(Subjective Bayesianism vs. Frequentism) VS. Formalism

One of the core aims of the philosophy of probability is to explain the relationship between frequency and probability. The frequentist proposes identity as the relationship. This use of identity is highly dubious. We know how to check for identity between numbers, or even how to check for the weaker copula relation between particular objects; but how would we test the identity of frequency and probability? It is not immediately obvious that there is some simple value out there which is modeled by probability, like position and mass are values that are modeled by Newton's Principia. You can actually check if density * volume = mass, by taking separate measurements of mass, density and volume, but what would you measure to check a frequency against a probability?

There are certain appeals to frequentest philosophy: we would like to say that if a bag has 100 balls in it, only 1 of which is white, then the probability of drawing the white ball is 1/100, and that if we take a non-white ball out, the probability of drawing the white ball is now 1/99. Frequentism would make the philosophical justification of that inference trivial. But of course, anything a frequentist can do, a Bayesian can do (better). I mean that literally: it's the stronger magic.

A Subjective Bayesian, more or less, says that the reason frequencies are related to probabilities is because when you learn a frequency you thereby learn a fact about the world, and one must update one's degrees of belief on every available fact. The subjective Bayesian actually uses the copula in another strange way:

Probability is subjective degree of belief.

and subjective Bayesians also claim:

Probabilities are not in the world, they are in your mind.

These two statements are brilliantly championed in Probability is Subjectively Objective. But ultimately, the formalism which I would like to suggest denies both of these statements. Formalists do not ontologically commit themselves to probabilities, just as they do not say that numbers exist; hence we don't allocate probabilities in the mind or anywhere else; we only commit ourselves to number theory, and probability theory. Mathematical theories are simply repeatable processes which construct certain sequences of squiggles called "theorems", by changing the squiggles of other theorems, according to certain rules called "inferences". Inferences always take as input certain sequences of squiggles called premises, and output a sequence of squiggles called the conclusion. The only thing an inference ever does is add squiggles to a theorem, take away squiggles from a theorem, or both. It turns out that these squiggle sequences mixed with inferences can talk about almost anything, certainly any computable thing. The formalist does not need to ontologically commit to numbers to assert that "There is a prime greater than 10000.", even though "There is x such that" is a flat assertion of existence; because for the formalist "There is a prime greater than 10000." simply means that number theory contains a theorem which is interpreted as "there is a prime greater than 10000." When you say a mathematical fact in English, you are interpreting a theorem from a formal theory. If under your suggested interpretation, all of the theorems of the theory are true, then whatever system/mechanism your interpretation of the theory talks about, is said to be modeled by the theory.

So, what is the relation between frequency and probability proposed by formalism? Theorems of probability, may be interpreted as true statements about frequencies, when you assign certain squiggles certain words and claim the resulting natural language sentence. Or for short we can say: "Probability theory models frequency." It is trivial to show that Komolgorov models frequency, since it also models fractions; it is an algebra after all. More interestingly, probability theory models rational distributions of subjective degree of believe, and the optimal updating of degree of believe given new information. This is somewhat harder to show; dutch-book arguments do nicely to at least provide some intuitive understanding of the relation between degree of belief, betting, and probability, but there is still work to be done here. If Bayesian probability theory really does model rational belief, which many believe it does, then that is likely the most interesting thing we are ever going to be able to model with probability. But probability theory also models spatial measurement? Why not add the position that probability is volume to the debating lines of the philosophy of probability?

Why are frequentism's and subjective Bayesianism's misuses of the copula not as obvious as volumeism's? This is because what the Bayesian and frequentest are really arguing about is statistical methodology, they've just disguised the argument as an argument about what probability is. Your interpretation of probability theory will determine how you model uncertainty, and hence determine your statistical methodology. Volumeism cannot handle uncertainty in any obvious way; however, the Bayesian and frequentest interpretations of probability theory, imply two radically different ways of handling uncertainty.

The easiest way to understand the philosophical dispute between the frequentist and the subjective Bayesian is to look at the classic biased coin:

A subjective Bayesian and a frequentist are at a bar, and the bartender (being rather bored) tells the two that he has a biased coin, and asks them "what is the probability that the coin will come up heads on the first flip?" The frequentist says that for the coin to be biased means for it not have a 50% chance of coming up heads, so all we know is that it has a probability that is not equal 50%. The Bayesain says that that any evidence I have for it coming up heads, is also evidence for it coming up tails, since I know nothing about one outcome, that doesn't hold for its negation, and the only value which represents that symmetry is 50%.

I ask you. What is the difference between these two, and the poor souls engaged in endless debate over realism about sound in the beginning of Making Beliefs Pay Rent?

If a tree falls in a forest and no one hears it, does it make a sound? One says, "Yes it does, for it makes vibrations in the air." Another says, "No it does not, for there is no auditory processing in any brain."

One is being asked: "Are there pressure waves in the air if we aren't around?" the other is being asked: "Are there auditory experiences if we are not around?" The problem is that "sound" is being used to stand for both "auditory experience" and "pressure waves through air". They are both giving the right answers to these respective questions. But they are failing to Replace the Symbol with the Substance and they're using one word with two different meanings in different places. In the exact same way, "probability" is being used to stand for both "frequency of occurrence" and "rational degree of belief" in the dispute between the Bayesian and the frequentist. The correct answer to the question: "If the coin is flipped an infinite amount of times, how frequently would we expect to see a coin that landed on heads?" is "All we know, is that it wouldn't be 50%." because that is what it means for the coin to be biased. The correct answer to the question: "What is the optimal degree of belief that we should assign to the first trial being heads?" is "Precisely 50%.", because of the symmetrical evidential support the results get from our background information. How we should actually model the situation as statisticians depends on our goal. But remember that Bayesianism is the stronger magic, and the only contender for perfection in the competition.

For us formalists, probabilities are not anywhere. We do not even believe in probability technically, we only believe in probability theory. The only coherent uses of "probability" in natural language are purely syncategorematic. We should be very careful when we colloquially use "probability" as a noun or verb, and be very careful and clear about what we mean by this word play. Probability theory models many things, including degree of belief, and frequency. Whatever we may learn about rationality, frequency, measure, or any of the other mechanisms that probability models, through the interpretation of probability theorems, we learn because probability theory is isomorphic to those mechanisms. When you use the copula like the frequentist or the subjective Bayesian, it makes it hard to notice that probability theory modeling both frequency and degree of belief, is not a contradiction. If we use "is" instead of "model", it is clear that frequency is not degree of belief, so if probability is belief, then it is not frequency.  Though frequency is not degree of belief, frequency does model degree of belief, so if probability models frequency, it must also model degree of belief.

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In other words, the OP has mixed up the quotation and the referent (or the representation and the referent).

It seems to me that I am the one proposing a sharp distinction between probability theory (the representation), and rational degree of belief (the referent). If you say that probability is degree of belief, you destroy all the distinction between the model and the modeled. If by "probability" you mean subjective degree of belief, I don't really care what you call it. But know that "probability" has been used in ways which are not consistent with that synonymy claim. By the fact that we do not have 100% belief that bayes does model ideal inference with uncertainty, this means that bayesian probability is not identical to subjective belief given out knowledge. If X is identical to Y, then X is isomorphic-to/models Y. Because we can still conceive of bayes not perfectly modeling rationality, without implying a contradiction, this means that our current state of knowledge does not include that bayes is identical to subjective degree of belief.

We learn that something is probability by looking at probability theory, not by looking at subjective belief. If rational subjective belief turned out to not be modeled by probability theory, then we would say that subjective degree of belief was not like probability, not that probability theory does not define probability.

The first person to make bayes, may have been thinking about rationality when he/she first created the system, or he/she may have been thinking about spatial measurements, or he/she may have been thinking about finite frequencies, and he/she would have made the same formal system in every case. Their interpretations would have been different, but they would all be the one identical probability theory. Which one the actual creator was thinking of, is irrelevant. What spaces, beliefs, finite frequencies all have in common is that they are modeled by probability theory. To use "probability" to refer to one of these, over another, is a completely arbitrary choice (mind you I said finite frequency).

