0 And 1 Are Not Probabilities

hmm... I feel even more confident about the existence of probability-zero statements than I feel about the existence of probability-1 statements. Because not only do we have logical contradictions, but we also have incoherent statements (like Husserl's "the green is either").

Can one form subjective probabilities over the truth of "the green is either" at all? I don't think so, but I remember a some-months-ago suggestion of Robin's about "impossible possible worlds," which might also imply the ability to form probability estimates over incoherencies. (Why not incoherent worlds? One might ask.) So the idea is at least potentially on the table.

And then it seems obvious that we will forever, across all space and time, have no evidence to support an incoherent proposition. That's as good an approximation of infinite lack of evidence as I can come up with. P("the green is either")=0?

One answer would be that an incoherent proposition is not a proposition, and so doesn't have any probability (not even zero, if zero is a probability.)

Another answer would be that there is some probability that you are wrong that the proposition is incoherent (you might be forgetting your knowledge of English), and therefore also some probability that "the green is either" is both coherent and true.

It's difficult to assign probability to incoherent statements, because since we can't mean anything by them, we can't assert a referent to the statement -- in that sense, the probability is indeterminate (additionally, one could easily imagine a language in which a statement such as "the green is either" has a perfectly coherent meaning -- and we can't say that's not what we meant, since we didn't mean anything). Recall also that each probability zero statement implies a probability one statement by its denial and vice versa, so one is equally capable of imagining them, if in a contrived way.

j.edwards, I think your last sentence convinced me to withdraw the objection -- I can't very well assign a probability of 1 to ~"the green is either" can I? Good point, thanks.

that anecdote wasn't amusing at all.

and it wasn't an anecdote.

and it doesn't prove the point. all it shows is that a single person didn't know their 17 times tables off the top of their head. there's no reason to expect someone to be as confident that 51 is or is not prime than 7 is or is not prime - and anyway, the point of the story should have been that, eventually, 7 might NOT be prime. which it's always going to be.

i didn't get it.

Probabilities of 0 and 1 are perhaps more like the perfectly massless, perfectly inelastic rods we learn about in high school physics - they are useful as part of an idealized model which is often sufficient to accurately predict real-world events, but we know that they are idealizations that will never be seen in real life.

However, I think we can assign the primeness of 7 a value of "so close to 1 that there's no point in worrying about it".

In stark contrast to this time last week, I now internally believe the title of this post.

I did enjoy "something, somewhere, is having this thought," Paul, despite all its inherent messiness.

'Green is either' doesn't tell us much. As far as we know it's a nonsensical statement, but I think that makes it more believable than 'green is purple', which makes sense, but seems extremely wrong. You might as well try to assign a probability to 'flarg is nardle'. I can demonstrate that green isn't purple, but not that green isn't either, nor that flarg isn't nardle.

Is there anything truer than '7 is prime'? What's the truest statement anyone can come up with? Can we definitely get no closer to 0 than 1, based on J Edwards & Paul, above?

I think you can still have probabilities sum to 1: probability 1 would be the theoretical limit of probability reaching infinite certitude. Just like you can integrate over the entire real line, i.e -∞ to ∞ even though those numbers don't actually exist.

i didn't get it.

Easy: it's a demonstration of how you can never be certain that you haven't made an error even on the things you're really sure about.

It's a cheap, dirty demonstration, but one nevertheless.

You seem to think probabilities of 0 and 1 are mysterious or contradictory when discussing randomness; they aren't. When you're talking about randomness, you need to define your support. that mere action gives you places where the probability is zero. For example: Can the time to run 100m ever be negative? No? Then P(t=0) = 1.

No puzzle there. But you're transfrormation to log-odds has some regularity conditions you're violating in those cases: the transform is only defined for probabilities in (0,1). But that doesn't mean log-odds or probabilities are flawed. Probabilities or 0 and 1 -- like log-odds of plus-and-minus infinity -- are just filling in the boundaries on the system you've created. Mathematically, you want to be able to handle limits; that means handling limits as a probability approaches 0 or 1. That's it.

This shouldn't be some huge philosophical puzzle; it's merely the need to have any mathematical system you use be complete. Sir David Cox would be the first to tell you that.

Cumulant - can you state, with infinite certainty, that no-one will ever run faster than light?

Another way to think about probabilities of 0 and 1 is in terms of code length.

Shannon told us that if we know the probability distribution of a stream of symbols, then the optimal code length for a symbol X is: l(X) = -log p(X)

If you consider that an event has zero probability, then there's no point in assigning a code to it (codespace is a conserved quantity, so if you want to get short codes you can't waste space on events that never happen). But if you think the event has zero probability, and then it happens, you've got a problem - system crash or something.

