about the need for “priors”, but you just gave seemingly complete priors and it still doesn’t satisfy me.

I think there is a fundamental metaphysical choice here, and I don’t know which way to choose.  (P.S. what are SSA and SIA?)

• Silent Cal

Good point.

Let’s make E be “Planet 3 has life” and F be “I am on planet 3”. We observe both E and F. P(X | E) and P(Y | E) are uncontroversially as I calculated. The question, I guess, is how to handle F. If P(F) is taken to be 0.1, then updating on both E and F will be the same as updating just on F (as it clearly should be), and bring you back to P(X) = 0.5, P(Y) = 0.5, essentially because P(F | E, Y) is one and P(F | E, X) is 0.1. To say P(F) = 0.1, though, you need to identify “I” with “a randomly chosen observer from the multiverse” or “a randomly chosen observer from a randomly chosen world” or some such. But note that if we choose “a randomly chosen observer from the multiverse”, then the relevant P(X) is the prior probability that “a randomly chosen observer from the multiverse” is in an X world, which is 0.9090, so the calculation ends up with the same result. If we choose “a randomly chosen observer from a randomly chosen world”, though, we get P(X | F) = .5. This is precisely the difference between SIA and SSA:

http://meteuphoric.wordpress.com/anthropic-principles/

I wonder if there’s a good deep reason why inferring “This particular planet has life” gives the SIA answer, while inferring “some planet has life” gives the SSA answer.

I think I am with you, but in a different language.  If your multiverse plays the role of a sample space in the Kolmogorov axioms, then E is a well defined subset of it.  That’s good.  But F=”I am on planet 3″ is not, because the definition of the sample space does not involve “me”.

Now we must expand the sample space beyond the initial multiverse.  Some ways of expanding it correspond then to SSA and others to SIA (thank you for the link BTW, it will take me a while to absorb it all.)

It is quite similar to SIA vs. SSA, except that SSA is only in favor of treating the first information you have about your existence as not evidence – if you learn later that you are human for instance, SSA usually recommends updating on that in the same way as SIA does. Treating ‘I observe X about me’ as ‘X is true of someone’ means that you never update except when there is a possible world in which nobody has X. You can get the same result by using SSA with the narrowest possible reference class, changing the reference class to be narrower every time you get new info.

• LR

The stipulation you make in the beginning that “half the worlds are X worlds and half are Y worlds” is what drives this result. I think the artificiality of this assumption is what causes the result to feel wrong at first.

If there are 11 total possible planets that you can be on with equal probability, and 10 of them happen to be in world X, it’s pretty intuitive to think you’re in world X with probability 10/11 (the .90909…) result you got above.

Of course in real life the distribution of possible worlds and their percentage of life-containing planets is likely much more complicated and currently unknown (unknowable?). I think this is what is driving your visceral since of incongruity.

• cas

hurry up and finish ur book mang

• http://entitledtoanopinion.wordpress.com TGGP

As Robin mentioned in his post where he said he was writing a book, he now has co-bloggers. This post was written by one of his co-bloggers, Katja Grace.

What you have written here sounds very intuitive to me.

• Arch1

I tentatively believe

P(Q|there is life on this planet) = P(Q|there exists a planet with life)

because I cannot see how “there is life on this planet” (“this planet” denoting the planet I, a life form, am on) sheds further light on the likelihood of Q being true than “there exists a planet with life.”   That said, I can’t prove it, so I think I don’t yet fully grok the situation yet.

It would help greatly if someone who DISbelieves the above equation would assume specific values for all relevant quantities (Q’s a priori probability, P(life on arb planet | Q), P(life on arb planet |~Q), N, etc), and then explicitly calculate numerical values for the LHS and the RHS of the above equation.

• Silent Cal

See my comment above (the one that begins “I have an example that I think might clarify things”). Q can be translated into the terms of that comment as “I am in an X world.”

• Arch1

Thanks Silent Cal, that actually changed my mind (I can be taught!).   I now think that the confusion revolves around the fact that Q (which I had been thinking of as an assertion about a universe) is, given Katja’s problem statement, really an assertion about a universe-as-experienced-by-an-observer.

So when computing P(Q) via Bayes’ formula we need to compute ratios of various kinds of experienced-universes, not ratios of various kinds of universes.  In other words, when counting universes we must weight them based on their respective number of observers.

If we assume that the number of intelligent observers in a universe is proportional to the number of planets w/ life, conditioning on “this planet has life” (or “planet 3 has life”, or, what I believe to be equivalent, “a randomly-chosen planet has life”) effectively weights each universe in just this way.

One can dream up variants of Katja’s scenario for which conditioning on “there exists a planet with life” (which effectively weights each universe equally independent of its number of observers) is the right approach, but given Katja’s scenario, together w/ the above assumption, I agree it makes sense to condition on a given planet having life.

• dmytryl

Well, “This planet has life”
has the information that e.g. planet in a zone with liquid water has
life, the information we are using when looking for other planets that may have life. Whereas “Some planet has life” would seem to strip that information unless you put all that you put what you know about ‘this’ into ‘some’. I have a feeling that the argument is entirely about semantics; the ‘some’ may effectively mean ‘this’.

• Jonathan Colvin

Are we talking about the “this universe” vs. “some universe” objection to anthropic fine tuning? For example  http://arxiv.org/ftp/arxiv/papers/0802/0802.4013.pdf

I’ve always found the “this universe” objection highly unconvincing, because it appears no different than the clearly mistaken “this planet” objection (the “this planet” objection says that an ensemble of planets can not explain why *this* planet is anthropic). For example: http://www.apologeticsinthechurch.com/uploads/7/4/5/6/7456646/this_universe.pdf