If we loose nothing by using "models" instead of "is", why would we ever use "is"? "Is' is a much stronger claim than "models". And frankly, I know how to check whether or not a given argument is an animal, for instance; how do I check if a given argument is a probability? I see if it satisifies the probability axioms. Finite frequency, measure, and rational degree of belief all seem to follow the probability axioms and inferences under specific, though similar, interpretations of probability theory.

As far as I can understand you, you seem to think that because humans aren’t always rational in the sense described by Cox’s postulates, probability theory only “models” human reasoning under uncertainty. You also seem to think that probability theory is just “squiggles on paper”.

Only models? Just squiggles on paper?

You've misunderstood the article, I think. Probability theory (the Kolmogorov Axioms) does model correct degrees of belief and describes normatively what they should be. It also models "long-term frequencies" in the sense that the Kolmogorov Axioms also apply to such things.

None of this requires the word "probability" to refer to degrees of belief. You don't even need a word at all to do the math and get the right answer. It's convenient to use the word that way though, since we already have a word "frequency" that refers to the stupider idea.

(And also I suspect that most people learned the word at school mostly by being given examples of likely and unlikely things. For them, "probability" refers to the little progress bar in their mind that goes up for more likely things and down for less likely things [ie. degrees of belief]. And thus many frequentists may commit philosophical errors when they try to define it as frequencies then use the intuitive definition to draw a conclusion in the same argument. This alone is a good reason to use "probability" for beliefs and "frequencies" for, well, frequencies.)

Yes, we can use "probability is degree of belief" but we have to be very careful about this sort of word play, because what that really means is that "probability models degree of belief".

Probability doesn't come from attempting to model something out in the world. It comes from attempting to find a measure of degree of belief that's consistent with certain desiderata, like "you shouldn't believe both a thing and its opposite." So the phrase "probability models degree of belief" is false.

You're riht, I mean to say "probability theory models theoretically optimal degree of belief updates, gven other degrees of belief". Or "probability theory models ideally rational degrees of belief."

Right. So what the heck's the point of the article? Why not just continue to say that probability is degrees of belief, and is not frequency?

Because then you'll keep arguing for decades about which one it really is, to absolutely no fruitful conclusion. Why not just keep saying that sound is air pressure and not auditory experience, or vice versa? When you do that, it makes it harder to see what is really going on. Call me conservative, but I think we should use as precise of a terminology as possible. Also, it seems to me that "probability is degree of belief" is an unverifiable claim, or I at least do not know what experiences I should test it with. But really, even in your own writing you don't feel comfortable using the copula as the relation between probability and degree of belief without italicizing it, doesn't that make you think that maybe there is a better word for the relation which you wouldn't feel like you need to italicize? How about "models"? And really we shouldn't be using probability as a noun, it's a function not an object, but we can deal with that later.

Exactly what about my article suggests that we should change our terminology to legitimize frequentism? I am saying that frequentism and subjective bayesianism both fail the moment they use the copula with probability as the subject, that is a stupid thing to do in philosophy. It's as bad as hegel. "Probability" is not a noun, it is a function, it is syncategorematic like "the", "or", "sake", etc. it is not categorematic; "probability" does not have a physical extension. And there are things that Volume has in common with degree of belief, which we might call probability like behavior. Again, if we found that degree of belief wasn't modeled by probability theory, we would say that subjective bayesianism was wrong, not that probability theory does not really describe probability. If "aubjective belief" did mean probability instead, if we found that probability theory did not model ideally rational degree of belief, we would say that komolgorov's axioms need to be fixed, they don't really define probability.

That isn't why there's a frequentist/Bayesian dispute. Everyone agrees they are both "interpretations". As another commenter has pointed out, the semantic argument is just a proxy for the dispute over whether one or other interpretation is preferable either philosophically or in practical terms.

Calling them interpretations seems to imply that at most one of them can be correct. "Displacement of a falling object on earth" and "kinetic energy of an 18.6 kg object" aren't competing interpretations of the math f(x) = 9.8x^2, they're just two different things the equation applies to.

If the frequentists are making any error, it's denying that beliefs must be updated according to the Kolmogorov Axioms, not asserting that frequencies can also be treated with the same laws. It's denying the former that might lead them to apply incorrect methods in inference, which is the only problem that really matters.

The definitional dispute about sound is different in that air pressure and auditory experience are both useful concepts, and there is no competition between them.

There is a dispute, ever hear of the idealists and the realists? Luckily it is over now. But either way. It does not matter why you are using one word to stand for many things, you shouldn't do it if you can use a terminology that is more widely accepted. I still think that bayesianism is a better interpretation, a much better interpretation than frequentism, but what is it an interpretation of? Is it an interpretation of math? Seems to me like it as interpretation of typographical string manipulations applied to certain basic strings.

As another commenter has pointed out, the semantic argument is just a proxy for the dispute over whether one or other interpretation is preferable either philosophically or in practical terms.

That wasn't another commenter, that was in my article, I'm pretty sure.

If people switched to saying that probability models both subjective degrees of belief and imaginary long-run frequency, there would still be this argument; however, it would then be harder for the Bayesian revolution (with whom the momentum lies) to finally oust the cursed frequentists, because language would be used in such a way as to imply equal validity of the interpretations.

If bayesianism wins this argument, which it probably will, it should win because it is the ideal system of statistical inference, not because they managed to convince a bunch of people of a statement with absolutely no empirical consequences. If you argue about what probability is you argue about surface bubbles of your theory that are just irrelevant to the real dispute you are having, whether you are a realist and an idealist, or a frequentist and a bayesian.

I think the interpretation of probability and what methods to use for inference are two separate debates. There was a really good discussion post on this a while back.

I'm also curious as to who exactly these frequentists are that you are arguing against. Perhaps I am spoiled by hanging out with people who regularly have to solve statistical problems, and therefore need to have a reasonable conception of statistics, but most frequentist sentiments that I encounter are fairly well-reasoned, sometimes even pointing out legitimate issues with Bayesian statistics. It is true that I sometimes get incorrect claims that I have to correct, but I don't think becoming a Bayesian magically protects you from this.

EDIT: To clarify, the "frequentist sentiments" I referred to did not explicitly distinguish between interpretations of probability and inference algorithms, but as the goal was engineering I think the arguments were all implicitly pragmatic.

I think the interpretation of probability and what methods to use for inference are two separate debates. There was a really good discussion post on this a while back.

I completely agree with this. It seems to me that we should completely throw away the question of what probability is, and look at which form of inference is optimal.

I'm going by what I've read of Jaynes, Yudkowsky, and books by a couple of other writers on Bayesian statistics.

I don't believe there are any legitimate issues with Bayesian statistics, because Bayes's rule is derived from basic desiderata of rationality which I find entirely convincing, and it seems to me that the maximum entropy principle is the best computable approximation to Solomonoff induction (although I'd appreciate other opinions on that).

There may be legitimate issues with people failing to apply the simple mathematical laws of probability theory correctly, because the correct application can get very complicated - but that is not an issue with Bayesian statistics per se. I'm sure that in many cases, the wisest thing to do might be to use frequentist methods, but being a Bayesian does not prohibit someone from applying frequentist methods when they are a convenient approximation.

The two issues that come to mind are the difficulty of specifying priors and the computational infeasibility of performing Bayesian updates.

I don't think anyone can reasonably dispute that if the correct prior is handed to you, together with a black box for applying Bayes' rule, then you should perform Bayesian updates based on your data to get a posterior distribution. That is simply a mathematical theorem (Bayes' theorem). And yes, it is also a theorem (Cox's theorem) that any rational agent is implicitly using a prior. But we aren't yet in a position to create a perfectly rational agent, and until we are, worrying about the specific form of consistency that is invoked for Cox's theorem seems silly.

