Tag Archives: Anthropics

Sleeping Beauty should remain pure

Sleeping Beauty

Image via Wikipedia

Consider the Sleeping Beauty Problem. Sleeping Beauty is put to sleep on Sunday night. A coin is tossed. If it lands heads, she is awoken once on Monday, then sleeps until the end of the experiment. If it lands tails, she is woken once on Monday, drugged to remove her memory of this event, then awoken once on Tuesday, before sleeping till the end of the experiment. The awakenings during the experiment are indistinguishable to Beauty, so when she awakens she doesn’t know what day it is or how the coin fell. The question is this: when Beauty wakes up on one of these occasions, how confident should she be that heads came up?

There are two popular answers, 1/2 and 1/3. However virtually everyone agrees that if Sleeping Beauty should learn that it is Monday, her credence in Tails should be reduced by half, from whatever it was initially. So ‘Halfers’ come to think heads has a 2/3 chance, and ‘Thirders’ come to think they heads is as likely as tails. This is the standard Bayesian way to update, and is pretty uncontroversial.

Now consider a variation on the Sleeping Beauty Problem where Sleeping Beauty will be woken up one million times on  tails  and  only once on heads.  Again, the  probability  you  initially  put  on  heads  is  determined  by  the reasoning  principle  you  use,  but  the  probability  shift  if you are to  learn  that  you are in the first awakening will be the same either way. You will have to shift your odds by a million to one toward heads. Nick Bostrom points out that in this scenario, either before or after  this  shift you will have  to be extremely certain either of heads or of tails, and that such extreme certainty seems intuitively unjustified, either before or after knowing you are experiencing the first wakening.

Extreme Sleeping Beauty wakes up a million times on tails or once on heads. There is no choice of 'a' which doesn't lead to extreme certainty either before or after knowing she is at her first waking.

However the only alternative to this certainty is for Sleeping Beauty to  keep  odds  near  1:1 both  before  and  after  she learns she is at her first waking. This entails apparently giving up Bayesian conditionalization. Having excluded 99.9999% of the situations she may have been in where tails would have come up, Sleeping Beauty retains her previous credence in tails.

This is what Nick proposes doing however: his ‘hybrid model’ of  Sleeping  Beauty.  He  argues  that  this  does  not  violate  Bayesian conditionalization in cases such as this because Sleeping Beauty is in different indexical positions before and after knowing that she is at her first waking, so her observer-moments (thick time-slices of a person) at the different times need not agree.

I disagree, as I shall explain. Briefly, the disagreement between different observer-moments should not occur and is deeper than it first seems, the existing arguments against so called non-indexical conditioning also fall against the hybrid model, and Nick fails in his effort to show that Beauty won’t predictably lose money gambling.

Is hybrid Beauty Bayesian?

Nick argues first that a Bayesian may accept having 50:50 credences both before and after knowing that it is Monday, then claims that one should do so, given the absurdities of the Extreme Sleeping Beauty problem above and variants of it. His argument for the first part is as follows (or see p10). There are actually five rather than three relevant indexical positions in the Sleeping Beauty Problem. The extra two are Sleeping Beauty after she knows it is Monday under both heads and tails. He explains that it is the ignorant Beauties who should think the chance of Heads is half, and the informed Mondayers who should think the chance of Heads is still half conditional on it being Monday. Since these are observer-moments in different locations, he claims there is no inconsistency, and Bayesian conditionalization is upheld (presumably meaning that each observer-moment has a self-consistent set of beliefs).

He generalizes that one need not believe P(X) = A, just because one used to think P(X|E) = A and one just learned E. For that to be true the probability of X given that you don’t know E but will learn it would have to be equal to the probability of X given that you do know E but previously did not. Basically, conditional probabilities must not suddenly change just as you learn the conditions hold.

Why exactly a conditional probability might do this is left to the reader’s imagination. In this case Nick infers that it must have happened somehow, as no apparently consistent set of beliefs will save us from making strong updates in the Extreme Sleeping Beauty case and variations on it.

If receiving new evidence gives one leave to break consistency with any previous beliefs on grounds that ones conditional credences may have  changed with ones location, there would be little left of Bayesian conditioning in practice. Normal Bayesian conditioning is remarkably successful then, if we are to learn that a huge range of other inferences were equally well supported in any case of its use.

