Yesterday I spoke of the Mind Projection Fallacy, giving the example of the alien monster who carries off a girl in a torn dress for intended ravishing—a mistake which I imputed to the artist's tendency to think that a woman's sexiness is a property of the woman herself, `woman.sexiness`, rather than something that exists in the mind of an observer, and probably wouldn't exist in an alien mind.

The term "Mind Projection Fallacy" was coined by the late great Bayesian Master, E. T. Jaynes, as part of his long and hard-fought battle against the accursèd frequentists. Jaynes was of the opinion that probabilities were in the mind, not in the environment—that probabilities express ignorance, states of partial information; and if I am ignorant of a phenomenon, that is a fact about my state of mind, not a fact about the phenomenon.

I cannot do justice to this ancient war in a few words—but the classic example of the argument runs thus:

You have a coin.

The coin is biased.

You don't know which way it's biased or how much it's biased. Someone just told you, "The coin is biased" and that's all they said.

This is all the information you have, and the only information you have.

You draw the coin forth, flip it, and slap it down.

Now—before you remove your hand and look at the result—are you willing to say that you assign a 0.5 probability to the coin having come up heads?

The frequentist says, "No. Saying 'probability 0.5' means that the coin has an inherent propensity to come up heads as often as tails, so that if we flipped the coin infinitely many times, the ratio of heads to tails would approach 1:1. But we know that the coin is biased, so it can have any probability of coming up heads *except* 0.5."

The Bayesian says, "Uncertainty exists in the map, not in the territory. In the real world, the coin has either come up heads, or come up tails. Any talk of 'probability' must refer to the *information* that I have about the coin—my state of partial ignorance and partial knowledge—not just the coin itself. Furthermore, I have all sorts of theorems showing that if I don't treat my partial knowledge a certain way, I'll make stupid bets. If I've got to plan, I'll plan for a 50/50 state of uncertainty, where I don't weigh outcomes conditional on heads any more heavily in my mind than outcomes conditional on tails. You can call that number whatever you like, but it has to obey the probability laws on pain of stupidity. So I don't have the slightest hesitation about calling my outcome-weighting a probability."

I side with the Bayesians. You may have noticed that about me.

Even before a fair coin is tossed, the notion that it has an *inherent* 50% probability of coming up heads may be just plain wrong. Maybe you're holding the coin in such a way that it's just about guaranteed to come up heads, or tails, given the force at which you flip it, and the air currents around you. But, if you don't know which way the coin is biased on this one occasion, so what?

I believe there was a lawsuit where someone alleged that the draft lottery was unfair, because the slips with names on them were not being mixed thoroughly enough; and the judge replied, "To whom is it unfair?"

To make the coinflip experiment repeatable, as frequentists are wont to demand, we could build an automated coinflipper, and verify that the results were 50% heads and 50% tails. But maybe a robot with extra-sensitive eyes and a good grasp of physics, watching the autoflipper prepare to flip, could predict the coin's fall in advance—not with certainty, but with 90% accuracy. Then what would the *real* probability be?

There is no "real probability". The robot has one state of partial information. You have a different state of partial information. The coin itself has no mind, and doesn't assign a probability to anything; it just flips into the air, rotates a few times, bounces off some air molecules, and lands either heads or tails.

So that is the Bayesian view of things, and I would now like to point out a couple of classic brainteasers that derive their brain-*teasing* ability from the tendency to think of probabilities as inherent properties of objects.

Let's take the old classic: You meet a mathematician on the street, and she happens to mention that she has given birth to two children on two separate occasions. You ask: "Is at least one of your children a boy?" The mathematician says, "Yes, he is."

What is the probability that she has two boys? If you assume that the prior probability of a child being a boy is 1/2, then the probability that she has two boys, on the information given, is 1/3. The prior probabilities were: 1/4 two boys, 1/2 one boy one girl, 1/4 two girls. The mathematician's "Yes" response has probability ~1 in the first two cases, and probability ~0 in the third. Renormalizing leaves us with a 1/3 probability of two boys, and a 2/3 probability of one boy one girl.

But suppose that instead you had asked, "Is your eldest child a boy?" and the mathematician had answered "Yes." Then the probability of the mathematician having two boys would be 1/2. Since the eldest child is a boy, and the younger child can be anything it pleases.

Likewise if you'd asked "Is your youngest child a boy?" The probability of their being both boys would, again, be 1/2.

Now, if at least one child is a boy, it must be either the oldest child who is a boy, or the youngest child who is a boy. So how can the answer in the first case be different from the answer in the latter two?

Or here's a very similar problem: Let's say I have four cards, the ace of hearts, the ace of spades, the two of hearts, and the two of spades. I draw two cards at random. You ask me, "Are you holding at least one ace?" and I reply "Yes." What is the probability that I am holding a pair of aces? It is 1/5. There are six possible combinations of two cards, with equal prior probability, and you have just eliminated the possibility that I am holding a pair of twos. Of the five remaining combinations, only one combination is a pair of aces. So 1/5.