It's possible that we don't really disagree. As a purely abstract statement about what you should do given unlimited computational resources, sure, Solomonoff induction is the way to go. I definitely agree with that. But if you need to actually solve a specific practical problem, additional considerations come into play.

By the way, what do you mean by "the maximum entropy principle is the best computable approximation to Solomonoff induction"? That sounds intriguing, so I'd be interested to have you elaborate a bit.

Regarding frequentism vs. Bayesianity in practical applications, the message I take from Yudkowsky and Jaynes is that frequentists have tended historically to lack apprehension of the fact that their methods are ad-hoc, and in general they fail to use Bayesian power when it is in fact advisable to do so - whereas Bayesians feel they can use ad-hoc approximate methods or accurate methods, whichever is appropriate to the task. This is a case in which a questionable philosophy needn't hamstring someone's thinking in principle, but appears to do so fairly predictably as a matter of fact.

Incidentally I'm surprised that there appears to be so much disagreement about this, given that LW is basically a forum brought into existence on the strength of Yudkowsky's abilities as a thinker, writer and populariser, and he clearly holds frequentism in contempt. It's not necessarily a bad thing that some people here are sympathetic to frequentism - intellectual diversity is good - I'm just surprised that there are so many on a Bayesian rationality forum!

About Maxent: I had in mind chapter 5 of this book by Li and Vitanyi.

We can formulate scientific theories in two steps. First, we formulate a set of possible alternative hypotheses, based on scientific observations or other data. Second, we select one hypothesis as the most likely one. Statistics is the mathematics of how to do this. A relatively recent paradigm in statistical inference was developed by J.J. Rissanen and by C.S. Wallace and his coauthors. The method can be viewed as a computable approximation to the incomputable approach in Section 5.2 [i.e. Solomonoff induction] and was inspired by it. In accordance with Occam’s dictum, it tells us to go for the explanation that compresses the data the most. [...]

This is the MDL (minimum description length) principle.

The ideal MDL principle selects the hypothesis H that minimizes K(H) + K(D|H) [...]

Where K is Kolmogorov complexity.

Unfortunately, the function K is not computable (Section 3.4). For practical applications one must settle for easily computable approximations. [...]

So ideal MDL, like Solomonoff induction, is also incomputable!

They go on to discuss approximations, and on page 390 (I don’t know if you have a copy of the book) they provide a usable approximation to be referred to as “MDL”. Later on page 398 they discuss Maxent, and conclude that that too is an approximation to ideal MDL.

As far as I can see, Maxent is more useful in practical applications than their approximate MDL. I felt that Maxent needed to be defended, since Jaynes considered it to be a major element of Bayesian probability theory; and as far as I can see there is no clearly better practical method of generating priors at this point in time such that Maxent could be considered to be one of Bayesianity’s “legitimate issues” vis a vis frequentism.

Incidentally I'm surprised that there appears to be so much disagreement about this, given that LW is basically a forum brought into existence on the strength of Yudkowsky's abilities as a thinker, writer and populariser, and he clearly holds frequentism in contempt. It's not necessarily a bad thing that some people here are sympathetic to frequentism - intellectual diversity is good - I'm just surprised that there are so many on a Bayesian rationality forum!

My intuition here is that you are not observing so many people who are sympathetic to frequentism, so much as people who are unsympathetic to holding contempt.

In much of the comments here you seem to be missing a simple point about mathematics and reference due to its relationship to tribal signaling between the "Bayesians" and the "Frequentists".

I've yet to see anything in this article, or the resulting comments thread, to suggest that the OP has anything to say apart from "let's say 'models' instead of 'is' (but mean the same thing)". And the only consequence of this is to puff up frequentism.

I tried (and apparently failed miserably) to make the case that in the interests of sanity, we should define our terms such that probability ≡ subjective degrees of belief. That's all it is, a definition - there's no philosophical significance to this "is" beyond that. It is not a claim that the frequency interpretation doesn't fit Cox's postulates - this is a naive interpretation of how language is used on the OP's part.

The definitional dispute about sound is inapt, because there is nothing to be gained by defining sound as one thing or the other. In this case however there is a real benefit to defining our terms in one particular way.

I will however delete the downvoted posts in this thread, to honour the great disapproval with which my conception of rationality has apparently met in this case.

I will however delete the downvoted posts in this thread, to honour the great disapproval with which my conception of rationality has apparently met in this case.

Generally, deleting posts with responses is impolite, as the discussion may be helpful to future readers.

I tried (and apparently failed miserably) to make the case that in in the interests of sanity, we should define our terms such that probability ≡ subjective degrees of belief.

I don't think you ever supplied a term other than "probability" that we should use for what the OP thought "probability" means. So we're still left with three entities and two words.

I don't think you ever supplied a term other than "probability" that we should use for what the OP thought "probability" means. So we're still left with three entities and two words.

Seems like a non-problem. Just say "I am entering these frequencies into Bayes's theorem", "I am using the mathematical tools of probability theory" or something like that.

Or perhaps say "probability is a measure of subjectively objective degrees of belief", and "probability theory is the set of mathematical tools used to compute probabilities, which can also be used to compute frequencies as the case may be".

Which is pretty much what happens already! This is why I object to such an article - it's a solution looking for a problem, which creates the illusion of a problem by a) being illiterate, so making itself hard to pin down b) nitpicking the use of words.

Generally, deleting posts with responses is impolite, as the discussion may be helpful to future readers.

They were also steadily generating an amount of negative karma days after posting that I felt was disproportionate, considering they were a sincere attempt to reach agreement with a less-than-articulate interlocutor.

They were also steadily generating an amount of negative karma days after posting that I felt was disproportionate, considering they were a sincere attempt to reach agreement with a less-than-articulate interlocutor.

Would not retraction have served?

They were also steadily generating an amount of negative karma days after posting that I felt was disproportionate, considering they were a sincere attempt to reach agreement with a less-than-articulate interlocutor.

I did not find User:potato less-than-articulate.

a) being illiterate, so making itself hard to pin down

I'm not sure what you mean by "illiterate" here, nor (thus) how it would make itself 'hard to pin down'.

b) nitpicking the use of words.

The dispute was about the proper use of words. I did not see anything that looked like 'nitpicking' in that context.

The advantage of "Formalism" over "Bayesianism" or "Frequentism" is that it clearly marks the mathematical toolkit, makes it clear what Bayesians and Frequentists are separately talking about, gets rid of the slippage Frequentists allegedly make between "degrees of belief" and "frequencies", and removes the question of what "probability" is "really" about, all without having to raise a flag in the mind-killing tribal warfare between "Bayesians" and "Frequentists".

But then, it's been noted that "a philosopher has never met a distinction he didn't like", so perhaps I'm just biased in favor of making clearer the distinction.

So in "formalism", I understand that we are to say: "probability models frequency", "probability models subjective degrees of belief" and "probability is the set of mathematical discoveries we have made, which deal with [ ], including such things as Bayes's theorem".

Whereas at the moment, Bayesians say: "probability is a measure of subjective degrees of belief", "probability isn't frequency", and "probability theory is the set of mathematical discoveries we have made, which deal with probability, including such things as Bayes's theorem".

And frequentists say: "probability is long-run frequency", "probability isn't subjective degrees of belief", and "probability theory is the set of mathematical discoveries we have made, which deal with probability, including such things as Bayes's theorem".

I like the Bayesian version. But the frequentist version doesn't confuse me; I understand perfectly well that these are merely competing interpretations, and I've never felt the urge to argue specifically about whether probability is degrees of belief or is frequency - nor have I ever seen anyone else do so. Clearly that would be a stupid argument, just like the definitional dispute about sound. However, sensible people do use these terms, arguing about whether probability 'is' one or the other, as a proxy for a more substantive argument about which is the better - i.e. more philosophically parsimonious, and having better practical outcomes - interpretation. (Actually they are more likely to phrase the argument as "probability should be considered to be X", and then say probability is X when they aren't having the argument, but hey.)