Nick’s calling Beauty’s unchanging belief in even odds consistent for a Bayesian is not because these beliefs meet some sort of Bayesian constraint, but because he is assuming there are not constraints on the relationship between the beliefs of different Bayesian observer-moments. By this reasoning, any set of internally consistent belief sets can be ‘Bayesian’. In the present case we chose our beliefs by a powerful disinclination toward making certain updates. We should admit it is this intuition driving our probability assignments then, and not call it a variant of Bayesianism. And once we have stopped calling it Bayesianism, we must ask if the intuitions that motivate it really have the force behind them that the intuitions supporting Bayesianism in temporally extended people do.

Should observer-moments disagree?

Nick’s argument works by distinguishing every part of Beauty with different information as a different observer. This is used to allow them to safely hold inconsistent beliefs with one another. So this argument is defeated if Bayesians should agree with one another, when they know one anothers’ posteriors, share priors and know one another to be rational. Aumann‘s agreement theorem does indeed show this. There is a slight complication in that the disagreement is over probabilities conditional on different locations, but the locations are related in a known way, so it appears they can be converted to disagreement over the same question. For instance past Beauty has a belief about the probability of heads conditional on her being followed by a Beauty who knows it is Monday, and Future Beauty has a belief conditional on the Beauty in her past being followed by one who knows it is Monday (which she now knows it was).

Intuitively, there is still only one truth, and consistency is a tool for approaching it. Dividing people into a lot of disagreeing parts so that they are consistent by some definition is like paying someone to walk your pedometer in order to get fit.

Consider the disagreement between observer-moments in more detail. For  instance,  suppose  before  Sleeping  Beauty  knows what  day  it  is  she assigns  50  percent  probability  to  heads  having  landed.  Suppose  she  then  learns  that  it  is Monday, and still believes she has a 50 percent chance of heads. Lets call the ignorant observer-moment Amy and the later moment who knows it is Monday Betty.

Amy and Betty do not merely come to different conclusions with different indexical  information. Betty believes Amy was wrong, given only the information Amy had. Amy thought that conditional on being followed by an observer-moment who knew it was Monday, the chances of Heads were 2/3. Betty knows this, and knows nothing else except that Amy was indeed followed by an observer-moment who knows it is Monday, yet believes the chances of heads are in fact half. Betty agrees with the reasoning principle Amy used. She also agrees with Amy’s priors. She agrees that were she in Amy’s position, she would have the same beliefs Amy has. Betty also knows that though her location in the world  has changed, she is in the same objective world as Amy – either Heads or Tails came up for both of them. Yet Betty must knowingly disagree with Amy about how likely that  world is to be one where Heads landed. Neither Betty nor Amy can argue that her belief about their shared world is more likely to be correct than the other’s. If this principle is even a step in the right direction then, these observer-moments could do better by aggregating their apparently messy estimates of reality.

Identity with other unlikely anthropic principles

Though I don’t think Nick mentions it, the hybrid model reasoning is structurally identical to SSSA using the reference class of ‘people with exactly one’s current experience’, both before and after  receiving  evidence  (different  reference  classes  in  each  case since they have different information). In both cases every member of Sleeping Beauty’s reference class shares the same experience. This means the proportion of her reference class who share her current experiences is always one. This allows Sleeping Beauty to stick with the fifty percent chance given by  the coin, both before and after knowing she is in her first waking, without any interference from changing potential locations.

SSSA with such a narrow reference class is exactly analogous to non-indexical conditioning, where ‘I observe X’ is interpreted as ‘X is observed by someone in the world’. Under both, possible worlds where your experience occurs nowhere are excluded and all other worlds retain their prior probablities, normalized. Nick has criticised non-indexical conditioning because it leads to an inability to update on most evidence, thus prohibiting science for instance. Since most people are quite confident that it is possible to do science, they are implicitly confident that non-indexical conditioning is well off the mark. This implies that SSSA using the narrowest reference class is just as implausible, except that it may be more readily traded for SSSA with other reference classes when it gives unwanted results. Nick has suggested SSA should be used with a broader reference class for this reason (e.g. see Anthropic Bias p181), though he also supports using different reference classes at different times.