Now suppose that instead you asked me, "Are you holding the ace of spades?" If I reply "Yes", the probability that the other card is the ace of hearts is 1/3. (You know I'm holding the ace of spades, and there are three possibilities for the other card, only one of which is the ace of hearts.) Likewise, if you ask me "Are you holding the ace of hearts?" and I reply "Yes", the probability I'm holding a pair of aces is 1/3.

But then how can it be that if you ask me, "Are you holding at least one ace?" and I say "Yes", the probability I have a pair is 1/5? Either I must be holding the ace of spades or the ace of hearts, as you know; and either way, the probability that I'm holding a pair of aces is 1/3.

How can this be? Have I miscalculated one or more of these probabilities?

If you want to figure it out for yourself, do so now, because I'm about to reveal...

That all stated calculations are correct.

As for the paradox, there isn't one. The *appearance *of paradox comes from thinking that the probabilities must be properties of the cards themselves. The ace I'm holding has to be either hearts or spades; but that doesn't mean that your *knowledge about* my cards must be the same as if you *knew* I was holding hearts, or *knew* I was holding spades.

It may help to think of Bayes's Theorem:

P(H|E) = P(E|H)P(H) / P(E)

That last term, where you divide by P(E), is the part where you throw out all the possibilities that have been eliminated, and renormalize your probabilities over what remains.

Now let's say that you ask me, "Are you holding at least one ace?" *Before* I answer, your probability that I say "Yes" should be 5/6.

But if you ask me "Are you holding the ace of spades?", your prior probability that I say "Yes" is just 1/2.

So right away you can see that you're *learning* something very different in the two cases. You're going to be eliminating some different possibilities, and renormalizing using a different P(E). If you learn two different items of evidence, you shouldn't be surprised at ending up in two different states of partial information.

Similarly, if I ask the mathematician, "Is at least one of your two children a boy?" I expect to hear "Yes" with probability 3/4, but if I ask "Is your eldest child a boy?" I expect to hear "Yes" with probability 1/2. So it shouldn't be surprising that I end up in a different state of partial knowledge, depending on which of the two questions I ask.

The only reason for seeing a "paradox" is thinking as though the probability of holding a pair of aces is *a property of cards* that have at least one ace, or a property *of cards* that happen to contain the ace of spades. In which case, it would be paradoxical for card-sets containing at least one ace to have an inherent pair-probability of 1/5, while card-sets containing the ace of spades had an inherent pair-probability of 1/3, and card-sets containing the ace of hearts had an inherent pair-probability of 1/3.

Similarly, if you think a 1/3 probability of being both boys is an *inherent property* of child-sets that include at least one boy, then that is not consistent with child-sets of which the eldest is male having an *inherent* probability of 1/2 of being both boys, and child-sets of which the youngest is male having an inherent 1/2 probability of being both boys. It would be like saying, "All green apples weigh a pound, and all red apples weigh a pound, and all apples that are green or red weigh half a pound."

That's what happens when you start thinking as if probabilities are *in* things, rather than probabilities being states of partial information *about *things.

Probabilities express uncertainty, and it is only agents who can be uncertain. A blank map does not correspond to a blank territory. Ignorance is in the mind.

It seems to me you're using "perceived probability" and "probability" interchangeably. That is, you're "defining" probability as the probability that an observer assigns based on certain pieces of information. Is it not true that when one rolls a fair 1d6, there is an actual 1/6 probability of getting any one specific value? Or using your biased coin example: our information may tell us to assume a 50/50 chance, but the man may be correct in saying that the coin has a bias--that is, the coin may really come up heads 80% of the... (read more)

"Is it not true that when one rolls a fair 1d6, there is an actual 1/6 probability of getting any one specific value?"

No. The unpredictability of a die roll or coin flip is not due to any inherent physical property of the objects; it is simply due to lack of information. Even with quantum uncertainty, you could predict the result of a coin flip or die roll with high accuracy if you had precise enough measurements of the initial conditions.

Let's look at the simpler case of the coin flip. As Jaynes explains it, consider the phase space for the coin's motion at the moment it leaves your fingers. Some points in that phase space will result in the coin landing heads up; color these points black. Other points in the phase space will result in the coin landing tails up; color these points white. If you examined the phase space under a microscope (metaphorically speaking) you would see an intricate pattern of black and white, with even a small movement in the phase space crossing many boundaries between a black region and a white region.

If you knew the initial conditions precisely enough, you would know whether the coin was in a white or black region of phase space, and you... (read more)

Case in point:

There are dice designed with very sharp corners in order to improve their randomness.

