Tag Archives: Anthropics

Suspicious arguments regarding cow counting

People sometimes think that the doomsday argument is implausible because it always says we are more likely to die out sooner than our other reasoning suggests, regardless of the situation. There’s something dubious about an argument that has the same conclusion about the world regardless of any evidence about it. Nick Bostrom paraphrases, “But isn’t the probability that I will have any given rank always lower the more persons there will have been? I must be unusual in some respects, and any particular rank number would be highly improbable; but surely that cannot be used as an argument to show that there are probably only a few persons?” (he does not agree with this view).

That this reasoning is wrong is no new insight. Nick explains for instance that in any given comparison of different length futures, the doomsday reasoning doesn’t always give you the same outcome. You might have learned that your birth rank ruled out the shorter future. It remains the case though that the shift from whatever you currently believe to what the doomsday argument tells you to believe is always one toward shorter futures. I think it is this that seems fishy to people.

I maintain that the argument’s predictable conclusion is not a problem at all, and I would like to make this vivid.

Once a farmer owned a group of cows. He would diligently count them, to ensure none had escaped, and discover if there were any new calves. He would count them by lining them up and running his tape measure along the edge of the line.

“One thousand cows” he exclaimed one day. “Fifty new calves!”

His neighbour heard him from a nearby field, and asked what he was talking about. The farmer held out his tape measure. The incredulous neighbour explained that since cows are more than an inch long, his figures would need some recalculation. Since his cows were about five foot long on average, the neighbour guessed he would need to divide his number by 60. But the farmer quickly saw that this argument must be bogus. If his neighbour was right, whatever number of cows he had the argument would say he had fewer. What kind of argument would that be?

A similar one to the Doomsday Argument’s claim that the future should always be shorter than we otherwise think. In such cases the claim is that your usual method of dealing with evidence is biased, not that there is some particular uncommon evidence that you didn’t know about.

Similarly, the Self Indication Assumption‘s ‘bias’ toward larger worlds is taken as reason against it. Yet it is just a claim that our usual method is biased toward small worlds.

On the Anthropic Trilemma

Eliezer’s Anthropic Trilemma:

So here’s a simple algorithm for winning the lottery: Buy a ticket.  Suspend your computer program just before the lottery drawing – which should of course be a quantum lottery, so that every ticket wins somewhere.  Program your computational environment to, if you win, make a trillion copies of yourself, and wake them up for ten seconds, long enough to experience winning the lottery.  Then suspend the programs, merge them again, and start the result.  If you don’t win the lottery, then just wake up automatically. The odds of winning the lottery are ordinarily a billion to one.  But now the branch in which you win has your “measure”, your “amount of experience”, temporarily multiplied by a trillion.  So with the brief expenditure of a little extra computing power, you can subjectively win the lottery – be reasonably sure that when next you open your eyes, you will see a computer screen flashing “You won!”  As for what happens ten seconds after that, you have no way of knowing how many processors you run on, so you shouldn’t feel a thing.

See the original post for assumptions, what merging minds entails etc. He proposes three alternative bullets to bite: accepting that this would work, denying that there is “any meaningful sense in which I can anticipate waking up as myself tomorrow, rather than Britney Spears” so undermining any question about what you should anticipate, and Nick Bostrom’s response, paraphrased by Eliezer:

…you should anticipate winning the lottery after five seconds, but anticipate losing the lottery after fifteen seconds. To bite this bullet, you have to throw away the idea that your joint subjective probabilities are the product of your conditional subjective probabilities.  If you win the lottery, the subjective probability of having still won the lottery, ten seconds later, is ~1.  And if you lose the lottery, the subjective probability of having lost the lottery, ten seconds later, is ~1.  But we don’t have p(“experience win after 15s”) = p(“experience win after 15s”|”experience win after 5s”)*p(“experience win after 5s”) + p(“experience win after 15s”|”experience not-win after 5s”)*p(“experience not-win after 5s”).

I think I already bit the bullet about there not being a meaningful sense in which I won’t wake up as Britney Spears. However I would like to offer a better, relatively bullet biting free solution.