These reasoning principles are more appealing in the Extreme Sleeping Beauty case, because our intuition there is to not update on evidence. However if we pick different principles for different circumstances according to which conclusions suit us, we aren’t using those principles, we are using our intuitions. There isn’t necessarily anything inherently wrong with using intuitions, but when there are reasoning principles available that have been supported by a mesh of intuitively correct reasoning and experience, a single untested intuition would seem to need some very strong backing to compete.

Beauty will be terrible at gambling

It first seems that Hybrid Beauty can be Dutch-Booked (offered a collection of bets she would accept and which would lead to certain loss for her), which suggests she is being irrational. Nick gives an example:

Upon awakening, on both Monday and Tuesday,
before either knows what day it is, the bookie offers Beauty the following bet:

Beauty gets $10 if HEADS and MONDAY.
Beauty pays $20 if TAILS and MONDAY.
(If TUESDAY, then no money changes hands.)

On Monday, after both the bookie and Beauty have been informed that it is
Monday, the bookie offers Beauty a further bet:

Beauty gets $15 if TAILS.
Beauty pays $15 if HEADS.

If Beauty accepts these bets, she will emerge $5 poorer.

Nick argues that Sleeping Beauty should not accept the first bet, because the bet will have to be made twice if tails comes up and only once if heads does, so that Sleeping Beauty isn’t informed about which waking she is in by whether she is offered a bet. It is known that when a bet on A vs. B will be made more times conditional on A than conditional on B, it can be irrational to bet according to the odds you assign to A vs. B. Nick illustrates:

…suppose you assign credence 9/10 to the proposition that the trillionth digit in the decimal expansion of π is some number other than 7. A man from the city wants to bet against you: he says he has a gut feeling that the digit is number 7, and he offers you even odds – a dollar for a dollar. Seems fine, but there is a catch: if the digit is number 7, then you will have to repeat exactly the same bet with him one hundred times; otherwise there will just be one bet. If this proviso is specified in the contract, the real bet that is being offered you is one where you get $1 if the digit is not 7 and you lose $100 if it is 7.

However  in these cases the problem stems from the bet being paid out many times under one circumstance. Making extra bets that will never be paid out cannot affect the value of a set of bets. Imagine the aforementioned city man offered his deal, but added that all the bets other than your first one would be called off once you had made your first one. You would be in the same situation as if the bet had not included his catch to begin with. It would be an ordinary bet, and you should be willing to bet at the obvious odds. The same goes for Sleeping Beauty.

We can see this more generally. Suppose E(x) is the expected value of x, P(Si) is probability of situation i arising, and V(i) is the value to you if it arises. A bet consists of a set of gains or losses to you assigned to situations that may arise.

E(bet) = P(S1)*V(S1) + P(S2)*V(S2) + …

The City Man’s offered bet is bad because it has a large number of terms with negative value and relatively high probability, since they occur together rather than being mutually exclusive in the usual fashion. It is a trick because it is presented at first as if there were only one term with negative value.

Where bets will be written off in certain situations, V(Si) is zero in the terms corresponding to those situations, so the whole terms are also zero, and may as well not exist. This means the first bet Sleeping Beauty is offered in her Dutch-booking test should be made at the same odds as if she would only bet once on either coin outcome. Thus she should take the bet, and will be Dutch booked.


In sum, Nick’s hybrid model is not a new kind of Bayesian updating, but use of a supposed loophole where Bayesianism is supposed to have few requirements. There doesn’t even seem to be a loophole there however, and if there were it would be a huge impediment to most practical uses of updating. Reasoning principles which are arguably identical to the hybrid model in the relevant ways have been previously discarded by most due to their obstruction of science among other things.  Last, Sleeping Beauty really will lose bets if she adopts the hybrid model and is otherwise sensible.

SIA and the Two Dimensional Doomsday Argument

This post might be technical. Try reading this if I haven’t explained everything well enough.

When the Self Sampling Assumption (SSA) is applied to the Great Filter it gives something pretty similar to the Doomsday Argument, which is what it gives without any filter. SIA gets around the original Doomsday Argument. So why can’t it get around the Doomsday Argument in the Great Filter?

The Self Sampling Assumption (SSA) says you are more likely to be in possible worlds which contain larger ratios of people you might be vs.  people know you are not*.