If randomness were an inherent property of dice, simply refining the shape shouldn't change the randomness, they are still plain balanced dice, after all.

But when you think of a "random" throw of the dice as a combination of the position of the dice in the hand, the angle of the throw, the speed and angle of the dice as they hit the table, the relative friction between the dice and the table, and the sharpness of the corners as they tumble to a stop, you realize that if you have all the relevant information you can predict the roll of the dice with high certainty.

It's only because we don't have the relevant information that we say the probabilities are 1/6.

GBM:

Q: What is the probability for a pseudo-random number generator to generate a specific number as his next output?

A: 1 or 0 because you can actually calculate the next number if you have the available information.

Q: What probability do you assign to a specific number as being it's next output if you don't have the information to calculate it?

Replace pseudo-random number generator with dice and repeat.

Even more important, I think, is the realization that, to decide how much you're willing to bet on a specific outcome, all of the following are essentially the same:

The bottom line is that you

don't know what the next value will be, and that's the only thing that matters.So therefore a person with perfect knowledge would not need probability. Is this another interpretation of "God does not play dice?" :-)

Alas, the coin was part of an erroneous stamping, and is blank on both sides.

In other words, probability is not likelihood.

Here is another example me, my dad and my brother came up with when we were discussing probability.

Suppose there are 4 card, an ace and 3 kings. They are shuffled and placed face side down. I didn't look at the cards, my dad looked at the first card, my brother looked at the first and second cards. What is the probability of the ace being one of the last 2 cards. For me: 1/2 For my dad: If he saw the ace it is 0, otherwise 2/3. For my brother: If he saw the ace it is 0, otherwise 1.

How can there be different probabilities of the same event? It is because probability is something in the mind calculated because of imperfect knowledge. It is not a property of reality. Reality will take only a

singlepath. We just don't know what that path is. It is pointless to ask for "the real likelihood" of an event. The likelihood depends on how much information you have. If you had all the information, the likelihood of the event would be 100% or 0%.The competent frequentist would presumably not be befuddled by these supposed paradoxes. Since he would not be befuddled (or so I am fairly certain), the "paradoxes" fail to prove the superiority of the Bayesian approach. Frankly, the treatment of these "paradoxes" in terms of repeated experiments seems to straightforward that I don't know how you can possibly think there's a problem.

"Probabilities express uncertainty, and it is only agents who can be uncertain. A blank map does not correspond to a blank territory. Ignorance is in the mind."

Eliezer, in quantum mechanics, one does not say that one does not have knowledge of both position and momentum of a particle simultaneously. Rather, one says that one CANNOT have such knowledge. This contradicts your statement that ignorance is in the mind. If quantum mechanics is true, then ignorance/uncertainty is a part of nature and not just something that agents have.

"Probabilities express uncertainty, and it is only agents who can be uncertain. A blank map does not correspond to a blank territory. Ignorance is in the mind."

Eliezer, in quantum mechanics, one does not say that one does not have knowledge of both position and momentum of a particle simultaneously. Rather, one says that one CANNOT have such knowledge. This contradicts your statement that ignorance is in the mind. If quantum mechanics is true, then ignorance/uncertainty is a part of nature and not just something that agents have.

Constant:

The competent frequentist would presumably not be befuddled by these supposed paradoxes.Not the last two paradoxes, no. But the first case given, the biased coin whose bias is not known, is indeed a classic example of the difference between Bayesians and frequentists. The frequentist says:

According to the frequentist, apparently there is no

rationalway to manage your uncertainty about a single flip of a coin of unknown bias, since whatever you do, someone else will be able to criticize your belief as "subjective" - such a devastating criticism that you may as well, um, flip a coin. Or consul... (read more)Maybe I'm stupid here... what difference does it make?

Sure, if we had a coin-flip-predicting robot with quick eyes it might be able to guess right/predict the outcome 90% of the time. And if we were precognitive we could clean up at Vegas.

In terms of non-hypothetical real decisions that confront people, what is the outcome of this line of reasoning? What do you suggest people do differently and in what context? Mark cards?

B/c currently, as far as I can see, you're saying, "The coin won't end up 'heads or tails' -- it'll end up heads, or it'll end u... (read more)

Sudeep: the inverse certainy of the position and momentum is a mathematical artifact and does not depend upon the validity of quantum mechanics. (Er, at least to the extent that math is independent of the external world!)

PK: I like your posts, and don't take this the wrong way, but, to me, your example doesn't have as much shocking unintuitiveness as the ones Eliezer Yudkowsky (no underscore) listed.

I'd like to understand: Are frequentist "probability" and subjective "probability" simply two different concepts, to be distinguished carefully? Or is there some true debate here?

I think that Jaynes shows a derivation follownig Bayesian principles of the frequentist probability from the subjective probability. I'd love to see one of Eliezer's lucid explanations on that.