First notice that you will have to bite Bostrom’s bullet if you even accept Eliezer’s premise that arranging to multiply your ‘amount of experience’ in one branch in the future makes you more likely to experience that branch. Call this principle ‘follow-the-crowd’ (FTC). And let’s give the name ‘blatantly obvious principle’ (BOP) to the notion that P(I win at time 2) is equal to P(I win at time 2|I win at time 1)P(I win at time 1)+P(I win at time 2|I lose at time 1)P(I lose at time 1). Bostrom’s bullet is to deny BOP.

We can set aside the bit about merging brains together for now; that isn’t causing our problem. Consider a simpler and smaller (for the sake of easy diagramming) lottery setup where after you win or lose you are woken for ten seconds as a single person, then put back to sleep and woken as four copies in the winning branch or one in the losing branch. See the diagram below. You are at Time 0 (T0). Before Time 1 (T1) the lottery is run, so at T1 the winner is W1 and the loser is L1. W1 is then copied to give the multitude of winning experiences at T2, while L2 remains single.

Now using the same reasoning as you would to win the lottery before, FTC, you should anticipate an 80% chance of winning the lottery at T2. There is four times as much of your experience winning the lottery as not then. But BOP says you still only have a fifty percent chance of being a lottery winner at T2:

P(win at T2) = P(win at T2|win at T1).P(win at T1)+P(win at T2|lose at T1).P(lose at T1) = 1 x 1/2 + 0 x 1/2 = 1/2

FTC and BOP conflict. If you accept that you should generally anticipate futures where there are more of you more strongly, it looks like you accept that P(a) does not always equal P(a|b)P(b)+P(a|-b)P(-b). How sad.

Looking at the diagram above, it is easy to see why these two methods of calculating anticipations disagree.  There are two times in the diagram that your future branches, once in a probabilistic event and once in being copied. FTC and BOP both treat the probabilistic event the same: they divide your expectations between the outcomes according to their objective probability. At the other branching the two principles do different things. BOP treats it the same as a probabilistic event, dividing your expectation of reaching that point between the many branches you could continue on. FTC treats it as a multiplication of your experience, giving each new branch the full measure of the incoming branch. Which method is correct?

Neither. FTC and BOP are both approximations of better principles. Both of the better principles are probably true, and they do not conflict.

To see this, first we should be precise about what we mean by ‘anticipate’. There is more than one resolution to the conflict, depending on your theory of what to anticipate: where the purported thread of personal experience goes, if anywhere. (Nope, resolving the trilemma does not seem to answer this question).

Resolution 1: the single thread

The most natural assumption seems to be that your future takes one branch at every intersection. It does this based on objective probability at probabilistic events, or equiprobably at copying events. It follows BOP. This means we can keep the present version of BOP, so I shall explain how we can do without FTC.

Consider diagram 2. If your future takes one branch at every intersection, and you happen to win the lottery, there are still many T2 lottery winners who will not be your future. They are your copies, but they are not where your thread of experience goes. They and your real future self can’t distinguish who is actually in your future, but there is some truth of the matter. It is shown in green.

Diagram 2

Now while there are only two objective possible worlds, when we consider possible paths for the green thread there are five possible worlds (one shown in diagram 2). In each one your experience follows a different path up the tree. Since your future is now distinguished from other similar experiences, we can see the weight of your experience at Tin a world where you win is no greater than the weight in a world where you lose, though there are always more copies who are not you in the world where you win.

The four worlds where your future is in a winning branch are each only a quarter as likely as one where you lose, because there is a fifty percent chance of you reaching W1, and after that a twenty five percent chance of reaching a given W2. By the original FTC reasoning then, you are equally likely to win or lose. More copies just makes you less certain exactly where it will be.

I am treating the invisible green thread like any other hidden characteristic. Suppose you know that you are and will continue to be the person with the red underpants, though many copies will be made of you with green underpants. However many extra copies are made, a world with more of them in future should not get more of your credence, even if you don’t know which future person actually has the red pants. If you think of yourself as having only one future, then you can’t also consider there to be a greater amount of your experience when there are a lot of copies. If you did anticipate experiences based on the probability that many people other than you were scheduled for that experience, you would greatly increase the minuscule credence you have in experiencing being Britney Spears when you wake up tomorrow.

