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Posts Tagged ‘self-indication assumption’

Unlike Robin Hanson, I am not surprised by who I am.   Sure, most things that exist are not alive, are not human and are not statisticians, but that doesn’t make it surprising that I am.  What I am is the only thing I could have been.

It’s true that Robin is smarter than most people, and most people don’t write a popular blog.  So should he be surprised that he is those things?   The only reason he noted those particular features is because those features already exist.  The question was generated by the result.  Everyone has things about them that are unusual.  Should we all be surprised?  For example, Brenda might be one of the few left-handed female plumbers in Texas.  Should she be surprised?  If everyone has unique things they can point to, then shouldn’t that fail to surprise us?

Consider the t-shirt experiment:

20 t-shirts, each a unique color, are placed in a box.  You are blindfolded.  A shirt is randomly selected from the box and placed on you.  You then remove the blindfold.

Suppose you participate in the experiment, and after you remove the blindfold you observe that your t-shirt is blue.  Your reaction could be: “I’m surprised to be wearing a blue t-shirt.  Only 1 out of 20 shirts was blue.”  But of course, you could say the same thing no matter which t-shirt was selected.  There was a probability of 1 that a shirt that was unlike the other 19 would be selected.  We see the result and then start thinking about how unique that result is.

This kind of reasoning leads to bad inference, such as the self-indication assumption or the doomsday argument.  The wikipedia version of the doomsday argument is: “supposing the humans alive today are in a random place in the whole human history timeline, chances are we are about halfway through it.”  In other words, if there was a time-traveling stork that selects humans from all humans that will ever exist, and randomly places them at various places in the human history timeline, then we are probably about halfway through human existence.  People then debate whether the doomsday conclusion is correct, but do not challenge the assumption that we know is wrong.   The doomsday argument can be rejected by simply noting that the assumption is bad (we are not in a random place in the human history timeline).

We shouldn’t be surprised that we exist, since we had to exist to notice that we exist and ask questions about our existence.  It would be more surprising if we noticed that we didn’t exist.

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In thinking about the self-indication assumption, let’s consider some experiments.

Experiment 1a

Suppose there are 1 million balls in an urn.  1 ball is orange and the rest are blue.

The algorithm goes like this:  flip a coin.  If heads, Large World wins and 999,999 balls will be randomly selected from the urn.  If tails, Small World wins and 1 ball will be drawn from the urn.

Once the ball(s) have been drawn, we are told whether the orange ball was drawn.

Prior probability of Large World: P(heads)=0.5

Posterior probability of Large World: P(heads|orange ball)≈1 and P(heads|orange ball not drawn)≈0

So, knowledge about whether the orange ball was drawn tells us a great deal about what world we are in.

Experiment 1b

Suppose there are 1 million balls in an urn.  All of the balls are blue.

The algorithm goes like this:  flip a coin.  If heads, Large World wins and 999,999 balls will be randomly selected from the urn and then painted orange.  If tails, Small World wins and 1 ball will be drawn from the urn and then painted orange.

Once the ball(s) have been drawn, we are told whether a ball that has subsequently been painted orange was drawn.

Prior probability:  P(heads)=0.5

Posterior probability:  P(heads|at least one blue ball painted orange)=P(heads)=0.5

Because regardless of the result of the coin flip at least one ball would be painted orange, knowing that at least one ball was painted orange tells us nothing about the result of the coin flip.  So in this experiment, the prior probability equals the posterior probability.

Experiment 2a

1,000,000 people are in a giant urn.  Each person is labeled with a number (number 1 through number 1,000,000).

A coin will be flipped.  If heads, Large World wins and 999,999 people will be randomly selected from the urn.  If tails, Small World wins and 1 person will be drawn from the urn.

Ahead of time, we label person #5,214 as special.  After the coin flip, and after the sample is selected, we are told whether special person #5214 was selected.

Prior probability of Large World: P(heads)=0.5

Posterior probability of Large World: P(heads|person #5,214 selected)≈1 and P(heads|person #5,214 not selected)≈0

Experiment 2b

1,000,000 people are in a giant urn.  Each person is labeled with a number (number 1 through number 1,000,000).

A coin will be flipped.  If heads, Large World wins and 999,999 people will be randomly selected from the urn.  If tails, Small World wins and 1 person will be drawn from the urn.

After the coin flip, and after the sample is selected, we are told that person #X was selected (where X is an integer between 1 and 1,000,000).