You can derive frequentist probabilities from subjective probabilities but not the other way around.

Silas: My post wasn't meant to be "shockingly unintuitive", it was meant to illustrate Eliezer's point that

probability is in the mind and not out there in realityin a ridiculously obvious way.Am I somehow talking about something entirely different than what Eliezer was talking about? Or should I complexificationafize my vocabulary to seem more academic? English isn't my first language after all.

Alas, no. Here's the deal: implicit in all the coin toss toy problems is the idea that the observations may be modeled as exchangeable. It really really helps to have a grasp on what the math looks like when we assume exchangeability.

In models where (infinite) exchangeability is assumed, the concept of long-run frequen... (read more)

Eliezer, I have no argument with the Bayesian use of the probability calculus and so I do not side with those who say "there is no rational way to manage your uncertainty", but I think I probably do have an argument with the insistence that it is the one true way. None of the problems you have so far outlined, including the coin one, really seem to doom either frequentism specifically, or more generally, an objective account of probability. I agree with this:

... (read more)(Replace the link to "removable singularity" with one to removable discontinuity.)

No way to do it other way around? Nothing along the lines of, say, considering a set of various "things to be explained" and for each a hypothesis explaining it, and then talk about subsets of those? ie, a subset in which 1/10 of the hypothesies in that subset are objectively true would be a set of hypothesies assigned .1 probability, or something?

Yeah, the notion of how to do this exactly is, admittedly, fuzzy in my head, but I have to say that it sure does seem like there ought to be some way to use the notion of frequentist probability to construct subjective probability along these lines.

I may be completely wrong though.

"Suppose our information about bias in favour of heads is equivalent to our information about bias in favour of tail. Our pdf for the long-run frequency will be symmetrical about 0.5 and its expectation (which is the probability in any single toss) must also be 0.5. It is quite possible for an expectation to take a value which has zero probability density."

What I said: if all you know is that it's a trick coin, you can lay even odds on heads.

"We can refuse to believe that the long-run frequency will converge to exactly 0.5 while simultaneou... (read more)

In other words, they are talking about tail events. That a frequentist probability (i.e., a long-run frequency) even exists can be a zero-probability event -- but you have to give axioms for probability before you can even make this claim. (Furthermore, I'm never going to observe a tail event, so I don't much care about them.)

Conrad,

Okay, so unpack "ungrounded" for me. You've used the phrases "probability" and "calculated or measured likelihood of heads coming up", but I'm not sure how you're defining them.

I'm going to do two things. First, I'm going to Taboo "probability" and "likelihood" (for myself -- you too, if you want). Second, I'm going to ask you exactly which specific observable event it is we're talking about. (First toss? Twenty-third toss? Infinite collection of tosses?) I have a definite feeling that our disagreement is about word usage.

If you honestly subscribe to this view of probability, please never give the odds for winning the lottery again. Or any odds for anything else.

What does telling me your probability that you assign something actually tell me about the world? If I don't know the information you are basing it on, very little.

I'm also curious about a formulation of probability theory that completely ignores random numbers and other theories that are based upon them (e.g. The law of large numbers, Central limit theorem).

Heck a re-write of http://en.wikipedia.org/wiki/Probability_theory with all mention of probabilities in the external world removed might be useful.

I'm not sure the many-worlds interpretation fully eliminates the issue of quantum probability as part of objective reality. You can call it "anthropic pseudo-uncertainty" when you get split and find that your instances face different outcomes. But what determines the

probabilityyou will see those various outcomes? Just your state of knowledge? No, theory says it is an objective element of reality, the amplitude of the various elements of the quantum wave function. This means that probability, or at least its close cousin amplitude, is indeed an ... (read more)Will Pearson, I'm having trouble determining to whom your comment is addressed.

Roland and Ian C. both help me understand where Eliezer is coming from. And PK's comment that "Reality will only take a single path" makes sense. That said, when I say a die has a 1/6 probability of landing on a 3, that means: Over a series of rolls in which no effort is made to systematically control the outcome (e.g. by always starting with 3 facing up before tossing the die), the die will land on a 3 about 1 in 6 times. Obviously, with perfect information, everything can be calculated. That doesn't mean that we can't predict the probability of... (read more)

::Okay, so unpack "ungrounded" for me. You've used the phrases "probability" and "calculated or measured likelihood of heads coming up", but I'm not sure how you're defining them.::

Ungrounded: That was a good movie. Grounded: That movie made money for the investors. Alternatively: I enjoyed it and recommend it. -- is for most purposes grounded enough.

