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.
Since this discussion was reopened, I've spent some time - mostly while jogging - pondering and refining my stance on the points expressed. I just got around to writing them down. Since there is no other way to do it, I'll present them boldly, apologizing in advance if I seem overly harsh. There is no such intention.
1) "Accursed Frequentists" and "Self-righteous Bayesians" alike are right, and wrong. Probability is in your knowledge - or rather, the lack thereof - of what is in the environment. Specifically, it is the measure of the ambiguity in the situation.
2) Nothing is truly random. If you know the exact shape of a coin, its exact weight distribution, exactly how it is held before flipping, exactly what forces are applied to flip it, the exact properties of the air and air currents it tumbles through, and exactly how long it is in the air before being caught in you open palm, then you can calculate - not predict - whether it will show Heads or Tails. Any lack in this knowledge leaves multiple possibilities open, which is the ambiguity.
3) Saying "the coin is biased" is saying that there is an inherent property, over all of the ambiguous ways you could hold the coin, the ambiguous forces you could use to flip it, the ambiguous air properties, and the ambiguous tumbling times, for it to land one way or another. (Its shape and weight are fixed, so they are unambiguous even if they are not known, and probably the source of this "inherent property.")
4) Your state of mind defines probability only in how you use it to define the ambiguities you are accounting for. Eliezer's frequentist is perfectly correct to say he needs to know the bias of this coin, since in his state of mind the ambiguity is what this biased coin will do. And Eliezer is also perfectly correct to say the actual bias is unimportant. His answer is 50%, since in his mind the ambiguity is what any biased coin do. They are addressing different questions.
5) A simple change to the coin question puts Eliezer in the same "need the environment" situation he claims belongs only to the frequentist: Fli[p his coin twice. What probability are you willing to assign to getting the same result on both flips?
6) The problem with the "B9" question discussed recently, is that there is no framework to place the ambiguity within. No environmental circumstances that you can use to assess the probability.
7) The propensity for some frequentists to want probability to be "in the environment" is just a side effect of practical application. Say you want to evaluate a statistical question, such as the effectiveness of a drug. Drug effectiveness can vary with gender, age, race, and probably many other factors that are easily identified; that is, it is indeed "in the environment." You could ignore those possible differences, and get an answer that applies to a generic person just as Eliezer's answer applies to a generic biased coin. But it behooves you to eliminate whatever sources of ambiguity you easily can.
8) In geometry, "point" and "line" are undefined concepts. But we all have a pretty good idea what they are supposed to mean, and this meaning is fairly universal.
"Length" and "angle" are undefined measurements of what separates two different instances of "point" and "line," respectively. But again, we have a pretty clear idea of what is intended.
In probability, "outcome" is an undefined concept. But unlike geometry, where the presumed meaning is universal, a meaning for "outcome" is different for each ambiguous situation. But an "event" is defined - as a set of outcomes.
"Relative likelihood" is an undefined measurement what separates two different instances of "event." And just like "length," we have a pretty clear idea of what it is supposed to mean. It expresses the relative chances that either event will occur in any expression of the ambiguities we consider.
9) "Probability" is just the likelihood relative to everything. As such, it represents the fractional chances of an event's occurrence. So if we can repeat the same ambiguities exactly, we expect the frequency to approach the probability. But note: this is not a definition of probability, as Bayesians insist frequentists think. It is a side effect of what we want "likelihood" to mean.
10) Eliezer misstated the "classic" two-child problem. The problem he stated is the one that corresponds to the usual solution, but oddly enough the usual solution is wrong for the question that is usually asked. And here I'm referring to, among others, Martin Gardner's version and Marilyn vos Savant's more famous version. The difference is that Eliezer asks the parent if there is a boy, but the classic version simply states that one child is a boy. Gardner changed his answer to 1/2 because, when the reason we have this information is not known, you can't implicitly assume that you will always know about the boy in a boy+girl family.
And the reason I bring this up, is because the "brain-teasing ability" of the problem derives more from effects of this implied assumption, than from any "tendency to think of probabilities as inherent properties of objects." This can be seen by restating the problem as a variation of Bertrand's Box Paradox:
The probability that, in a family of two children, both have the same gender is 1/2. But suppose you learn that one child is in scouts - but you don’t know if it is Boy Scouts or Girl Scouts. If it is Boy Scouts, those who answer the actual "classic" problem as Eliezer answered his variation will say the probability of two boys is 1/3. They'd say the same thing, about two girls, if it is Girl Scouts. So it appears you don’t even need to know what branch of Scouting it is to change the answer to 1/3.
The fallacy in this logic is the same as the reason Eliezer reformulated the problem: the answer is 1/3 only if you ask a question equivalent to "is at least one a boy," not if you merely learn that fact. And the "brain-teaser ability" is because people sense, correctly, that they have no new information in the "classic" version of the problem which would allow the change from 1/2 to 1/3. But they are told, incorrectly, that the answer does change.