Bayesian Statistics: The three cultures

So my theory is that probability is an ill-defined, unfalsifiable concept. And yet, it _seems_ to model aspects of the world pretty well, empirically. However, might it be leading us astray?

Consider the statement p(X) = 0.5 (probability of event X is 0.5). What does this actually mean? It it a proposition? If so, is it falsifiable? And how?

If it is not a proposition, what does it actually mean? If someone with more knowledge can chime in here, I'd be grateful. I've got much more to say on this, but only after I hear from those with a rigorous grounding the theory.

> If it is not a proposition, what does it actually mean?

It's a measure of plausibility - enabling plausible reasoning.

https://www.lesswrong.com/posts/KN3BYDkWei9ADXnBy/e-t-jaynes...

https://en.wikipedia.org/wiki/Cox%27s_theorem

So here's a sort of hard-nosed answer: probability is just as well-defined as any other mathematics.

> Consider the statement p(X) = 0.5 (probability of event X is 0.5). What does this actually mean?

It means X is a random variable from some sample space to a measurable space and P is a probability function.

> If so, is it falsifiable? And how?

Yes, by calculating P(X) in the given sample space. For example, if X is the event "you get 100 heads in a row when flipping a fair coin" then it is false that P(X) = 0.5.

It's a bit like asking whether 2^2 = 4 is falsifiable.

There are definitely meaningful questions to ask about whether you've modeled the problem correctly, just as it's meaningful to ask what "2" and "4" mean. But those are separate questions from whether the statements of probability are falsifiable. If you can show that the probability axioms hold for your problem, then you can use probability theory on it.

There's a Wikipedia article on interpretations of probability here: https://en.wikipedia.org/wiki/Probability_interpretations. But it is pretty short and doesn't seem quite so complete.

> So my theory is that probability is an ill-defined, unfalsifiable concept

Probability isn’t a single concept, it is a family of related concepts - epistemic probability (as in subjective Bayesianism) is a different concept from frequentist probability - albeit obviously related in some ways. It is unsurprising that a term looks like an “ill-defined, unfalsifiable concept” if you are mushing together mutually incompatible definitions of it.

> Consider the statement p(X) = 0.5 (probability of event X is 0.5). What does this actually mean?

From a subjective Bayesian perspective, p(X) is a measure of how much confidence I - or any other specified person - have in the truth of a proposition, or my own judgement of the weight of evidence for or against it, or my judgement of the degree of my own knowledge of its truth or falsehood. And 0.5 means I have zero confidence either way, I have zero evidence either way (or else, the evidence on each side perfectly cancels each other out), I have a complete lack of knowledge as to whether the proposition is true.

> It it a proposition?

It is a proposition just in the same sense that “the Pope believes that God exists” is a proposition. Whether or not God actually exists, it seems very likely true that the Pope believes he does

> If so, is it falsifiable? And how?

And obviously that’s falsifiable, in the same sense that claims about my own beliefs are trivially falsifiable by me, using my introspection. And claims about other people’s beliefs are also falsifiable, if we ask them, and if assuming they are happy to answer, and we have no good reason to think they are being untruthful.

> For example, if X is the event "you get 100 heads in a row when flipping a fair coin" then it is false that P(X) = 0.5

I think you haven't thought about this deeply enough yet. You take it as self evident that P(X) = 0.5 is false for that event, but how do you prove that? Assuming you flip a coin and you indeed get 100 heads in a row, does that invalidate the calculated probability? If not, then what would?

I guess what I'm driving at is this notion (already noted by others) that probability is recursive. If we say p(X) = 0.7, we mean the probability is high that in a large number of trials, X occurs 70% of the time. Or that the proportion of times that X occurs tends to 70% with high probability as the number of trials increase. Note that this second order probability can be expressed with another probability ad infinitum.

This is the truly enlightened answer. Pick some reasonably defined concept of it if forced. Mainly though, you notice it works and apply the conventions.

So you response actually strengthens my point, rather than rebuts it.

> From a subjective Bayesian perspective, p(X) is a measure of how much confidence I - or any other specified person - have in the truth of a proposition, or my own judgement of the weight of evidence for or against it, or my judgement of the degree of my own knowledge of its truth or falsehood.

See how inexact and vague all these measures are. How do you know your confidence is (or should be) 0.5 ( and not 0.49) for example? Or, how to know you have judged correctly the weight of evidence? Or how do you know the transition from "knowledge about this event" to "what it indicates about its probability" you make in your mind is valid? You cannot disprove these things, can you?

Unless you you want to say the actual values do not actually matter, but the way the probabilities are updated in the face of new information is. But in any case, the significance of new evidence still has to be interpreted; there is no objective interpretation, is there?.

> I think you haven't thought about this deeply enough yet.

On the contrary, I've thought about it quite deeply. Or at least deeply enough to talk about it in this context.

> You take it as self evident that P(X) = 0.5 is false for that event, but how do you prove that?