Doesn’t this conflict with the use of FTC to avoid the Bolzmann brain problem, Eliezer’s original motivation for accepting it? No. The above reasoning means there is a difference between where you should anticipate going when you are at T0, and where you should think you are if you are at T2.

If you are at T0 you should anticipate a 50% chance of winning, but if you are at T2 you have an 80% chance of being a winner. Sound silly? That’s because you’ve forgotten that you are potentially talking about different people. If you are at T2, you are probably not the future of the person who was at T0, and you have no way to tell. You are a copy of them, but their future thread is unlikely to wend through you. If you knew that you were their future, then you would agree with their calculations.

That is, anyone who only knows they are at T2 should consider themselves likely to have won, because there are many more winners than losers. Anyone who knows they are at T2 and are your future, should give even odds to winning. At T0, you know that the future person whose measure you are interested in is at T2 and is your future, so you also give even odds to winning.

Avoiding the Bolzmann brain problem requires a principle similar to FTC which says you are presently more likely to be in a world where there are more people like you. SIA says just that for instance, and there are other anthropic principles that imply similar things. Avoiding the Bolzmann brain problem does not require inferring from this that your future lies in worlds where there are more such people. And such an inference is invalid.

This is exactly the same as how it is invalid to infer that you will have many children from the fact that you are more likely to be from a family with many children. Probability theory doesn’t distinguish between the relationship between you and your children and the relationship between you and your future selves.

Resolution 2

You could instead consider all copies to be your futures. Your thread is duplicated when you are. In that case you should treat the two kinds of branching differently, unlike BOP, but still not in the way FTC does. It appears you should anticipate a 50% chance of becoming four people, rather than an 80% chance of becoming one of those people. There is no sense in which you will become one of the winners rather than another. Like in the last case, it is true that if you are presently one of the copies in the future, you should think yourself 80% likely to be a winner. But again ‘you’ refers to a different entity in this case to the one it referred to before the lottery. It refers to a single future copy. It can’t usefully refer to a whole set of winners, because the one considering it does not know if they are part of that set or if they are a loser. As in the last case, your anticipations at T0 should be different from your expectations for yourself if you know only that you are in the future already.

In this case BOP gives us the right answer for the anticipated chances of winning at T0. However it says you have a 25% chance of becoming each winner at T2 given you win at T1, instead of 100% chance of becoming all of them.

Resolution 3:

Suppose that you want to equate becoming four people in one branch as being more likely to be there. More of your future weight is there, so for some notions of expectation perhaps you expect to be there. You take ‘what is the probability that I win the lottery at T1?’ to mean something like ‘what proportion of my future selves are winning at T1?’. FTC gives the correct answer to this question – you aren’t especially likely to win at T1, but you probably will at T2. Or in the original problem, you should expect to win after 5 seconds and lose after 15 seconds, as Nick Bostrom suggested. If FTC is true, then we must scrap BOP. This is easier than it looks because BOP is not what it seems.

Here is BOP again:

P(I win at T2) is equal to P(I win at T2|I win at T1)P(I win at T1)+P(I win at T2|I lose at T1)P(I lose at T1)

It looks like a simple application of

P(a) = P(a|b)P(b)+P(a|-b)P(-b)

But here is a more extended version:

P(win at 15|at 15) = P(win an 15|at 15 and came from win at 5)P(win at 5|at 5)+P(win at 15|at 15 and came from loss at 5)P(lose at 5|at 5)

This is only equal to BOP if the probability of having a win at 5 in your past when you are at 15 is equal to the probability of winning at 5 when you are at 5. To accept FTC is to deny that. FTC says you are more likely to find the win in your past than to experience it because many copies are descended from the same past. So accepting FTC doesn’t conflict with P(a) being equal to P(a|b)P(b)+P(a|-b)P(-b), it just makes BOP an inaccurate application of this true principle.