::I'm going to do two things. First, I'm going to Taboo "probability" and "likelihood" (for myself -- you too, if you want). Second, I'm going to ask you... (read more)

GBM:: ..That said, when I say a die has a 1/6 probability of landing on a 3, that means: Over a series of rolls in which no effort is made to systematically control the outcome (e.g. by always starting with 3 facing up before tossing the die), the die will land on a 3 about 1 in 6 times.::

--Well, no: it does mean that, but don't let's get tripped up that a measure of probability requires a series of trials. It has that same probability even for one roll. It's a consequence of the physics of the system, that there are 6 stable distinguishable end-states and explosively many intermediate states, transitioning amongst each other chaotically.

Conrad.

I have to say that it sure does seem like there ought to be some way to use the notion of frequentist probability to construct subjective probability along these lines.Assign a measure to each possible world (the prior probabilities). For some state of knowledge K, some set of worlds Ck is consistent with K (say, the set in which there is a brain containing K). For some proposition X, X is true in some set of worlds Cx. The subjective probability P(X|K) = measure(intersection(Ck,Cx)) / measure(Ck). Bayesian updating is equivalent to removing worlds from K. To make it purely frequentist, give each world measure 1 and use multisets.

Does that work?

Who else thinks we should Taboo "probability", and replace it two terms for objective and subjective quantities, say "frequency" and "uncertainty"?

The frequency of an event depends on how narrowly the initial conditions are defined. If an atomically identical coin flip is repeated, obviously the frequency of heads will be either 1 or 0 (modulo a tiny quantum uncertainty).

Oops, removing worlds from Ck, not K.

GBM, I think you get the idea. The reason we don't want to say that the gomboc has an

inherentprobability of one for righting itself (besides that we, um, don't use probability one), is that as it is with the gomboc, so it is with the die or anything else in the universe. The premise is that determinism, in the form of some MWI, is (probably!) true, and so no matter what you or anyone else knows, whatever will happen is sure to happen. Therefore, when we speak of probability, we can only be referring to a state of knowledge. It is still of course the case... (read more)Cyan, sorry. My comment was to Eliezer and statements such as

"that probabilities express ignorance, states of partial information; and if I am ignorant of a phenomenon, that is a fact about my state of mind, not a fact about the phenomenon."

I think there's still room for a concept of objective probability -- you'd define it as anything that obeys David Lewis's "Principal Principle" which this page tries to explain (with respect to some natural distinction between "admissible" and "inadmissible" information).

Before accepting this view of probability and the underlying assumptions about the nature of reality one should look at the experimental evidence. Try Groeblacher, Paterek, et al arXiv.0704.2529 (Aug 6 2007) These experiments test various assumptions regarding non=local realism and conclude= "...giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned"

Standard reply from MWIers is that MWI keeps realism and locality by throwing away a different hidden assumption called "counterfactual definiteness".

Nick Tarleton:

Who else thinks we should Taboo "probability", and replace it two terms for objective and subjective quantities, say "frequency" and "uncertainty"?I second that, this would probably clear a lot of the confusion and help us focus on the real issues.

The "probability" of an event is how much anticipation you have for that event occurring. For example if you assign a "probability" of 50% to a tossed coin landing heads then you are half anticipating the coin to land heads.

You've already given me information by using the word medication -- implicity, you're asking me to recall what I know about medications before I render an answer. So no, those outcomes aren't

necessarilyequally plausible to me. Here's a situation which is a much better approximation(!) of total absence of information: either event Q or event J has happened just now, and I will tell you which in my next comment. The... (read more)Now, if at least one child is a boy, it must be either the oldest child who is a boy, or the youngest child who is a boy. So how can the answer in the first case be different from the answer in the latter two?Because they obviously aren't exclusive cases. I simply don't see mathematically why it's a paradox, so I don't see what this has to do with thinking that "probabilities are a property of things."

The "paradox" is that people want to compare it to a different problem, the problem where the cards are ordered. In that case, if you ... (read more)

Or, I suppose, I would compare it to the other noted statistical paradox, whereby a famous hospital has a better survival rate for both mild and severe cases of a disease than a less-noted hospital, but a worse overall survival rate because it sees more of the worst cases. Merely because people don't understand how to do averages has little to do with them requiring an agent.

The estimated Bayesian probability has nothing to do with the coin. If it did, assigning a probability of 0.5 to one of the two possible outcomes would be necessarily incorrect, because one of the few things we know about the coin is that it's not fair.

The estimate is of our confidence in using that outcome as an answer. "How confident can I be that choosing this option will turn out to be correct?" We know that the coin is biased, but we don't know which outcome is more likely. As far as we know, then, guessing one way is as good as guessing... (read more)

Another way to look at it: if you repeatedly select a coin with a random bias (selected from any distribution symmetric about .5) and flip it, H/T will come out 50/50.

Silas: The uncertainty principle comes from the fact that position and momentum are related by Fourier transform. Or, in laymans terms, the fact that particles act like waves. This is one of the fundamental principles of QM, so yeah, it sort of does depend on the validity thereof. Not the Schrodinger equation itself perhaps, but other concepts.