By definition a fair coin is one for which P(H) = P(T) = 1/2. See e.g. https://en.wikipedia.org/wiki/Fair_coin. Fair coins flips are also by definition independent, so you have a series of independent Bernoulli trials. So P(H^k) = P(H)^k = 1/2^k. And P(H^k) != 1/2 unless k = 1.

> Assuming you flip a coin and you indeed get 100 heads in a row, does that invalidate the calculated probability? If not, then what would?

Why would that invalidate the calculated probability?

> If not, then what would?

P(X) = 0.5 is a statement about measures on sample spaces. So any proof that P(X) != 0.5 falsifies it.

I think what you're really trying to ask is something more like "is there really any such thing as a fair coin?" If you probe that question far enough you eventually get down to quantum computation.

But there is some good research on coin flipping. You may like Persi Diaconis's work. For example his Numberphile appearance on coin flipping https://www.youtube.com/watch?v=AYnJv68T3MM

> By definition a fair coin is one for which P(H) = P(T) = 1/2. See e.g. https://en.wikipedia.org/wiki/Fair_coin.

But that's a circular tautology, isn't it?

You say a fair coin is one where the probability of heads or tails are equal. So let's assume the universe of coins is divided into those which are fair, and those which are not. Now, given a coin, how do we determine it is fair?

If we toss it 100 times and get all heads, do we conclude it is fair or not? I await your answer.

> See how inexact and vague all these measures are. How do you know your confidence is (or should be) 0.5 ( and not 0.49) for example?

Well, you don't, but does it matter? The idea is it is an estimate.

Let me put it this way: we all informally engage in reasoning about how likely it is (given the evidence available to us) that a given proposition is true. The idea is that assigning a numerical estimate to our sense of likelihood can (sometimes) be a helpful tool in carrying out reasoning. I might think "X is slightly more likely than ~X", but do I know whether (for me) p(X) = 0.51 or 0.501 or 0.52? Probably not. But I don't need a precise estimate for an estimate to be helpful. And that's true in many other fields, including things that have nothing to do with probability – "he's about six feet tall" can be useful information even though it isn't accurate to the millimetre.

> Or, how to know you have judged correctly the weight of evidence?

That (largely) doesn't matter from a subjective Bayesian perspective. Epistemic probabilities are just an attempt to numerically estimate the outcome of my own process of weighing the evidence – how "correctly" I've performed that process (per any given standard of correctness) doesn't change the actual result.

From an objective Bayesian perspective, it does – since objective Bayesianism is about, not any individual's actual sense of likelihood, rather what sense of likelihood they ought to have (in that evidential situation), what an idealised perfectly rational agent ought to have (in that evidential situation). But that's arguably a different definition of probability from the subjective Bayesian, so even if you can poke holes in that definition, those holes don't apply to the subjective Bayesian definition.

> Or how do you know the transition from "knowledge about this event" to "what it indicates about its probability" you make in your mind is valid?

I feel like you are mixing up subjective Bayesianism and objective Bayesianism and failing to carefully distinguish them in your argument.

> But in any case, the significance of new evidence still has to be interpreted; there is no objective interpretation, is there?.

Well, objective Bayesianism requires there be some objective standard of rationality, subjective Bayesianism doesn't (or, to the extent that it does, the kind of objective rationality it requires is a lot weaker, mere avoidance of blatant inconsistency, and the minimal degree of rationality needed to coherently engage in discourse and mathematics.)

> But that's a circular tautology, isn't it?

No it's not a tautology... it's a definition of fairness.

> If we toss it 100 times and get all heads, do we conclude it is fair or not?

This is covered in any elementary stats or probability book.

> Now, given a coin, how do we determine it is fair?

I addressed this in my last two paragraphs. There's a literature on it and you may enjoy it. But it's not about whether statistics is falsifiable, it's about the physics of coin tossing.

> This is covered in any elementary stats or probability book.

No, it is really not. That you are avoiding giving me a straightforward answer says a lot. If you mean this:

> So any proof that P(X) != 0.5 falsifies it

Then the fact that we got all heads does not prove P(X) != 0.5. We could get a billions heads and still that is not proof that P(X) != 0.5 (although it is evidence in favor of it).

> I addressed this in my last two paragraphs...

No you did not. Again you are avoiding giving a straightforward answer. That tell me you are aware of the paradox and are simply avoiding grappling with it.

You’re right that a particular claim like p(X=x)=a can’t be falsified in general. But whole functions p can be compared and we can say one fits the data better than another.

For example, say Nate Silver and Andrew Gelman both publish probabilities for the outcomes of all the races in the election in November. After the election results are in, we can’t say any individual probability was right or wrong. But we will be able to say whether Nate Silver or Andrew Gelman was more accurate.

As a mathematical theory, probability is well-defined. It is an application of a larger topic called measure theory, which also gives us the theoretical underpinnings for calculus.

Every probability is defined in terms of three things: a set, a set of subsets of that set (in plain language: a way of grouping things together), and a function which maps the subsets to numbers between 0 and 1. To be valid, the set of subsets, aka the events, need to satisfy additional rules.