As for the "formalist" version, firstly it puts the frequentist and Bayesian interpretations on a level footing. Even if sensible people were wasting time and effort arguing specifically over a mere definition, the cost of conceding ground to the problematic frequentist interpretation outweighs any benefit from ending that argument, in comparison to the option of simply carrying on using the language of the Bayesian.

Furthermore it appears to me that probability theory, given this use of language, lacks a referent. Probability theory has been renamed (simply) probability, and it no longer appears to be the theory of anything. Whether or not this use of language could be considered wrong per se, it hardly seems to be clearing up any philosophical confusion! If I ask "what is this thing that I am computing using Bayes's theorem?", the answer is no longer "the posterior probability" - if probability is the new word for the mathematical tools of probability theory, the phrase posterior probability no longer means anything. So perhaps I'll have to invent a new word to refer to the same thing that the word probability used to refer to.

Do you begin to see why I think this is a waste of time?

NB: I think we're making much more progress than I made with user:potato. That's what I mean about the difficulty of having to argue with someone who is inarticulate, i.e. can't state his case properly.

"probability is the set of mathematical discoveries we have made, which deal with [ ], including such things as Bayes's theorem".

Probably better put in terms of being a formal system, rather than "a set of mathematical discoveries". But I fear that tends towards begging the question!

As for the "formalist" version, firstly it puts the frequentist and Bayesian interpretations on a level footing. Even if sensible people were wasting time and effort arguing specifically over a mere definition, the cost of conceding ground to the problematic frequentist interpretation outweighs any benefit from ending that argument, in comparison to the option of simply carrying on using the language of the Bayesian.

This treatment (notably the use of terms like "conceding ground") suggests that you are engaging in a "political"/"debate" mode rather than a "truth-seeking" mode. This leads me to believe that we have more to lose by accepting the "Bayesian/Frequentist" duality than by dissolving it entirely and changing our terminology to match. This matches my impression of previous forays into the "Bayesian/Frequentist" 'holy wars'.

If politics is mind-killing, then it must certainly be avoided even at great cost with respect to our most basic tools of rationality.

Do you begin to see why I think this is a waste of time?

Indeed, though in that case you've spent far more time on this than most who exercised the default 'ignore' option.

If I ask "what is this thing that I am computing using Bayes's theorem?", the answer is no longer "the posterior probability" - if probability is the new word for the mathematical tools of probability theory, the phrase posterior probability no longer means anything. So perhaps I'll have to invent a new word to refer to the same thing that the word probability used to refer to.

A good point.

That's what I mean about the difficulty of having to argue with someone who is inarticulate, i.e. can't state his case properly.

I understood what you meant - I just did not see any inarticulateness on the part of User:potato.

I've never felt the urge to argue specifically about whether probability is degrees of belief or is frequency - nor have I ever seen anyone else do so.

I normally see this being explicitly the subject on Bayesian/Frequentist debates, and many long conversations with philosophers have revolved around whether "equating probability with subjective belief" is an "ontological confusion".

This treatment (notably the use of terms like "conceding ground") suggests that you are engaging in a "political"/"debate" mode rather than a "truth-seeking" mode.

Duly noted. I'll try not to give this impression in future.

I normally see this being explicitly the subject on Bayesian/Frequentist debates, and many long conversations with philosophers have revolved around whether "equating probability with subjective belief" is an "ontological confusion".

I may have simply failed to notice these arguments taking place. In order to dissolve any such ostensible ontological question, I'd recommend pointing out that to say probability is one or other thing is merely a statement to the effect that one interpretation is preferred for some reason by the writer - since both interpretations satisfy the Cox postulates or Kolmogorov axioms, we could define probability to be either subjective degrees of belief or long-run frequency, and make sound and rational inferences in either case (albeit perhaps not with the same efficiency). This should be enough to persuade an otherwise sensible person that he's engaged in a futile argument about definitions.

Formalism attempts to solve the problem by effectively tabooing the concept of probability such that it no longer has a definition. Although we might be able to get around the problem that I mentioned by answering the question ""what is this thing that I am computing using Bayes's theorem?" by saying "the posterior subjective degree of belief" or "the posterior frequency", it's easy to see how the same kind of philosophers would end up arguing over whether, in the case of a coin flip for example, we are really talking about prior and posterior subjective degrees of belief, or about prior and posterior long-run frequencies. And we would have lost the use of the word "probability", which makes our messages shorter than they would otherwise be.

To the extent that there is such a thing as the proper use of words, to delete useful words from our vocabulary in order to (probably unsuccessfully) prevent people from having a definitional argument that could best be dispelled by introducing them to such notions as "dissolving the question" and reductionism isn't it. On the other hand I'll give user:potato credit for exposing an issue that may be more problematic than I at first believed.

I expect that we are substantially in agreement at this point.

FWIW, I think my three preferred terms are "Probabilities", "Frequencies", and "Normed Measure Theory". That's what Kolmogorov's formalization amounts to anyway, and as the OP said it truly need not be connected to either probabilities or frequencies in a given use.

I don't understand. Based on reading through the passages you referenced in PtLoS, maximum entropy is a way of choosing a distribution out of a family of distributions (which, by the way, is a frequentist technique, not a Bayesian one). Solomonoff induction is a choice of prior. I don't really understand in what sense these are related to each other, or in what sense Maxent generates priors at all.

Incidentally I'm surprised that there appears to be so much disagreement about this, given that LW is basically a forum brought into existence on the strength of Yudkowsky's abilities as a thinker, writer and populariser, and he clearly holds frequentism in contempt.

I've always felt that the frequentists that Eliezer argues against are straw men. As I said earlier, I've never met a frequentist who is guilty of the accusations that you keep making, although I have met Bayesians whose philosophy interfered with their ability to do good statistical modeling / inference. Have you actually run into the people who you seem to be arguing against? If not, then I think you should restrict yourself to arguing against opinions that people are actually trying to support, although I also think that whether or not some very foolish people happen to be frequentists is irrelevant to the discussion (something Eliezer himself discussed in the "Reversed Stupidity is not Intelligence" post).

If you know nothing about a variable except that it's in the interval [a, b] your probability distribution must be from the class of distributions where p(x) = 0 for x outside of [a, b]. You pick the distribution of maximal entropy from this class as your prior, to encode ignorance of everything except that x ∈ [a,b].

That is one way Maxent may generate a prior, anyway.

a way of choosing a distribution out of a family of distributions (which, by the way, is a frequentist technique, not a Bayesian one).

We can call dibs on things now? Ooh, I call dibs on approximating a slowly varying function as a constant!

I'm pretty sure almost all of freqeuntist methods are derivable as from bayes, or close approximations of bayes. Do they have any tool which is radically un-bayesian?

See paulfchristiano's examples elsewhere in this thread.

Another example would be support vector machines, which work really well in practice but aren't Bayesian (although it's possible that they are actually Bayesian and I just can't figure out what prior they correspond to).

There are also neural networks, which are sort of Bayesian but (I think?) not really. I'm not actually that familiar with neural nets (or SVMs for that matter) so I could just be wrong.

ETA: It is the case that every non-dominated decision procedure is either a Bayesian procedure or the limit of Bayesian procedures (which I think could alternately be thought of as a Bayesian procedure with a potentially improper prior). So in that sense, for any frequentist procedure that is not Bayesian, there is another procedure that gets higher expected utility in all possible worlds, and is therefore strictly better. The only problem is that this is again an abstract statement about decision procedures, and doesn't take into account the computational difficulty of actually finding the better procedure.

This paper is the closest I've ever seen to a fully Bayesian interpretation of SVMs; mind you, the authors still use "pseudo-likelihood" to describe the data-dependent part of the optimization criterion.

Neural networks are just a kind of non-linear model. You can perform Bayes upon them if you want.

The definitional dispute about sound is different in that air pressure and auditory experience are both useful concepts, and there is no competition between them.