In summary:

1. If your future is (by definitional choice or underlying reality) a continuous non-splitting thread, then something like SIA should be used instead of FTC, and BOP holds. Who you anticipate being differs from who you should think you are when you get there. Who you should think you are when you get there remains as something like SIA and avoids the Bolzmann brain problem.

2. If all your future copies are equally your future, you should anticipate becoming a large number of people with the same probability as that you would have become one person if there were no extra copies. In which case FTC does not hold, because you expect to become many people with a small probability instead of one of those many people with a large probability. BOP holds in a modified form where it doesn’t treat being copied as being sent down a random path. But if you want to know what a random moment from your future will hold, a random moment from T1 is more likely to include losing than a random moment from T2. For working out what a random T2 moment will hold, BOP is a false application of a correct principle.

3. If for whatever reason you conceptualise yourself as being more likely to go into future worlds based on the number of copies of you there are in those worlds, then FTC does hold, but BOP becomes false.

I think the most important point is that the question of where you should anticipate going need not have the same answer as where a future copy of you should expect to be (if they don’t know for some reason). A future copy who doesn’t know where they are should think they are more likely to be in world where there are many people like themselves, but you should not necessarily think you are likely to go into such a world. If you don’t think you are as likely to go into such a world, then FTC doesn’t hold. If you do, then BOP doesn’t hold.

It seems to me the original problem uses FTC while assuming there will be a single thread, thereby making BOP look inevitable. If the thread is kept, FTC should not be, which can be conceptualised as in either of resolutions 1 or 2. If FTC is kept, BOP need not be, as in resolution 3. Whether you keep FTC or BOP will give you different expectations about the future, but which expectations are warranted is a question for another time.

Person moments make sense of anthropics

Often people think that various forms of anthropic reasoning require you to change your beliefs in ways other than conditionalizing on evidence. This is false, at least in the cases I know of. I shall talk about Frank Arntzenius‘ paper Some Problems for Conditionalization and Reflection [gated] because it explains the issue well, though I believe his current views agree with mine.

He presents five thought experiments: Two Roads to Shangri La, The Prisoner, John Collins’s Prisoner, Sleeping Beauty and Duplication. In each of them, it seems the (arguably) correct answer violates van Fraassen’s reflection principle, which basically says that if you expect to believe something in the future without having been e.g. hit over the head between now and then, you should believe it now. For instance the thirder position in Sleeping Beauty seems to violate this principle because before the experiment Beauty believes there is a fifty percent chance of heads, and that when she wakes up she will think there is a thirty three percent chance. Arntzenius argued that these seemingly correct answers really are the correct ones, and claimed that they violate the reflection principle because credences can evolve in two ways other than by conditionalization.

First he said credences can shift, for instance through time. I know that tomorrow I will have a higher credence in it being Monday than I do today, and yet it would not be rational for me to increase my credence in it being Monday now on this basis. They can also ‘spread out’. For instance if you know you are in Fairfax today, and that tomorrow a perfect replica of your brain experiencing Fairfax will be made and placed in a vat in Canberra, tomorrow your credence will go from being concentrated in Fairfax to being spread between there and Canberra. This is despite no damage having been done to your own brain. As Arntzenius pointed out, such an evolution of credence looks like quite the opposite of conditionalization, since conditionalization consists of striking out possibilities that your information excludes – it never opens up new possibilities.

I agree that beliefs should evolve in these two ways. However they are both really conditionalization, just obscured. They make sense as conditionalization when you think of them as carried out by different momentary agents, based on the information they infer from their connections to other momentary agents with certain beliefs (e.g. an immediately past self).

Normal cases can be considered this way quite easily. Knowing that you are the momentary agent that followed a few seconds after an agent who knew a certain set of facts about the objective world, and who is (you assume) completely trustworthy, means you can simply update the same prior with those same facts and come to the same conclusion. That is, you don’t really have to do anything. You can treat a stream of moments as a single agent. This is what we usually do.