Likewise, if you think an event has probability of one, there's no point in sending ANY bits. The receiver will also know that the event is certain, so he can just insert the symbol into the stream without being told anything (this could happen in a symbol stream where three As are always followed by a fourth). But again, if you think the event is certain and then it turns out not to be, you've got a problem: the receiver doesn't get the code you want to send.

If you refuse to assign zero or unity probabilities to events, then you have a strong guarantee that you will always be able to encode the symbols that actually appear. You might not get good code lengths, but you'll be able to send your message. So Eliezer's stance can be interpreted as an insistence on making sure there is a code for every symbol sequence, regardless of whether that sequence appears to be impossible.

Brent,

From what I understood on reading the Wikipedia article on Bayesian probability and inferring from how he writes (and correct me if I'm wrong), Eliezer is talking about your "subjective probability." You are a being, have consciousness, and interpret input as information. Given a lot of this information, you've formed an idea that 7 is prime. You've also formed an idea that other people exist, and that the sky is blue, which also have a high subjective probability in your mind because you have a lot of direct information to sustain that belief.

Moreover, if you've ever been wrong before, hopefully you've noticed that you have been wrong before. That's a little information that "you are sometimes wrong about things that you are very sure of". So, you might apply this information to your formula of your probability of the idea that "7 is prime", so you still end up with a high probability, but not 1.

Now, you might not think that "you are sometimes wrong about things that you are sure of" about every single subject, such as primeness. But, what if you had the information that other humans, smart people, have at some point in the past, incorrectly understood the primeness of a number (the anecdote). You might state, that "human beings are sometimes wrong about the primeness of a number," and "I am a human being." Again, if you include that information in your calculation of the probability that the idea that "7 is prime" is true, then you end up with a high probability, but not 1.

(Oh, but what if you didn't make the statement "human beings are sometimes wrong about the primeness of a number", but instead, "this idiot is sometimes wrong about the primeness of a number, but I am never" Well, you can. That's one big problem with Bayesian subjective probabilities. How do we generalize? How can we formalize it so that two people with the same information deterministically get the same probability? Logical (or objective epistemic) probability attempts to answer these questions.)

So, you're right that it is just "a single person" getting it wrong, that his cerainty was incorrect. But that's Eliezer's point. We are not supreme beings lording over all reality, we are humans who have memorized some information from the past and made some generalizations, including generalizations that sometimes our generalizations are wrong.

I agree with cumulant. The mathematical subject of probability is based on measure theory, which loses a ton of convergence theorems if we exclude 0 and 1. We can agree that things that are not known a priori can't have probability 0 or 1, but I think we must also agree that "an impossible thing will happen soon" has probability 0, because it's a contradiction. An alternate universe in which the number 7 (in the same kind of number system as ours, etc.) is prime is damn-near inconceivable, but an alternate universe in which impossible things are possible is purely absurd.

If our mathematical reasoning is coherent enough for it to be meaningful to make probability assignments then certainly we are not so fundamentally flawed that what we consider tautologies could be false. If you are willing to accept that maybe 0 is 1, then you can't do any of your probability adjustments, or use Bayes' Theorem, or anything of the sort without having a (possibly unstated) caveat that probability theory might be complete nonsense. But what's the probability that probability theory is nonsense (i.e. false or inconsistent)? What does that even mean? We can only assign a probability if that makes sense, so conditioned on the sentence making sense, probability theory must be nonsense with probability 0, no? So averaged over all possible universes (those where probability theory makes sense, and those where it doesn't) the sentence "probability makes sense with probability 1" better approximates the truth value of probability making sense than "probability makes sense with probability p" for p0. If it's not, it's still not worse, but what the hell are we even saying?

Speaking of measure theory, what probability should we assign to a uniformly distributed random real number on the interval [0, 1] being rational? Something bigger than 0? Maybe in practice we would never hold a uniform distribution over [0, 1] but would assign greater probability to "special" numbers (like, say, 1/2). But regardless of our probability distribution, there will exist subsets of [0, 1] to which we must assign probability 0.

The only way I can see around this is to refuse to talk about infinite (or at least uncountable) sets. Are there others?

I suspect Eliezer would object to my post claiming that I'm confusing map and territory, but I don't think that's fair. If there's a map you're trying to use all over the place (and you do seem to), then I claim it makes no sense to put a little region on the map labelled "maybe this map doesn't make any sense at all". If the map seems to make sense and you're still following it for everything, you'll have to ignore that region anyway. So is it really reasonable to claim that "the probability that probability makes sense is <1"?