There is a dispute, ever hear of the idealists and the realists? Luckily it is over now. But either way. It does not matter why you are using one word to stand for many things, you shouldn't do it if you can use a terminology that is more widely accepted. I still think that bayesianism is a better interpretation, a much better interpretation than frequentism, but what is it an interpretation of? Is it an interpretation of math? Seems to me like it as interpretation of typographical string manipulations applied to certain basic strings.

As another commenter has pointed out, the semantic argument is just a proxy for the dispute over whether one or other interpretation is preferable either philosophically or in practical terms.

That wasn't another commenter, that was in my article, I'm pretty sure.

If people switched to saying that probability models both subjective degrees of belief and imaginary long-run frequency, there would still be this argument; however, it would then be harder for the Bayesian revolution (with whom the momentum lies) to finally oust the cursed frequentists, because language would be used in such a way as to imply equal validity of the interpretations.

If bayesianism wins this argument, which it probably will, it should win because it is the ideal system of statistical inference, not because they managed to convince a bunch of people of a statement with absolutely no empirical consequences. If you argue about what probability is you argue about surface bubbles of your theory that are just irrelevant to the real dispute you are having, whether you are a realist and an idealist, or a frequentist and a bayesian.

I don't think that's where I meant to put that comment.

Probability theory is maths, and although I agree that questions like “where is maths?” and “what is maths?” and “does maths exist?” are confusing

See, these questions are not confusing to me at all. Hofstadter's formalism deals with them perfectly. Have you ever read G.E.B.? I assumed so, but I wasn't sure, maybe you haven't.

Yes, I do think that probability theory is a repeatable process of typographical string manipulations. What do you think it is?

Ultimately language should be useful, and I don’t see the point of changing the word “is” to the word “models”. This wouldn’t change my beliefs about probability theory; I’d just be using the word “models” to mean the same thing as the word “is”. And I would then lack a means of saying that the Bayesian interpretation of probability is good, and the frequentist interpretation is stupid and counter-productive – I want to be able to say that probability is X, and probability isn't Y, because this is the most useful way of using language to talk about probability theory – why would I want to put the good interpretation and the dumb interpretation on an equal footing by saying that probability “models” X and also “models” Y?

Here I completely disagree, and almost wonder if you haven't been reading my comments. Bayesianism is stronger, more capable, perfecter, stronger, more rational, more useful than frequentism, first of all, and all of that has nothing to do with the commitment to conceptualism that subjective bayes requires. This is all still true if you are a formalist.

Bayesianism is not righter than frequentism because probabilities are really subjective beliefs, and the frequentists were wrong, it's not frequency. Bayesains are righter than frequentists because bayes-inferences are deductively demonstrable to win more than frequentist-inferences. Again, the argument about what probability really is is just a way to disguise the argument about who's statistical method is more successful, the only way the frequentist even has a shot at such an argument if it is disguised as a question about what probability is instead of a question about who's inferences are theoretically ideal.

So um, platonism? Really? Why? What does it get you that formalism doesn't with less ontological commitment?

Why can't a frequentist say: "Bayesians are conflating probability with subjective degree of belief." ? They were here first after all.

Probability does model frequency, and it does model subjective degree of believe, and this is not a contradiction. Using the copula is the problem, obviously: if subjective degree of believe is not frequency, and probability is frequency, then probability is not subjective degree of belief. Analogously, if subjective degree of believe is not frequency, and probability is subjective degree of belief, then probability is not frequency.

The problem is that they all conflate "probability" with "subjective degree of belief" and "frequency", the bayesian conflates subjective degree of belief and probability. The frequentist conflates probability and frequency.

The frequentist/Bayesian dispute is of real import, because ad-hoc frequentist statistical methods often break down in extreme cases, throw away useful data, only work well with Gaussian sampling distributions etc.

The debate over whether to use Bayesian methods or frequentest methods is of import. I think potato was trying to say this here:

How we should actually model the situation as a probability distribution depends on our goal. But remember that Bayesianism is the stronger magic.

But the question of whether probability is frequency, or if probability is subjective degree of belief, is just as silly as a dispute over whether numbers are quantity, or if they are orders. The answer is that numbers model both, and are neither.

Probability "models" frequency in the sense that sometimes frequency data dominates all of our other knowledge about some phenomenon.

No, probability models frequency in the sense that there is an interpretation of komologorov which only mentions terms from the part of our language used to talk about frequency, and all komologorov theorems come out as true statements about frequency under this interpretation.

I mean, literally, Bayes is an arithmetic of odds and fractions, of course it models frequency. At least as well as fractions and odds do. Probability is a frequency as often as it is a fraction or an odds.

So you are saying that as long as frequentists understand that Bayesian methods are theoretically ideal and cannot be improved upon, whereas frequentist methods may be useful approximations, they shouldn't run into any real life mistakes. This is nearly true, were it not for the fact that frequentists don't actually believe this.

They don't, but could and should.

If someone manages simultaneously to believe that frequentist philosophy (probability ≡ frequency) is sound, yet frequentist methods are fallible ad-hoc methods and Bayesian methods provide the best inferences possible given our state of knowledge, then he is performing quite a feat.

I agree, this is why instead of saying that probability is identical to frequency, or that it is frequency, we should say that it models frequency.

What is this comment supposed to add? Is it an ad hominem, or are you asking for clarification? If you don't understand that comment perhaps you should try rereading my original post, I have updated it a bit since you first commented, perhaps it is clearer.

(edit) clarification:

The reason that probabilities model frequency is not because our data about some phenomena are dominated by facts of frequency. If you take 10 chips, 6 of them red, 4 of them blue, 5 red ones and 1 blue one on the table, and the rest not on the table, you'll find that bayes can be used to talk about the frequencies of these predicates in the population. You only need to start with theorems that when interpreted produce the assumptions I just provided, e.g., P(red and on the table) = 1/2, P(~red and on the table) = 1/10, P(red and ~on the table) = 2/5. From those basic statements we can infer using bayes all the following results: P(red|on the table) = 5/6, P(~red|on the table) = 1/6, P( (red and on the table) or blue) = 9/10, P(red) = P(red|on the table) P(on the table) + P(red|~on the table) P(~on the table) = 6/10, etc. These are all facts about the FREQUENCY distributions of these chips' predicates, which can be reached using bayes, and the assumptions above. We can interpret P(red) as the frequency of red chips out of all the chips, and P(red|on the table) as the frequency of red chips out of chips on the table. You'll find that anything you proof about these frequencies using bayesian inference will be true claims about the frequencies of these predicates within the chips. Hence, bayes models frequency. This is all I meant by bayes models frequency. You'll also find that it works just as well with volume or area. (I am sorry I wasn't that concrete to begin with.)

In the same exact way, you can interpret probability theorems as talking about degrees of belief, and if you ask a bayesian, all those interpreted theorems will come out as true statements about rational degree of belief. In this way bayes models rational belief. You can also interpret probability theory as talking about boston's night life, but not everyone of those interpreted theorems will be true, so probabiliity theory does not model boston's night life under that interpretation. To model something, means to produce only true statements under a given interpretation about that something.

Frequentists may not treat their tool box as a set of mostly unrelated approximations to perfect learning, or treat bayes as the optimal laws of inference, but they should as far as I can tell. And if they did, they would not cease to be frequentists, they would still use the same methods, use "probability" the same way, and still focus on long run frequency over evidential support. The only difference is that rather than saying probability is frequency and that probability is not subjective degree of belief, they would say that probability models both frequency and subjective degree of belief. Subjective bayesians should make a similar update, though I am sure they don't swing the copula around as liberally as frequentists. This is what i meant when i said that frequentists could and should believe that frequentism is just a useful approximation, and that bayes is in some sense optimal. I was never really arguing about the practical advantages of bayesianism over frequntism, but about how they both seem to make a similar philosophical mistake in using identity or the copula when the relation of modeling is more applicable. A properly Hofstadterish formalism seems like the best way to deal with all of this comprehensively.