However sometimes being connected in a certain way to another agent does not make everything that is true for them true for you. Most obviously, if they are a past self and know it is 12 o clock, your connection via being their one second later self means you should exclude worlds where you are not at time 12:00:01. You have still learned from your known relationship to that agent and conditionalized, but you have not learned that what is true of them is true of you, because it isn’t. This is the first way Arntzenius mentioned that credences seem to evolve through time not by by conditionalization.

The second way occurs when one person-moment is at location X, and another person moment has a certain connection to the person at X, but there is more than one possible connection of that sort. For instance when two later people both remember being an earlier person because the earlier person was replicated in some futuristic fashion. Then while the earlier person moment could condition on their exact location, the later one must condition on being in one of several locations connected that way to the earlier person’s location, so their credence spreads over more possibilities than that of the earlier self. If you call one of these later momentary agents the same person as the earlier one, and say they are conditionalizing, it seems they are doing it wrong. But considered as three different momentary people learning from their connections they are just conditionalizing as usual.

What exactly the later momentary people should believe is a matter of debate, but I think that can be framed entirely as a question of what their state spaces and priors look like.

Momentary humans almost always pass lots of information from one to the next, chronologically along chains of memory through non-duplicated people, knowing their approximate distance from one another. So most of the time they can treat themselves as single units who just have to update on any information coming from outside, as I explained. But conditionalization is not specific to these particular biological constructions; and when it is applied to information gained through other connections between agents, the resulting time series of beliefs within one human will end up looking different to that in a chain with no unusual extra connections.

This view also suggests that having cognitive defects, such as memory loss, should not excuse anyone from having credences, as for instance Arntzenius argued it should in his paper Reflections on Sleeping Beauty: “in the face of forced irrational changes in one’s degrees of belief one might do best simply to jettison them altogether”. There is nothing special about credences derived from beliefs of a past agent you identify with. They are just another source of information. If the connection to other momentary agents is different to usual, for instance through forced memory loss, update on it as usual.

Agreement on anthropics

Aumann’s agreement theorem says that Bayesians with common priors who know one another’s posteriors must agree. There’s no apparent reason this shouldn’t apply to posteriors arrived at using indexical information. This does not mean that you and I should both believe we are as likely to be the author of this blog, but that we should agree on the chances that I am.

The Self-Sampling Assumption (SSA) does not allow for this agreement between people with different reference classes, as I shall demonstrate. Consider the figure below. Suppose A people and B people both begin with an equal prior over the two worlds. Everyone knows their type (A or B), but other than that they do not know their location. For instance an A person may be in any of eight places, as far as they know. A people consider their reference class to be A people only. B people consider their reference class to be B people only. The people who are standing next to each other in the diagram meet and exchange their knowledge. For instance an A person meeting a B person will learn that the B person is a B person, and that they don’t know anything much else.

When A people meet B people, they both come to know what the other person’s posterior is. For instance an A person who meets a B person knows that the B person doesn’t know anything except that they are a B person who met an A person. From this the A person can work out the B person’s posterior over which world they are in.

Suppose everyone uses SSA. When an A person and a B person meet, the A people come to think they are four times as likely to be in World 1. This is because in world two, only a quarter of A people meet a B person, whereas in world 1 they all do. The B people they meet cannot agree – in either world they expected to talk with an A person, and for that A person to be pretty sure they are in world 1. So despite knowing one another’s posteriors and having common priors over which world exists, the A and B people who meet must disagree. Not only on one another’s locations within the world, but over which world they are in*.

An example of this would be a husband and wife celebrating their wedding in a Chinese town with poor census data and an ongoing gender gap. The husband exclaims ‘wow, I am a husband! The disparity between gender populations in this town is probably smaller than I thought’. His wife expected in any case that she would end up with a husband who would make this inference from their marriage, and so cannot update and agree with him. Notice that neither partner need think the other has chosen the ‘wrong’ reference class in any way, it might be the reference class they would have chosen were they in that alternative indexical position.

In both of these cases the Self-Indication Assumption (SIA) allows for perfect agreement. Recall SIA weights the probability of worlds by the number of people in them in your situation. When A and B knowingly communicate, they are in symmetrical positions – either side of a communicating A and B pair. Both parties weight their hypotheses by the number of such pairs, and so they agree. Incidentally, when they first found out that they existed, and later when they learned their type, they did disagree. Communicating resolves this, instead of creating a disagreement as with SSA.