As for whether QM proves that all probabilities are inherent in a system, it doesn't. It just prevents mutual information in certain situations. In coin flips or dice rolls, theoretically you could predict the o... (read more)

Z. M. Davis: Thank you. I get it now.

Follow-up question: If Bob believes he has a >50% chance of winning the lottery tomorrow, is his belief objectively wrong? I would tentatively propose that his belief is unfounded, "unattached to reality", unwise, and unreasonable, but that it's not useful to consider his belief "objectively wrong".

If you disagree, consider this: suppose he wins the lottery after all by chance, can you still claim the next day that his belief was objectively wrong?

Nick Tarleton: Not sure I entirely correctly understood your suggestion, need to think about it more.

However, my initial thought is that it may require/assume logical omnicience.

ie, what of updating based on "subjective guesses" of which worlds are consistent or inconsistent with the data. That is, as consistent as you can tell, given bounded computational resources. I'm not sure, but your model, at least at first glance, may not be able to say useful stuff about those that are not logically ominicent.

Also, I'm unclear, could you clarify what it ... (read more)

Hal, I'd say probability could be both part of objective physics and a mental state in this sense: Given our best understanding of objective physics, for any given mental state (including the info it has access to) there is a best rational set of beliefs. In quantum mechanics we know roughly the best beliefs, and we are trying to use that to infer more about the underlying set of states and info.

Rolf Nelson: "Follow-up question: If Bob believes he has a >50% chance of winning the lottery tomorrow, is his belief objectively wrong? I would tentatively propose that his belief is unfounded, "unattached to reality", unwise, and unreasonable, but that it's not useful to consider his belief "objectively wrong"."

It all depends on what information Bob has. He might have carefully doctored the machines and general setup of the lottery draw to an extent that he might have enough information to have that probability. Now if Bo... (read more)

However, my initial thought is that it may require/assume logical omnicience.Probably. Bayes is also easier to work with if you assume logical omniscience (i.e. knowledge of P(evidence|X) and P(E|~X)).

Also, I'm unclear, could you clarify what it is you'd be using a multiset for? Do you mean "increase measure only by increasing number of copies of this in the multiset, and no other means allowed" or did you intend something else?Yes, using multisets of worlds with identical measure is equivalent to (for rational measures only) but 'more frequentis... (read more)

You have to lay Â£1 on heads or tails on a biased coin toss. Your probability is in your mind, and your mind has no information either way. Hence, you lay the pound on either. Hence you assign a 0.5 probability to heads, and also to tails.

If your argument is 'I don't mean

my personalprobability, I meanthe actual probability', abandon all hope. All probability is 'perceived'. Unless you think you have all the evidence.Some probabilities are objective, inherent properties of bits of the universe, and the universe

doeshave all the evidence. The coin possesses an actual probability independent of what anyone knows or believes about it.if the vast majority of the measure of possible worlds given Bob's knowledge is in worlds where he loses, he's objectively wrong.That's a self-consistent system, it just seems to me more useful and intuitive to say that:

"P" is true => P

"Bob believes P" is true => Bob believes P

but not

"Bob's belief in P" is true => ...er, what exactly?

Also, I frequently need to attach probabilities to facts, where probability goes from [0,1] (or, in Eliezer's formulation, (-inf, inf)). But it's rare for me to have to any reason to att... (read more)

I second tabooing probability, but I think that we need more than two words to replace it. Casually, I think that we need, at the least, 'quantum measure', 'calibrated confidence', and 'justified confidence'. Typically we have been in the habit of calling both "Bayesian", but they are very different. Actual humans can try to be better approximations of Bayesians, but we can't be very close. Since we can't be Bayesian, due to our lack of logical omniscience, we can't avoid making stupid bets and being Dutch Booked by smarter minds. It's there... (read more)

just fyi, there's no such thing as the 'eldest' of

twoboys; there's just an elder and a younger. superlatives are reserved for groups of three or more.as i'm a midget among giants here, i'm afraid that's all i have to add. :)

Enginerd: The uncertainty inherent in determining a pair of conjugate variables - such as the length and pitch of a sound - is indeed a core part of QM, but is not probabilistic. In this case, the term "uncertainty" is not about probabilities, even if QM is probabilistic in general, rather a consequence of describing states in terms of wave functions, which can be interpreted probabilistically. This causes many to mistakenly think the Heisenberg's "Uncertainty Principle" is the probabilistic part of QM. As Wikipedia[1] puts it: &quo... (read more)

You're equating perceived probability with physical probability, and this is false, when either you or anyone else ignores that distinction.

However, your whole argument depends on a deterministic universe. Research quantum mechanics; we can't really say that we have a deterministic universe, and physics itself can only assign a probability at a certain point.