You understand what I was saying now? I really want to know. That you are confused by what seem to me to be my most basic claims, and that you are also as familiar with E. T. Jaynes as your comments suggests is worrying to me. Does this clarification make you less confused?

Fine, let's make up a new frequentism, which is probably already in existence: finite frequentism. Bayes still models finite frequencies, like the example i gave of the chips.

When a normal frequentest would say "as the number of trials goes to infinity" the finite frequentest can say "on average" or "the expectation of". Rather than saying, as the number of die rolls goes to infinity the fraction of sixes is 1/6, we can just say that as the number rises it stabilizes around and gets closer to 1/6. That is a fact which is finitely verifiable. If we saw that the more die rolls we added to the average, the closer the fraction of sixes approached 1/2, and the closer it hovered around 1/2, the frequentest claim would be falsified.

There may be no infinite populations. But the frequentist can still make due with finite frequencies and expected frequencies, and i am not sure what he would loose. There are certainly finite frequencies in the world, and average frequencies are at least empirically testable. What can the frequentist do with infinite populations or trials, that he/she can't do with expected/average frequencies.

Also, are you a finitist when it comes to calculus? Because the differential calculus requires much more commitment to the idea of a limit, infinity, and the infinitesimal, than frequentists require, if frequentests require these concepts at all. Would you find a finitist interpretation of the calculus to be more philosophically sound than the classical approach?

potato,

I don't think there's much value in replying to Phlebas' latest reply.

Except one makes it seem like the stuff does exist and the other makes it seem like it doesn't. If we interpret the law of large numbers as saying that after an infinite number of trials, the average value of that sequence of results will equal the expected value of the random variable, then any finite amount of evidence is not enough to be evidence of this interpretation, let alone to verify it. But if we interpret the law as saying that the more trials we add, the more closely the average result should hover around the expected value of the variable. That interpretation can be falsified and evidenced empirically using only finite observations.

Now for this part:

This assumes an "expected value" which could only be known by some other means, i.e. accepting the Bayesian notion of probability as subjective degrees of belief, or supposing an infinite number of trials. Such a definition of frequentism begs the question.

Well it is actually in the bartender's premise that the coin is biased, so they both know that whatever heads/trials hovers around as trials rises, it is not 1/2.

But assuming they didn't have that premise, what could the frequentist do, without requiring non-empirically verifiable claims as assumptions?

Only thing i can think of: He/she could resort to ranges. Never actually defining the probability of heads, just determining the probability with which the actual probability i.e. frequency of heads, is within a given range. There is some ideal actual frequency, which would be the outcome given infinitely many trials, but you can only find a range within which it is, and it would require infinite amounts of evidence to constrain heads/trials to a point; and we don't have that kind of time. Bayes can be extended to ranges of probability trivially. THis would make it so that finite observables act as evidence for some hypotheses which include the term "infinity". But it wouldn't justify the whole of frequentest methodology.

But again, even if the frequentest interpretation fails in ways which the bayesian interpretation does not, this is not evidence of probability being degree of belief. It is evidence of probability modeling degree of belief, and of Bayesianism having sounder ontological commitments than frequentism. This would not surprise me.

Infinite frequency is not real. But our intuitions about it are real. Komolgorov may then be said to model actual finite frequencies, and our intuitions about infinite frequencies which are finitely and axiomatically describable. Let us not forget that there are not circles or squares anywhere either, but we should still hold that you can't square the circle. Not all models have to be out there, some may be in here Frequentism requires infinite frequencies for its interpretation to be true, which don't exist. The subjective bayes interpretation of bayes does not require anything that really doesn't exist (though degrees of belief are plenty mysterious). This is a good reason to be a subjective Bayesian, and not a frequentest, which I was not aware of consciously, but it is not a good reason to stop being a formalist.

Who cares if frequentists, or non-LW bayesians, use the copula like a bunch of sillies, even after G.E.B. is published. We LWers, should use "identity" if we are claiming identity, and "modeling" if we are claiming a model. But realistically, the claim that "Probability theory models rational belief systems." seems much more defensible, concrete, and useful, than the claim that "Probability is degree of belief."

Ok, I am a Bayesian, i.e., I use bayesianism over frequentism, and find frequentest methods rather silly. And I am at least what I would call a finite frequentist, i.e., I think komolgorov models finite frequency.

I am not here to say that Bayesianism is on equal ground with Frequentism, at all. If Bayes's interpreted sentences can be empirically verified, and freqeuntist interpretations cannot be empirically verified, than this is to bayesianism's favor. It means it is the more useful theory. But it is not grounds to use "is" where we should use "models" instead. It is not because bayesains need to be put on equal footing with frequentists that I propose this terminology in place of the copula; it is because rationalists should be clear, specially in philosophy. So just to be clear, we should use "model" instead of "is" because it is what is really going on; the concepts of Hofstadterish formalism and model theory, are the best way to understand how probability theory ends up telling us how to distribute beliefs. The relation between subjective degree of belief, and probability theory, is clearly not identity, or the copula.

Subjective degrees of belief are a part of your cognition. Probability theory is a repeatable process consisting in shuffling squiggles on paper, or some other medium. These are not identical. Q.e.d.

We might call this the "paper projection fallacy". Where you project some pattern in squiggles on a piece of paper into your mind. Analogous to the "mind projection fallacy".

Bayesian probability theory is the mathematical formulation representing ideal reasoning under uncertainty

The squiggles on the paper are our representation of this probability - they are "probability", not probability, if you like.

no, probability, or I assume you really mean rationality, is the void

Bayes is just playing with squiggles on paper. If when you interpreted bayes, you found some claim, which seemed to not work, you would have to abandon bayes, or be irrational.

The squiggles on the paper are our representation of this probability.

What probability where? If you start by saying degree of belief is probability, and then show that degreeo f belief is probability, I am not impressed. You can call them "the representation" instead of calling them "the theory" if you want. And you can use "is" instead of model if you want. And you can even use "probability" instead of "degree of belief", though I suspect that may all get rather confusing quickly. But do realize that every reason you give for saying that probability is degree of belief, a frequentest can give for saying that probability is frequency.

"Probability" is a really stupid noun, kind of like "red-hood", or "emergence". Notice how in the actual theory, we only ever talk about the probability of something. "Probability" is a function, not an object. Ask yourself: "what IS probability?" really probe, and you'll find that that is a stupid question. The right question would have been, "what does probability return given an argument?" The answer is that it might return the rational degree of belief of a proposition, the frequency of a predicate out of a finite population, the frequency of a value out of an infinite amount of trials, the volume of a space, the area of a shape, or even the length of a line. All of these are consistent with the komologorov axioms.

The frequentist/Bayesian dispute is of real import, because ad-hoc frequentist statistical methods often break down in extreme cases, throw away useful data, only work well with Gaussian sampling distributions etc.

I think you have this backwards. Frequentist techniques typically come with adversarial guarantees (i.e., "as long as the underlying distribution has bounded variance, this method will work"), whereas Bayesian techniques, by choosing a specific prior (such as a Gaussian prior), are making an assumption that will hurt them in an extreme cases or when the data is not drawn from the prior. The tradeoff is that frequentist methods tend to be much more conservative as a result (requiring more data to come to the same conclusion).

If you have a reasonable Bayesian generative model, then using it will probably give you better results with less data. But if you really can't even build the model (i.e. specify a prior that you trust) then frequentist techniques might actually be appropriate. Note that the distinction I'm drawing is between Bayesian and frequentist techniques, as opposed to Bayesian and frequentist interpretations of probability. In the former case, there are actual reasons to use both. In the latter case, I agree with you that the Bayesian interpretation is obviously correct.

Bayesian techniques, by choosing a specific prior (such as a Gaussian prior), are making an assumption that will hurt them in an extreme cases or when the data is not drawn from the prior. The tradeoff is that frequentist methods tend to be much more conservative as a result (requiring more data to come to the same conclusion).