Utilitarian:

Measure theory gives a clear answer to this: it's 0. Which is fine. For all x, the probability that your rv will take the value x is 0. Actually the probability that your rv is computable is also 0. (Computable numbers are the largest countable class I know of.) What's false is the tempting statement that probability 0 events are impossible. It's only the converse that's true: impossible events have probability 0. There's another tempting statement that's false, namely the statement that if S is an arbitrary collection of disjoint events, the probability of one of them happening is the sum of the probabilities of each one happening. Instead, this only holds for countable sets S. This is part of the definition of a measure.

If there's a map you're trying to use all over the place (and you do seem to), then I claim it makes no sense to put a little region on the map labelled "maybe this map doesn't make any sense at all". If the map seems to make sense and you're still following it for everything, you'll have to ignore that region anyway.

Janos, are you saying that it is in fact impossible that your map in fact doesn't make any sense? Because I do, indeed, have a little section of my map labelled "maybe this map doesn't make any sense at all", and every now and then, I think about it a little, because there are so many fundamental premises of which I am unsure even in their definitions. (E.g: "the universe exists", and "but why?") Just because this area of my map drops out of my everyday decision theory due to failure to generate coherent advice on preferences, does not mean it is absent from my map. "You must ignore" or rather "You should usually ignore" is decision theory, and probability theory should usually be firewalled off from preferences.

Computable numbers are the largest countable class I know of.

Either all countable sets are the same size anyway, or you can generate a larger set by saying "all computable reals plus the halting probability". How about computable with various oracles?

What's false is the tempting statement that probability 0 events are impossible. It's only the converse that's true: impossible events have probability 0.

If you cannot repose probability 1 in the statement "all events to which I assign probability 0 are impossible" you should apply a correction and stop reposing probability 0 to those events. Do you mean to say that all impossible events have probability 0, plus some more possible events also have probability 0? This makes no sense, especially as a justification for using "probability 0" in a meaningfully calibrated sense.

To use "probability 0" without a finite expectation of being infinitely surprised, you must repose probability 1 in the belief that you use "probability 0" only for actually impossible events; but not necessarily believe that you assign probability 0 to every impossible event (satisfying both conditions implies logical omniscience).

I should mention that I'm also an infinite set atheist.

I can admit the possibility that probability doesn't work, but not have to do anything about it. If probability doesn't work and I can't make rational decisions, I can expect to be equally screwed no matter what I do, so it cancels out of the equation.

The definable real numbers are a countable superset of the computable ones, I think. (I haven't studied this formally or extensively.)

If you don't want to assume the existence of certain propositions, you're asking for a probability theory corresponding to a co-intutionistic variant of minimal logic. (Cointuitionistic logic is the logic of affirmatively false propositions, and is sometimes called Popperian logic.) This is a logic with false, or, and (but not truth), and an operation called co-implication, which I will write a <-- b.

Take your event space L to be a distributive lattice (with ordering <), which does not necessarily have a top element, but does have dual relative pseudo-complements. Take < to be the ordering on the lattice. (a <-- b) if for all x in the lattice L,

for all x, b < (a or x) if and only if a <-- b < x

Now, we take a probability function to be a function from elements of L to the reals, satisfying the following axioms:

1. P(false) = 0
2. if A < B then P(A) <= P(B)
3. P(A or B) + P(A and B) = P(A) + P(B)

There you go. Probability theory without certainty.

This is not terribly satisfying, though, since Bayes's theorem stops working. It fails because conditional probabilities stop working -- they arise from a forced normalization that occurs when you try to construct a lattice homomorphism between an event space and a conditionalized event space.

That is, in ordinary probability theory (where L is a Boolean algebra, and P(true) = 1), you can define a conditionalization space L|A as follows:

L|A = { X in L | X < A } true' = A false' = false and' = and or' = or not'(X) = not(X) and A P'(X) = P(X)/P(A)

with a lattice homomorphism X|A = X and A

Then, the probability of a conditionalized event P'(X|A) = P(X and A)/P(A), which is just what we're used to. Note that the definition of P' is forced by the fact that L|A must be a probability space. In the non-certain variant, there's no unique definition of P', so conditional probabilities are not well-defined.