@Daniel:

You're attacking the wrong argument. Just look up the electron double-slit experiment. (http://en.wikipedia.org/wiki/Double-slit_experiment) Its not only about the observer effect, but how the probability that you say doesn't exist causes interference to occur unless an observer is present. The observer is the one who collapses the probability wave down to a deterministic bayesian value.

It sounds like both you and the author of this blog do not understand Schrodinger's cat.

Let me further explain my point. Somewhere earlier said that reality only takes one path. Unless an observer is present, the electron double slit experiment proves that this assumption is false.

Welcome to

Overcoming Bias, anon! Try to to avoid triple-posting. The author of this post has actually just written a series on quantum mechanics, which begins with "Quantum Explanations." He argues forcefully for a many-worlds interpretation, whichisdeterministic "from the standpoint of eternity," although not for any particular observer due to indexical uncertainty. (You might say that, yes, reality does not take only one path, but it might as well have, because neither do observers!)@Z. M. Davis

Thanks for the welcome. While I disagree with the etiquette, I'll try to follow it. A three post limit serves only to stifle discussion; there are other ways to deal with abusive posters than limiting the abilities of non-abusive posters. Also, I'm pretty sure my comment is still valid, relevant, and an addition to the discussion, regardless of whether I posted it now or a couple hours ago.

Back to the many worlds approach, as an individual observer of the universe myself, it seems to me that attempting to look at the universe "from the sta... (read more)

That the probability assigned to flipping a coin depends on what the assigner knows doesn't prove probability's subjectivity, only that probability isn't an objective property of the

coin. Rather, if the probability is objective, it must be a property of a system, including the throwing mechanism. Two other problems with Eliezer's argument. 1) Rejecting objective interpretations of probability in empirical science because, in everyday usage, probability is relative to what's known, is to provide an a priori refutation of indeterminism, reasoning which do... (read more)Stephen R. Diamond, there are two distinct things in play here: (i) an assessment of the plausibility of certain statements conditional on some background knowledge; and (ii) the relative frequency of outcomes of trials in a counterfactual world in which the number of trials is very large. You've declared that probability can't be (i) because it's (ii) -- actually, the Kolmogorov axioms apply to both. Justification for using the word "probability" to refer to things of type (i) can be found in the first two chapters of this book. I personally cal... (read more)

Like Butters in that South Park episode, I can't help after all these posts but to notice that I am confused.

"Renormalizing leaves us with a 1/3 probability of two boys, and a 2/3 probability of one boy one girl." help me with this one, i'm n00b. If one of the kids is known to be a boy (given information), then doesn't the other one has 50/50 chances to be either a boy or a girl? And then having 50/50 chances for the couple of kids to be either a pair of boys or one boy one girl?

Conrad wrote:

I think it would be worthwhile to examine this conclusion - as it might seem to be an obvious one to a lot of people. Let us assume that there is a very good mechanical arm that makes a completely fair toss of the coin in the opinion of all humans so that we can talk entirely about the bias of the coin.

Let's say that the mechanism makes one toss; all you know is that the coin is biased - not how. Assume... (read more)

"Or here's a very similar problem: Let's say I have four cards, the ace of hearts, the ace of spades, the two of hearts, and the two of spades. I draw two cards at random. You ask me, "Are you holding at least one ace?" and I reply "Yes." What is the probability that I am holding a pair of aces? It is 1/5. There are six possible combinations of two cards, with equal prior probability, and you have just eliminated the possibility that I am holding a pair of twos. Of the five remaining combinations, only one combination is a p... (read more)

That is quite debatable. For one thing, it is possible to for quantum indeterminism, if there is any, to leak into the macroscopic world. Even if it were not possible, there is still the issue of microscopic indeterminism. You cannot pro... (read more)

Hate to be a stickler for this sort of thing, but even in the bayesian interpretation there are probabilities in the world, it's just that they are facts about the world and the knowledge the agents have of the world in combination. It's a fact that a perfect bayesian given P(a), P(a|b), and P(a|~b) will ascribe P(b|a), a probability of P(a|b)P(a) / P(b), and that that is the best value to give P(b|a).

If an agent has perfect knowledge then it need not ascribe any non-1 probability to any proposition it holds. But it is a fact about agents in the world tha... (read more)

Does this mean that there is nothing that is inherently uncertain? I guess another way to put that would be, could Laplace's Demon infer the entire history of the universe back to front from a single moment? It might seem obvious that there are singularities moving backwards through time (i.e. processes whose result does not give you information about their origin), so couldn't the same thing exist moving forward through time?

Anyway, great article!

My first post, so be gentle. :)

I disagree that there is a difference between "Bayesian" and "Frequentist;" or at least, that it has anything to do with what is mentioned in this article. The field of Probability has the unfortunate property of appearing to be a very simple, well defined topic. But it actually is complex enough to be indefinable. Those labels are used by people who want to argue in favor of one definition - of the indefinable - over another. The only difference I see is where they fail to completely address a problem.