If you have a silly hat, SSA says you are more likely to be in world 2 - assuming Worlds 1 and 2 are equally likely to exist (i.e. you haven't looked aside at your companions), and your reference class is people.

The Doomsday Argument uses the Self Sampling Assumption. Briefly, it argues that if there are many generations more humans, the ratio of people who might be you (are born at the same time as you) to people you can’t be (everyone else) will be smaller than it would be if there are few future generations of humans. Thus few generations is more likely than previously estimated.

An unusually large ratio of people in your situation can be achieved by a possible world having unusually few people unlike you in it or unusually many people like you, or any combination of these.


Fewer people who can't be me or more people who may be me make a possible world more likely according to SSA.

For instance on the horizontal dimension, you can compare a set of worlds which all have the same number of people like you, and different numbers of people you are not. The world with few people unlike you has the largest increase in probability.



The top row from the previous diagram. The Doomsday Argument uses possible worlds varying in this dimension only.

The Doomsday Argument is an instance of variation in the horizontal dimension only. In every world there is one person with your birth rank, but the numbers of people with future birth ranks differ.

At the other end of the spectrum you could be comparing  worlds with the same number of future people and vary the number of current people, as long as you are ignorant of how many current people there are.

The vertical axis. The number of people in your situation changes, while the number of others stays the same. The world with a lot of people like you gets the largest increase in probability.

This gives a sort of Doomsday Argument: the population will fall, most groups won’t survive.

The Self Indication Assumption (SIA) is equivalent to using SSA and then multiplying the results by the total population of people both like you and not.

In the horizontal dimension, SIA undoes the Doomsday Argument. SSA favours smaller total populations in this dimension, which are disfavoured to the same extent by SIA, perfectly cancelling.

[1/total] * total = 1
(in bold is SSA shift alone)

In vertical cases however, SIA actually makes the Doomsday Argument analogue stronger. The worlds favoured by SSA in this case are the larger ones, because they have more current people. These larger worlds are further favoured by SIA.

[(total – 1)/total]*total = total – 1

The second type of situation is relatively uncommon, because you will tend to know more about the current population than the future population. However cases in between the two extremes are not so rare. We are uncertain about creatures at about our level of technology on other planets for instance, and also uncertain about creatures at some future levels.

This means the Great Filter scenario I have written about is an in between scenario. Which is why the SIA shift doesn’t cancel the SSA Doomsday Argument there, but rather makes it stronger.

Expanded from p32 of my thesis.

*or observers you might be vs. those you are not for instance – the reference class may be anything, but that is unnecessarily complicated for the the point here.

SIA says AI is no big threat

Artificial Intelligence could explode in power and leave the direct control of humans in the next century or so. It may then move on to optimize the reachable universe to its goals. Some think this sequence of events likely.

If this occurred, it would constitute an instance of our star passing the entire Great Filter. If we should cause such an intelligence explosion then, we are the first civilization in roughly the past light cone to be in such a position. If anyone else had been in this position, our part of the universe would already be optimized, which it arguably doesn’t appear to be. This means that if there is a big (optimizing much of the reachable universe) AI explosion in our future, the entire strength of the Great Filter is in steps before us.

This means a big AI explosion is less likely after considering the strength of the Great Filter, and much less likely if one uses the Self Indication Assumption (SIA).

The large minimum total filter strength contained in the Great Filter is evidence for larger filters in the past and in the future. This means evidence against the big AI explosion scenario, which requires that the future filter is tiny.

SIA implies that we are unlikely to give rise to an intelligence explosion for similar reasons, but probably much more strongly. As I pointed out before, SIA says that future filters are much more likely to be large than small. This is easy to see in the case of AI explosions. Recall that SIA increases the chances  of hypotheses where there are more people in our present situation. If we precede an AI explosion, there is only one civilization in our situation, rather than potentially many if we do not. Thus the AI hypothesis is disfavored (by a factor the size of the extra filter it requires before us).

What the Self Sampling Assumption (SSA), an alternative principle to SIA, says depends on the reference class. If the reference class includes AIs, then we should strongly not anticipate such an AI explosion. If it does not, then we strongly should (by the doomsday argument). These are both basically due to the Doomsday Argument.