All your example p(X) = 0.5 says is that some function assigns the value of 0.5 to some subset which you've called X.

That it seems to be good at modelling the real world can be attributed to the origins of the theory: it didn't arise ex nihilo, it was constructed exactly because it was desirable to formalize a model for seemingly random events in the real world.

> What does this actually mean? It it a proposition? If so, is it falsifiable? And how?

If you saw a sequence of 1000 coin tosses at say 99% heads and 1% tails, you were convinced that the same process is being used for all the tosses and you had an opportunity to bet on tails with 50% stakes, would you do it?

This is a pragmatic answer which rejects P(X)=0.5. We can try to make sense of this pragmatic decision with some theory. (Incidentally, being exactly 0.5 is almost impossible, it makes more sense to verify if it is an interval like (0.49,0.51)).

The CLT says that probability of X can be obtained by conducting independent trials and the in limit, the average number of times X occurs will approach p(X).

However, 'limit' implies an infinite number of trials, so any initial sequence doesn't determine the limit. You would have to choose a large N as a cutoff and then take the average.

But, is this unique to probability? If you take any statement about the world, "There is a tree in place G", and you have a process to check the statement ("go to G and look for a tree"), can you definitely say that the process will successfully determine if the statement is true? There will always be obstacles("false appearances of a tree" etc.). To rule out all such obstacles, you would have to posit an idealized observation process.

For probability checking, an idealization which works is infinite independent observations which gives us p(X).

PS: I am not trying to favour frequentism as such, just that the requirement of an ideal of observation process shouldn't be considered as an overwhelming obstacle. (Sometimes, the obstacles can become 'obstacles in principle' like position/momentum simultaneous observation in QM and if you had such obstacles, then indeed one can abandon the concept of probability).

> So my theory is that probability is an ill-defined, unfalsifiable concept. And yet, it seems to model aspects of the world pretty well, empirically.

I have privately come to the conclusion that probability is a well-defined and testable concept only in settings where we can argue from certain exact symmetries. This is the case in coin tosses, games of chance and many problems in statistical physics. On the other hand, in real-world inference, prediction and estimation, probability is subjective and much less quantifiable than statisticians (Bayesians included) would like it to be.

> However, might it be leading us astray?

Yes, I think so. I increasingly feel that all sciences that rely on statistical hypothesis testing as their primary empirical method are basically giant heaps of garbage, and the Reproduciblity Crisis is only the tip of the iceberg. This includes economics, social psychology, large swathes of medical science, data science, etc.

> Consider the statement p(X) = 0.5 (probability of event X is 0.5). What does this actually mean? It it a proposition? If so, is it falsifiable? And how?

I'd say it is an unfalsifiable proposition in most cases. Even if you can run lots of cheap experiments, like with coin tosses, a million runs will "confirm" the calculated probability only with ~1% precision. This is just lousy by the standards of the exact sciences, and it only goes downhill if your assumptions are less solid, the sample space more complex, or reproducibility more expensive.

I think ants_everywhere's statement was misinterpreted. I don't think they meant that flipping 100 heads in a row proves the coin is not fair. They meant that if the coin is fair, the chance of flipping heads 100 times in a row is not 50%. (And that is of course true; I'm not really sure it contributes to the discussion, but it's true).

ants_everywhere is also correct that the coin-fairness calculation is something you can find in textbooks. It's example 2.1 in "Data analysis: a bayesian tutorial" by D S Sivia. What it shows is that after many coin flips, the probability for the bias of a coin-flip converges to roughly a gaussian around the observed ratio of heads and tails, where the width of that gaussian narrows as more flips are accumulated. It depends on the prior as well, but with enough flips it will overwhelm any initial prior confidence that the coin was fair.

The probability is nonzero everywhere (except P(H) = 0 and P(H) = 1, assuming both heads and tails were observed at least once), so no particular ratio is ever completely falsified.

Thank you, yes you understood what I was saying :)

> I'm not really sure it contributes to the discussion, but it's true

I guess maybe it doesn't, but the point I was trying to make is the distinction between modeling a problem and statements within the model. The original claim was "my theory is that probability is an ill-defined, unfalsifiable concept."

To me that's a bit like saying the sum of angles in a triangle is an ill-defined, unfalsifiable concept. It's actually well-defined, but it starts to seem poorly defined if we confuse that with the question of whether the universe is Euclidean. So I'm trying to separate the questions of "is this thing well-defined" from "is this empirically the correct model for my problem?"

Sorry, I didn't mean to phrase my comment so harshly! I was just thinking that it's odd to make a claim that sounds so obvious that everyone should agree with it. But really it does make sense to state the obvious just in order to establish common ground, especially when everyone is so confused. (Unfortunately in this case your statement was so obviously true that it wrapped around; everyone apparently thought you must have meant something else, and misinterpreted it).

Oh I didn't take it harshly. Just wanted to clarify since you and I seemed on the same wavelength but that part didn't come across clearly :)