Prior probability of Large World: P(heads)=0.5

Posterior probability of Large World: P(heads|person #X selected)=P(heads)=0.5

Regardless of whether the coin landed heads or tails, we knew we would be told about some person being selected.  So, the fact that we were told that someone was selected tells us nothing about which world we are in.

Self-indication assumption (SIA)

Recall that the SIA is

Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.

“Given the fact that you exist…”  Why me?  Because I was already selected.  I am that ball that was painted orange.  I am person #X.  I only became the special ball and the special number after I was selected.

The mistake of the SIA is the data were generated from experiments like 1b and 2b, but is treated as if it’s from 1a and 2a.

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update:  an even more detailed argument here

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Rare events

Suppose you drew 5 cards and they were all hearts.  You might ask “what’s the probability of drawing 5 hearts at random from a 52 card deck?”  Well, sure, that’s easy enough to calculate.  The probability is 0.000495.  Wow!  Such a rare event!  Must be your lucky day.

But…  the reason you asked about 5 hearts is because that’s what you experienced.  You peeked at the data, and then asked your question. I am sure if you would have gotten 5 clubs you would have asked about that.  Or if you had gotten a straight.

So, one way to rephrase the question is as follows:  “what’s the probability of drawing 5 cards at random from a 52 card deck that, upon viewing these cards, would have gotten my attention and prompted me to ask a question about probability?”  I’m confident that any flush or straight would have gotten your attention, and four of a kind as well.  So let’s stick with those.  The probability of drawing either a flush, a straight or four of a kind is 0.006.  While this is still a very rare event, it’s about 10 times higher than that of getting 5 hearts.

My existence

I have heard it argued that

the probability that you are aware right now,  when your existence could have ended billions of years ago, or could have come into being billions of years in the future – this probability is so small, so insignificant, that it is practically non-existent… this could only mean the existence of God.

However, the probability calculation is incorrect.   Let’s define the event A as follows:

A:  I exist now out of all of the possible times I could have existed

The  argument is that P(A) is essentially 0.  However, the argument ignores the fact that I already do exist right now, which is why I am asking the question. I am asking a question based on data that I have already seen.  We have to condition on that data.  Therefore,  let’s define the event B as:

B:  I exist right now

What we are interested in is not P(A), but P(A|B).  We have to condition on B, because B is the reason we are asking the question.  It’s the data we peeked at.   Well, it turns out that P(A|B)=1.  Thus, it’s not a rare event and certainly cannot be an argument in favor of any religious beliefs.

Self-indication assumption

The self-indication assumption (SIA) is

Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.

Katja Grace presents a simple example of SIA:

For instance if you were born of an experiment where the flip of a fair coin determined whether one (tails) or two (heads) people were created, and all you know is that and that you exist, SIA says heads was twice as likely as tails.

For simplicity, let’s assume this action only happened once (one coin flip).  Thus, there is only one or two people in the world, depending on whether it was tails or heads.  Let’s assume if we are in two person world, we don’t see the other person.

Let’s define the event A as:

event A:  the coin came up tails (i.e., the one person world)

SIA reasoning is that since there were 3 possible people including myself, and I was selected, the probability that I’m on 1 person world is 1/3.

The flaw here is the focus on me existing.  I already exist, so it’s cheating to write questions about me existing after seeing that I exist.  Like the card player who formed the hypothesis after seeing the cards, we’re asking the wrong question.

Instead, let’s define the event B as:

event B:  at least one person exists

Event B is what we really want to condition on.   We want to condition on an arbitrary person existing — there is nothing special about me in this scenario (unless you cheat and use the data you peeked at).   Well, in either of these two worlds (heads or tails) there will exist someone that is wondering which world they are in.  So, the fact that there is someone wondering which world they are in tells us no information about which world we are in.  That is, P(A|B)=P(A)=1/2.

So, in my opinion SIA is wrong.  The fact that I exist tells me nothing about the number of observers.

Doomsday argument

The doomsday argument seems flawed for the same reason.  It basically says that the fact that you exist is evidence that the race will die out soon.

It’s correct that if you could randomly draw a human from the N that will exist, you will probably pick one that is towards the tail of the distribution (when the population is greatest).  However, we cannot think of ourselves as a random draw.  It’s peeking at the data. We already exist.

It should be obvious the argument is flawed based on the fact it always comes to the same conclusion.  Suppose, for example, the total number of humans to ever exist (past and future) will total N.  Every human, numbers 1, 2, … , N, will at some point exist, and could wonder if humans will face extinction soon.  So, if we condition on the fact that right now there is at least one human asking that question, we have no information about whether that human is close to number N.  All information about possible extinction would have to rely on other sources.

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