In summary, if you begin with some uncertainty about whether we precede an AI explosion, then updating on the observed large total filter and accepting SIA should make you much less confident in that outcome. The Great Filter and SIA don’t just mean that we are less likely to peacefully colonize space than we thought, they also mean we are less likely to horribly colonize it, via an unfriendly AI explosion.

Light cone eating AI explosions are not filters

Some existential risks can’t account for any of the Great Filter. Here are two categories of existential risks that are not filters:

Too big: any disaster that would destroy everyone in the observable universe at once, or destroy space itself, is out. If others had been filtered by such a disaster in the past, we wouldn’t be here either. This excludes events such as simulation shutdown and breakdown of a metastable vacuum state we are in.

Not the end: Humans could be destroyed without the causal path to space colonization being destroyed. Also much of human value could be destroyed without humans being destroyed. e.g. Super-intelligent AI would presumably be better at colonizing the stars than humans are. The same goes for transcending uploads. Repressive totalitarian states and long term erosion of value could destroy a lot of human value and still lead to interstellar colonization.

Since these risks are not filters, neither the knowledge that there is a large minimum total filter nor the use of SIA increases their likelihood.  SSA still increases their likelihood for the usual Doomsday Argument reasons. I think the rest of the risks listed in Nick Bostrom’s paper can be filters. According to SIA averting these filter existential risks should be prioritized more highly relative to averting non-filter existential risks such as those in this post. So for instance AI is less of a concern relative to other existential risks than otherwise estimated. SSA’s implications are less clear – the destruction of everything in the future is a pretty favorable inclusion in a hypothesis under SSA with a broad reference class, but as always everything depends on the reference class.

Anthropic principles agree on bigger future filters

I finished my honours thesis, so this blog is back on. The thesis is downloadable here.  I’ll blog some other interesting bits soon.

My main point was that two popular anthropic reasoning principles, the Self Indication Assumption (SIA) and the Self Sampling Assumption (SSA), as well as Full Non-indexical Conditioning (FNC)  basically agree that future filter steps will be larger than we otherwise think, including the many future filter steps that are existential risks.

Figure 1: SIA likes possible worlds with big populations at our stage, which means small past filters, which means big future filters.

SIA says the probability of being in a possible world is proportional to the number of people it contains who you could be. SSA says it’s proportional to the fraction of people (or some other reference class) it contains who you could be. FNC says the probability of being in a possible world is proportional to the chance of anyone in that world having exactly your experiences. That chance is more the larger the population of people like you in relevant ways, so FNC generally gets similar answers to SIA. For a lengthier account of all these, see here.

SIA increases expectations of larger future filter steps because it favours smaller past filter steps. Since there is a minimum total filter size, this means it favours big future steps. This I have explained before. See Figure 1. Radford Neal has demonstrated similar results with FNC.

Figure 2: A larger filter between future stages in our reference class makes the population at our own stage a larger proportion of the total population. This increases the probability under SSA.

SSA can give a variety of results according to reference class choice. Generally it directly increases expectations of both larger future filter steps and smaller past filter steps, but only for those steps between stages of development that are at least partially included in the reference class.

For instance if the reference class includes all human-like things, perhaps it stretches from ourselves to very similar future people who have avoided many existential risks. In this case, SSA increases the chances of large filter steps between these stages, but says little about filter steps before us, or after the future people in our reference class. This is basically the Doomsday Argument – larger filters in our future mean fewer future people relative to us. See Figure 2.

Figure 3: In the world with the larger early filter, the population at many stages including ours is smaller relative to some early stages. This makes the population at our stage a smaller proportion of the whole, which makes that world less likely. (The populations at each stage are a function of the population per relevant solar system as well as the chance of a solar system reaching that stage, which is not illustrated here).

With a reference class that stretches to creatures in filter stages back before us, SSA increases the chances of smaller past filters steps between those stages. This is because those filters make observers at almost all stages of development (including ours) less plentiful relative to at least one earlier stage of creatures in our reference class. This makes those at our own stage a smaller proportion of the population of the reference class. See Figure 3.

The predictions of the different principles differ in details such as the extent of the probability shift and the effect of timing. However it is not necessary to resolve anthropic disagreement to believe we have underestimated the chances of larger filters in our future. As long as we think something like one of the above three principles is likely to be correct, we should update our expectations already.

[Edited 2021 to change download options]