Tak... (read more)

I used to be a frequentist, and say that the probability of the unfair coin landing heads is either 4/5 or 1/5, but I don't know exactly which. But that is not to say that I saw probabilities on things instead of on information. I'll explain.

If someone asked me if it will it rains tomorrow, I would ask which information am I supposed to use? If it rained in the past few days? Or would I consider tomorrow as a random day and pick the frequency of rainy days in the year? Or maybe I should consider the season we are in. Or am I supposed to use all available i... (read more)

Thinking of probabilities as levels of uncertainty became very obvious to me when thinking about the Monty Hall problem. After the host has revealed that one of the three doors has a booby prize behind it, you're left with two doors, with a good prize behind one of them.

If someone walks into the room at that stage, and you tell them that there's a good prize behind one door and a booby prize behind another, they will say that it's a 50/50 chance of selecting the door with the prize behind it. They're right for themselves, however the person who had been... (read more)

I'm sorry, why isn't the prior probability that you say "why yes, I am holding the ace of spades" = 1/4?

Edit: unless you meant "draw a pair", in which case yes, the ace of spades would show up in three out of six possible pairings.

Maybe it isn't really 50%, and it isn't really 100% how-it-came up either. That it is rational to make estimates based on our own ignorance is not proof that the universe... (read more)

So, I've been on this site for awhile. When I first came here, I had never had a formal introduction to Bayes' theorem, but it sounded a lot like ideas that I had independently worked out in my high school and college days (I was something of an amateur mathematician and game theorist).

A few days ago I was reading through one of your articles - I don't remember which one - and it suddenly struck me that I may not

actuallyunderstand priors as well as I think I do.After re-reading some fo the series, and then working through the math, I'm now reasonably con... (read more)

Very low, because B9 has to hypothesize a causal framework involving colors without any way of observing anything but quantitatively varying luminosities. In other words, they must guess that they're looking at the average of three variables instead of at one variable. This may sound simple but there are many other hypotheses that could also be true, like two variables, four variables, or most likely of all, one variable. B9 will be surprised. This is right and proper. Most physics theories you make up with no evidence behind them will be wrong.

This was a very difficult concept for me, Eliezer. Not because I disagree with the Bayesian principle that uncertainty is in the mind, but because I lacked the inferential step to jump from that to why there were different probabilities depending on the question you asked.

Might a better (or additional) way to explain this be to point out an analogy to the differing probabilities of truth you might assign to confirmed experimental hypothesis that were either originally vague, and therefore have less weight when adjusting the overall probability of truth vs. specific, and therefore shift the probability of truth further.

Hopefully I'm actually understanding this correctly at all.

The problem with trying to split, the it must be the oldest child who is the boy or the youngest child who is the boy is that the two situations overlap. You need to split the situation into oldest, youngest and both. If we made the ruling that both should be excluded, then we'd be able to complete the argument that there shouldn't be a difference between knowing that one child is a boy or knowing that the oldest child is a boy.

I think that the main point of this is correct, but the definition of "mind" used by this phrase is unclear and might be flawed. I'm not certain, just speculating.

As an aside, I think it is equivocation to talk about this kind of probability as being the same kind of probability that quantum mechanics leads to. No, hidden variable theories are not really worth considering.

But projectivism has been written about for quite a long time (since at least the 1700s), and is very well known so I find it hard to believe that there are any significant proponents of 'frequentism' (as you call it).

To those who've not thought about it, everyday projectivism comes naturally, but it falls apart at the slightest consideration.

When it comes to Hempel's raven, though, even those who understand projectivism can have difficulty coming to terms with the probabilistic reality.

I think I can show how probability is not purely in the mind but also an inherent property of things, bear with me.

Lets take an event of seeing snow outside, for simplicity we know that snow is out there 3 month a year in winter, that fact is well tested and repeats each year. That distribution of snowy days is property of the reality. When we go out of bunker after spending there unknown amount of time we assign probability 1/4 to seeing a snow, and that number is function of our uncertainty about the date and our precise knowledge of when snow is out th... (read more)

E[x]=0.5

even for the frequentist, and that's what we make decisions with, so focusing on p(x) is a bit of misdirection. The whole frequentist-vs-bayesian culture war is fake. They're both perfectly consistent with well defined questions. (They have to be, because math works.)

And yes to everything else, except...

As to whether god plays dice with the universe... that is not in the scope of probability theory. It's math. Your Bayesian is really a pragmatist, and your frequentist is a straw person.

Great post!

Kinship, or more accurately the lack of it, is likewise in the mind. That's why it always annoys me to see the parenthetical phrase "no relation" in a newspaper or magazine article.