To regain something like this for cointuitionistic logic, we can switch to tracking degrees of disbelief, rather than degrees of belief. Say that:

1. D(false) = 1
2. for all A, D(A) > 0
3. if A < B then D(A) >= D(B)
4. D(A or B) + D(A and B) = D(A) + D(B)

This will give you the bounds you need to let you need to nail down a conditional disbelief function. I'll leave that as an exercise for the reader.

Hi guys you don't know me and I prefer to stay anonymous. I look at it backwards and get the very same result as Eliezer Y. What is total degeneracy? In practice, it is being total impervious to updating, regardless of the magnitude of the information seen (even infinity). That can only be achieved by unitary of nul probabilities as priors. Bayesian updating never takes you there (posteriors). And no updating can take place from that situation. Anonymous

If the map seems to make sense and you're still following it for everything, you'll have to ignore that region anyway.

Just cos it's not a very nice place to visit, doesn't mean it ain't on the map. ;)

"1, 2, and 3 are all integers, and so is -4. If you keep counting up, or keep counting down, you're bound to encounter a whole lot more integers. You will not, however, encounter anything called "positive infinity" or "negative infinity", so these are not integers."

This bothered me, more to the point, it hit on some stuff I've been thinking about. I realize I don't have a very good way to precisely state what I mean by "finite" or "eventually"

The above, for instance, basically says "if infinity is not an integer, then if I start at an integer and move an integer number of steps away from it, I will still be at an integer that's not infinity, therefore infinity isn't an integer"

But if we allowed infinity to be considered an integer, then we allow an infinite number of steps...

How about this: if N is a non infinite integer, SN is N's successor, PN is N's predecessor, neither SN nor PN will be infinite. Great, no matter where we start from, we can't reach an infinity in one step, so that seems to make this notion more solid.

but... if N is an infinity, then neither SN nor PN (thinking about ordinals now, btw, instead of cardinals) will be finite. Doh.

So the situation seems a bit symmetric here. This is really annoying to me.

I have as of late been getting the notion that the notions of "finite" and "eventually" are so tied to the idea of mathematical induction that it's probably best do define the former in terms of the latter... ie, the number of steps from A to A is finite if and only if induction arguments starting from A and going in the direction toward B actually validly prove the relevant proposition for B.

This is a vague notion, but near as I can tell, it comes closes to what I actually think I mean when I say something like "finite" or "eventually reach in a finite number of steps" or something like that.

ie, finite values are exactly those critters for which mathematical induction arguments can be used on. (maybe this is a bad definition. I'm more stating it as a "here's my suspicion of what may be the best basis to really represent the concept")

Anyways, as far as 0,1 not being probabilities... While I agree that one should't believe a proposition with probability 0 or 1, I'm not sure I'd consider them nonprobabilities. Perhaps "unreachable" probabilities instead. Disallowing stuff like sum to 1 normalizations and so on would seem to require "unnatural" hoops to jump through to get around that.

Unless, of course, someone has come up with a clean model without that. (If so, well, I'm curious too.)

Eliezer:

I'm not sure what an "infinite set atheist" is, but it seems from your post that you use different notions of probability than what I think of as standard modern measure theory, which surprises me. Utilitarian's example of a uniform r.v. on [0, 1] is perfect: it must take some value in [0, 1], but for all x it takes value x with probability 0. Clearly you can't say that for all x it's impossible for the r.v. to take value x, because it must in fact take one of those values. But the probabilities are still 0. Pragmatically the way this comes out is that "probability 0" doesn't imply impossible. If you perform an experiment countably-infinitely many times with the probability of a certain outcome being 0 each time, the probability of ever getting that outcome is 0; in this sense you can say the outcome is almost impossible. However it's possible that each outcome individually is almost impossible, even though of course the experiment will have an outcome.

You can object that such experiments are physically impossible e.g. because you can only actually measure/observe countably many outcomes. That's fine; that just means you can get by with only discrete measures. But such assumptions about the real world are not known a priori; I like usual measure theory better, and it seems to do quite a good job of encompassing what I would want to mean by "probability", certainly including the discrete probability spaces in which "probability 0" can safely be interpreted to mean "impossible".

You're right, it's not that hard to come up with larger countable classes of reals than the computables; I just meant that all of the usual, "rolls-off-the-tip-of-your-tongue" classes seem to be subsets of the computables. But maybe Nick is right, and the definables are broader. I haven't studied this either.

And yes, I also sometimes think about how assumptions I make about life and the perceptible universe could be wrong, but I do not do this much for mathematics that I've studied deeply enough, because I'm almost as convinced of its "truth" as I am of my own ability to reason, and I don't see the use in reasoning about what to do if I can't reason. This is doubly true if the statements I'm contemplating are nonsense unless the math works.