Bayesian methods with uninformative (possibly improper) priors agree with frequentist methods whenever the latter make sense.

Are you referring to the result that every non-dominated decision procedure is either a Bayesian procedure or a limit of Bayesian procedures? If so, one could imagine a frequentist procedure that is strictly dominated by other procedures, but where finding the dominating procedures is computationally infeasible. Alternately, a procedure could be non-dominated, and thus Bayesian for the right choice of prior, but the correct choice of prior could be difficult to find (the only proof I know of the "non-dominated => Bayesian" result is non-constructive).

Can you explain further? Casually, I consider results like compressed sensing and multiplicative weights to be examples of frequentist approaches (as do people working in these areas), which achieve their results in adversarial settings where no prior is available. I would be interested in seeing how Bayesian methods with improper priors recommend similar behavior.

I admit I'm not familiar with either of those... Can you make a simple example of an “adversarial setting where no prior is available”?

I let you choose some linear functionals, and then tell you the value of each one on some unknown sparse vector (compressed sensing).

We play an iterated game with unknown payoffs; you observe your payoff in each round, but nothing more, and want to maximize total payoff (multiplicative weights).

Put even more simply, what is the Bayesian method that plays randomly in rock-paper-scissors against an unknown adversary? Minimax play seems like a canonical example of a frequentist method; if you have any fixed model of your adversary you might as well play deterministically (at least if you are doing consequentialist loss minimization).

The minimax estimator can be related to Bayesian estimation through the concept of a "least-favorable prior".

User:potato smartly linked to Wei Dai's post "Frequentist Magic vs. Bayesian Magic", which along with its commentary would perhaps be the next thing to read after this post. (Wei Dai and user:potato think that (algorithmic) Bayesian magic is stronger but I agree with Toby Ord's and Vladimir Slepnev's points and would caution against hasty conclusions.)

More interestingly, probability theory models rational distributions of subjective degree of believe, and the optimal updating of degree of believe given new information. This is somewhat harder to show; dutch-book arguments do nicely to at least provide some intuitive understanding of the relation between degree of belief, betting, and probability, but there is still work to be done here.

In a sense there has been highly important and insightful work done in the vein of the Dutch book arguments, looking into e.g. how decisions should be made under what is called "indexical" or "anthropic" uncertainty, a situation where Bayesian reasoning seems like it ought to work but doesn't as such. This work has been done in large part by Less Wrong-cognizant folk. One easy-to-understand summary of some of the motivations for such research can be found found in User:ata's analysis of the classic "sleeping beauty" problem which is similar to User:potato's disambiguating approach used in the above post (and note Vladimir Nesov's similar warning against equivocation on the word "probability"). If you're interested in the foundations of justification (whether epistemic or moral, or what's the difference?), then you'd be wise to look into the big open questions in decision theory. There are a lot of 'em.

Based on reading the comment threads here, it seems as though some folks are missing something important in this post. So, I'll try to restate it differently in case that helps. Below, double-quotes are used to refer to the word, while single-quotes are used merely to highlight and separate tokens.

There is a mathematical formalism, which we can call "probability", which does a good job of modeling various sorts of things, like 'subjective degrees of belief', 'frequencies', 'volume', and 'area'.

'Bayesians' think that "probability" should refer to 'subjective degrees of belief'. 'Frequentists' think that "probability" should refer to 'frequencies'. One alleged problem with 'Frequentists' is that they seem to carelessly slip between 'subjective degrees of belief' and 'frequencies', due to the natural-language meaning of "probability".

One suggestion proposed in the comment threads was to simply use "probability" to refer to 'subjective degrees of belief' as the 'Bayesians' would have us do, since we already have the word "frequencies" to refer to 'frequencies'. I intuit that the OP's objection to this suggestion is that it leaves us with no word for the mathematical formalism called "probability" above, which can apply just as well to 'frequencies', 'subjective degrees of belief', and many other things.

I intuit that the OP's objection to this suggestion is that it leaves us with no word for the mathematical formalism called "probability" above, which can apply just as well to 'frequencies', 'subjective degrees of belief', and many other things.

"Probability theory"

Upvoted for resolving a mysterious question I resolved a few days ago myself.

What does "copula" mean?

EDIT: I sort-of get it from reading wikipedia, but I still don't really see what it means in the context of this post.

What does "copula" mean?

Well, when two probability theories love each other very much...

The "is" in "probability is degree of belief".

I think mathematical formalism is a limited route. Mathematics gives you formal structures that might be useful for something, but demonstrating just what they are useful for is the real trick. . I've found that mathematicians often quite breezily assume their formalism applies to the world.

"Probability is defined as some formal structure." Yawn. Until you show me that said mathematics actually solves the problems I'm trying to solve with concepts of probability, I'm uninterested.

we learn because probability theory is isomorphic to those mechanisms.

Showing how it is isomorphic is the real task.

This is fundamentally where Jaynes had it better than most. He starts by trying to solve the problem of which beliefs to have confidence and rely on. He's solving a real problem, instead of creating mathematical objects and just applying real world labels to elements of them after the fact.

The easiest way to understand the philosophical dispute between the frequentist and the subjective Bayesian is to look at the classic biased coin: ... I ask you. What is the difference between these two...

The difference between the two is that the subjective bayesian has a means to cope with the problem that at least most frequentists lack.

I mostly agree, finding why/how it is isomorphic is the important thing. But it is still isomorphic to more than one thing, frequency and subjective degree of belief included.

The two are still locked in a debate which is ultimately the result of interpreting one question in two different ways, and then answering the two seperate questions as if they were exclusive answers to one question. Exactly as the two argue about sound being there in the absence of observers.

The bayesian would give the same answer as the frequentist if he interpreted the question as the frequentist. Same goes for the sound realist, for the same reasons.

The two are still locked in a debate which is ultimately the result of interpreting one question in two different ways, and then answering the two seperate questions as if they were exclusive answers to one question. Exactly as the two argue about sound being there in the absence of observers.

Not exactly. The real argument is about what should be used for inference (both scientific and otherwise). The debate about "what probability actually is" is just another case of debating semantics as a proxy for debating what's actually at stake.

Quick edit: and your post helps make this clear.

Yes, I think we agree. Except that i don't think that the fact that there is meaningful argument to be had about bayesian inference v.s. frequentist inference, means that the debate has not been centered around arguing about what probability is, which is a mistake; the same class of mistake as the mistake being made by the realists and idealists arguing over sound. The bayesian and the frequentist have proposed ways to settle their debate. And there are observations which act as evidence for bayesian inference, or frequentist inference. But exactly what experience should i expect if i think "probability is frequency" as opposed to if I think "probability is subjective degree of belief" ? Arguing about which inferences are optimal, is perfectly reasonable, but arguing about what thing probability really is, is silly.

Yes, I think we agree. Except that i don't think that the fact that there is meaningful argument to be had about bayesian inference v.s. frequentist inference, means that the debate has not been centered around arguing about what probability is, which is a mistake...

Did I give the impression that I thought the argument about what probability is wasn't a mistake?

Oh, well in rereading my comment I could see why it was ambiguous. Yeah, I think we agree.

How isormorphic it is remains to be seen. The infinite set digressions have not been particularly helpful to real problems.

The objective bayesian is free to estimate frequencies, and has done so, a la Jaynes. He explicitly identifies that both questions are answering different questions, and answers both.

I'm not aware of anyone doing this, but I think a frequentist could just as well interpret subjective degrees of belief in frequentist terms, but the sample space would be in informational terms, looking for transformation groups in states of knowledge.

"Probability" is a word used in the interpretation of probability theory.

Sometimes. I think if we're trying to keep terms straight, you should separate probability_SubjectiveBayes, probability_Math, probability_Frequentist, and probability_HumanLanguage. You seem to conflate probability_Math and probability_HumanLanguage.

probability_SubjectiveBayes, probability_Math, probability_Frequentist, and probability_HumanLanguage. You seem to conflate probability_Math and probability_HumanLanguage.

probability\_SubjectiveBayes, probability\_Math, probability\_Frequentist, and probability\_HumanLanguage. You seem to conflate probability\_Math and probability\_HumanLanguage.

