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."

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

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 single path. 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%.

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.

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 reality in 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.

If I'm being asked to accept or reject a number meant to correspond to the calculated or measured likelihood of heads coming up, and I trust the information about it being biased, then the only correct move is to reject the 0.5 probability.

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)

(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)

But frequentists emphatically are not talking about individual tosses. They are talking about infinitely repeated tosses.

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 probability you 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.

::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?

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 inherent probability 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.

What about when you're dealing with a medication that might kill someone, or not: in the absence of any information, do you say that's 50-50?

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 necessarily equally 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 £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 personal probability, I mean the actual probability', abandon all hope. All probability is 'perceived'. Unless you think you have all the evidence.

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 does have 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 two boys; 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, which is deterministic "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.

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.

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)

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!

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.

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?

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)