*If this does not seem bad enough, they each agree that the other person reasoned as well as they did.

Another implausible implication of this application of SSA is that you will come to agree with creatures that are more similar to you, even if you are certain that a given creature inside your reference class is identical to one outside your reference class in every aspect of its data collection and inference abilities.

The Unpresumptuous Philosopher

Nick Bostrom showed that either position in Extreme Sleeping Beauty seems absurd, then gave a third option. I argued that his third option seems worse than either of the original pair. If I am right there that  the  case  for Bayesian  conditioning without updating on  evidence  fails, we have  a  choice  of  disregarding  Bayesian  conditioning in at least some situations,  or  distrusting the aversion to extreme updates as in Extreme Sleeping Beauty. The latter seems the necessary choice, given the huge disparity in evidence supporting Bayesian conditioning and that supporting these particular intuitions about large updates and strong beliefs.

Notice that both the Halfer and Thirder positions on Extreme Sleeping Beauty have very similar problems. They are seemingly opposed by the same intuitions against extreme certainty in situations where we don’t feel certain, and extreme updates in situations where we hardly feel we have any evidence. Either before or after discovering you are in the first waking, you must be very sure of how the coin came up. And between ignorance of the day and knowledge, you must change your mind drastically. If we must choose one of these positions then, it is not clear which is preferable on these grounds alone.

Now notice that the Thirder position in Extreme Sleeping Beauty is virtually identical to SIA and consequently the Presumptuous Philosopher’s position (as Nick explains, p64). From Anthropic Bias:

 

The Presumptuous Philosopher

39It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories, T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite, and there are a total of a trillion, trillion observers in the cosmos. According to T2, the world is very, very, very big but finite, and there are a trillion, trillion, trillion observers. The super-duper symmetry considerations are indifferent between these two theories. Physicists are preparing a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: “Hey guys, it is completely unnecessary for you to do the experiment, because I can already show to you that T2 is about a trillion times more likely to be true than T1 (whereupon the philosopher [...] appeals to SIA)!”

The Presumptuous Philosopher is like the Extreme Sleeping Beauty Thirder because they are both in one of two possible worlds with a known probability of existing, one of which has a much larger population than the other. They are both wondering which of these worlds they are in.

Is the Presumptuous Philosopher really so presumptuous? Analogous to the Extreme Sleeping Beauty Halfer then shall be the Unpresumptuous Philosopher. When the Unpresumptuous Philosopher  learns  there  are  a  trillion  times  as many  observers  in T2  she  remains  cautiously unmoved. However, when the physicists later discover where in the cosmos our planet is under  both  theories,  the  Unpresumptuous  Philosopher  becomes  virtually  certain  that  the sparsely populated T1 is correct while the Presumptuous Philosopher hops back on the fence.

The Presumptuous Philosopher is often chided for being sure the universe is infinite, given there is some chance of an infinite universe existing. It should be noted that this is only as long as he cannot restrict his possible locations in it to any finite region. The Unpresumptuous Philosopher is uncertain under such circumstances. However she believes with probability one that we are in a finite world if she knows her location is within any finite region. For instance if she knows the age  of  her spatially finite universe  she  is  certain  that  it will  not  continue  for  infinitely  long. Here her presumptuous friend is quite unsure.

Statue of an unknown Cynic philosopher from th...

This philosopher has a nice perch now, but where will he go if evidence moves him? Photo: Yair Haklai

It seems to me that as the two positions on Extreme Sleeping Beauty are as unintuitive as each other, the two philosophers seem as presumptuous as each other. The accusation of inducing a large probability shift and encouraging ridiculous certainty is hardly an argument that can be used against the SIA-Thirder-Presumptuous Philosopher position in favor of the SSA-Halfer-Unpresumptuous Philosopher side. Since the Presumptuous Philosopher is usually considered the big argument against SIA, and not considered an argument against SSA at all, an update in favor of SIA is in order.