Some sentences in the text have typos where "frequentest" replaces "frequentist."

All in all a decent post I thought. Why can't i see the score?

A score of "dot" means that the post is quite new, so it isn't showing the score yet.

They are both giving the right answers to these respective questions. But they are failing to Replace the Symbol with the Substance in their ordinary speech. In the exact same way, "probability" is being used to stand for both "frequency of occurrence" and "rational degree of belief" in the dispute between the Bayesian and the frequentist.

This is inaccurate. If frequentists stuck exactly to that definition, they could never get an answer from the real world, because we never have the infinite number of experiments required to get the limiting frequency. Definitions should be useful.

The answer the frequentist is giving is that we only have enough information to know that it is not 50%. Which is correct.

eidt: not 50% on average.

The answer the frequentist is giving is that we only have enough information to know that it is not 50%. Which is correct.

Nope. We also know that it's a coin with two sides. If it was a 4-sided die that was guaranteed not to have a 50% chance of '1', the situation would be quite different, don't you think? The problem has enough information to be solved.

If our straw frequentist wants a frequency of 1/2, they should consider "if infinite independent people give me this problem and then flip their coin, with what frequency will these trials give heads?"

Couldn't we just say that 1 was heads and ~1 was tails? Then it would be the same, right?

The problem has enough information to be solved.

really? Could you explain further, what do i know besides that if infinite independent people give me this problem and then flip the coin, it will not land heads 50% of the time? Knowing that it is two sided doesn't change anything as far as I can tell. And what problem are you solving exactly? Just to make sure we're on the same page.

I always thought of the coin bayes frequentest thing as being a dispute over definitions. It didn't seem like you really derive P(heads) ≠ 1/2 as a frequentest, it is kind of already in the premises of the situation given your interpretation of "probability", and your model of bias. On the other hand, the bayesian using his/her interpretation of "probability", makes a one step inference using the principle of indifference that P(heads) = 1/2.

Neither of these are deep theorems of the respective statistical disciplines, their proof is trivial in both traditions. They are the statistical consequences of interpreting probability in different ways, i.e., modeling different things with probability to deal with uncertainty. I didn't think of this difference as showing some deep difference between the bayesian and the frequentist; the frequentist and the bayesian are different in terms of their most basic surface apparatus; they use their tool (probability) to model different things (freqeuncy and degree of belief), which then gets them to two statistical methods, but still only one probability theory.

Could you explain further, what do i know besides that if infinite independent people give me this problem and then flip the coin, it will not land heads 50% of the time?

My wording was unclear - I've edited to fix. If people flip the same coin over and over, it won't land heads 50% of the time. "Independent" means different coins. The key idea is to imagine, rather than flipping the same coin over and over, being in the same state of information over and over.

I always thought of the coin bayes frequentist thing as being a dispute over definitions.

It is, in a sense. But if you want to make any decisions about the coin (say that "coin" is whether or not the next solar flare will knock out your satellite), and frequentist and bayesian estimators disagree, which should you use? If you have certain desiderata about your decisions (e.g. you won't take bets that are guaranteed to lose), this is a math problem with a right answer.

And then of course the question is, if this "frequentist probability" stuff is almost always the same as this "bayesian probability" stuff, and when it's different you shouldn't ever base decisions on it, why keep it as an alternate definition? Words should be useful.

Well, let's not keep frequentism as a statistical method, cause bayes almost always if not always does better. But it is a theoretically interesting fact that komolgorov models finite frequencies, and our intuitions about infinite frequencies, and a fact that it does.

Understanding exactly what degrees of belief are, becomes a lot easier (I suppose) if you know that for some reason they are isomorphic to frequencies, and that for some reason they are also isomorphic to spatial measures. This not only allows us to solve problems of degree of belief by solving problems of frequency and area. But also, if we understand what degree of belief has in common with frequency and area, then we understand what it has in common with bayes.

Understanding exactly what degrees of belief are, becomes a lot easier (I suppose) if you know that for some reason they are isomorphic to frequencies

If probabilities were systematically wrong about the frequency of success in independent trials, there would be some other method of reasoning from incomplete information that was better than probabilistic logic. But since the real world obeys all the requirements for probabilistic logic (basically, causality works), there is no such method, and so frequencies match probabilities.

for some reason they are also isomorphic to spatial measures

Read a introductory chapter on set theory that uses pictures to represent sets, and you will understand why.

It's certainly an interesting fact that these things behave the same. But it's not an unsolved problem. We don't have to keep a definition around that's useless in the real world because of any lurking mystery.

That's untrue - a biased coin might well still happen to produce 50% heads and 50% tails given a certain finite number of trials.

Manfred's point is that the frequentist is not using "probability" to stand for "frequency of occurrence", but to stand for "imaginary frequency of occurrence in an infinite number of trials" - otherwise the frequentist position would be blatantly false for the reason that I pointed out.

Ok, now I understand what you are saying.

I wrote my update here

Ok, so the frequentest is giving the right answer given the question he is being asked about hypothetical infinite frequencies.

Formalists do not commit themselves to probabilities, just as they do not commit themselves to numbers

And yet, if I set one apple next to one apple, there are two apples. Arithmetic predicts facts about the world with such reliability that it is perfectly reasonable to say that sentences about numbers have real-world truth values, regardless of whether numbers "exist". We come up with arithmetic because it enables us to make sense of the world, because the world actually does behave that way.

And yet, if I set one apple next to one apple, there are two apples.

And if I pour one bucket of water into another, do I now have two buckets?

(Yes, there's something being conserved in this example, but is it 'number of buckets'/'number of apples'?)

And if I pour one bucket of water into another, do I now have two buckets?

Yes? One empty bucket, one full bucket and a bunch of water that overflowed and went on the floor.

Is that so? I never said the buckets averaged more than half full.

Is that so? I never said the buckets averaged more than half full.

Yes. You definitely have two buckets.

If there is some parlance in which you can refer to "a bucket of water" and mean "less half a bucket of water" without just being misleading to the point of falsehood then one of those buckets will be an empty bucket and the other bucket has an unspecified amount of water in it. Unless, I suppose, you also failed to mention that after you emptied one bucket into the other bucket you replaced the water with a lemur. Then you would have two buckets, one with water in it and another with a lemur.

Regardless if there is some argument by analogy to be made which requires something which when added to another thing produces just one thing it is best to focus on the water, not the buckets. 'Buckets' is going to rfer to either the actual object or be an ad-hoc unit for water measurement - exactly the thing that the analogy wants to avoid!

If I am a pessimist and have a glass that is only half-full (or knowing this sorry world, less than that), and I throw it in your face, are you 'being misleading to the point of falsehood' when you yell that I threw a glass of water at you and then punch me?

Or would you say instead 'You threw a glass of uncertain description which may or may not have a lemur in it at me!'?

If I am a pessimist and have a glass that is only half-full (or knowing this sorry world, less than that), and I throw it in your face, are you 'being misleading to the point of falsehood' when you yell that I threw a glass of water at you and then punch me?

I now have one glass.

But it takes a machine besides the universe to count apples. Namely, humans. Arithmetic is turing complete, as is probability theory, so we should not be confused when we notice that it can practically talk about everything under the sun, including things out there in being.

Formalism, which appears in the title, is not mentioned anywhere in the post, as far as I can see.

Apart from that, I have to ask, what is the purpose of this post? What is the point you are making? It looks like a rehash of Sequence elements, put together somewhat at random. It's almost as though someone had coded a Markov-chain bot to imitate a typical LessWrong discussion, except that such a bot would be unlikely to mis-spell 'frequentist'.

The author is resolving a mysterious question ("Is probability frequentist or bayesian?") quite nicely. Maybe it is covered in that painfully long monster "Sequences" but is surely useful to a novice.