Eliezer:

I am curious as to why you asked Peter not to repeat his stunt.

Also, I would really like to know how confident you are in your infinite set atheism and for that matter in your non-standard philosophy of mathematics attitudes in general.

Regarding infinite set atheism:

Is the set of "possible landing sites of a struck golf ball" finite or infinite?

In other words, can you finitely parameterize locations in space? Physicists normally model "position" as n-tuples of real numbers in a coordinate system; if they were forced to model position discretely, what would happen?

I can claim to see an infinite set each time I use a ruler...

Doug S., I believe according to quantum mechanics the smallest unit of length is Planck length and all distances must be finite multiples of it.

Eliezer:

I should mention that I'm also an infinite set atheist.

You've mentioned this before, and I have always wondered: what does this mean? Does it mean that you don't believe there are any infinite sets? If so, then you have to believe that a mathematician who claims the contrary (and gives the standard proof) is making a mistake somewhere. What is it?

Frankly, even if you actually are a finitist (which I find hard to imagine), it doesn't seem relevant to this disucssion: every argument you have presented could equally well have been given by someone who accepts standard mathematics, including the existence of infinite sets.

The nature of 0 & 1 as limit cases seem to be fascinating for the theorists. However, in terms of 'Overcoming Bias', shouldn't we be looking at more mundane conceptions of probability ? EY's posts have drawn attention to the idea that the amount of information needed to add additional cetainty on a proposition increases exponentially while the probability increases linearly. This says that in utilitarian terms, not many situations will warrant chasing the additional information above 99.9% certainty (outside technical implementations in nuclear physics, rocket science or whatever). 99.9% as a number is taken out of a hat. In human terms, when we say 'I'm 99.9% sure that 2+2 always =4', where not talking about 1000 equivalent statements. We're talking about one statement, with a spatial representation of what '100% sure' means with respect to that statement, and 0.1% of that spatial representation allowed for 'niggling doubts', of the sort : what have I forgotten ? What don't I know ? What is inconceivable for me ? The interesting question for 'overcoming bias' is : how do we make that tradeoff between seeking additional information on the one hand and accepting a limited degree of certainty on the other ? As an example (cf. the Evil Lords of the Matrix), considering whether our minds are being controlled by magic mushrooms from Alpha Pictoris may someday increase the 'niggling doubt' range from 0.1% to 5%, but the evidence would have to be shoved in our faces pretty hard first.

Doug S., I believe according to quantum mechanics the smallest unit of length is Planck length and all distances must be finite multiples of it.

Not in standard quantum mechanics. Certain of the many theories unsupported hypotheses of quantum gravity (such as Loop Quantum Gravity) might say something similar to this, but that doesn't abolish every infinite set in the framework. The total number of "places where infinity can happen" in modern models has tended to increase, rather than decrease, over the centuries, as models have gotten more complex. One can never prove that nature isn't "allergic to infinities" (the skeptic can always claim, "wait, but if we looked even closer or farther, maybe we would see a heretofore unobserved brick wall"), but this allergy is not something that has been empirically observed.

I think Eliezer's "infinite set atheism" is a belief that infinite sets, although well-defined mathematically, do not exist in the "real world"; in other words, that any physical phenomenon that actually occurs can be described using a finite number of bits. (This can include numbers with infinite decimal expansions, as long as they can be generated by a finitely long computer program. Therefore, using pi in equations is not prohibited, because you're using the symbol "pi" to represent the program, which is finite.)

A consequence of "infinite set atheism" seems to be that the universe is a finite state machine (although one that is not necessarily deterministic). Am I understanding this properly?

What do you mean by "infinite set atheism"? You are essentially stating that you don't believe in mathematical limits -- because that is one of the major consequences of infinite sets (or sequences).

If you don't believe in those... well, you lose calculus, you lose the density of real numbers, you lose the need or understanding of man events with probability 0 or 1, and you lose the point of Zeno's Paradox.

Janos is spot on about measure zero not implying impossibility. What is the probability of a golf ball landing at any exact point? Zero. But it has to land somewhere, so no one point is impossible.

Impossibility would mean absence from your sigma algebra. What's that you ask? Without making this painful, you need three things for probability: an idea of what constitutes "the space of everything", an idea of what constitutes possible events out of that space which we can confirm or deny, and an assignment of numbers to those events. (This is often LaTeX'ed as (\Omega, \mathcal{F}, P).) The conversation here seems to be confusing the filtration/sigma-algebra F with the numbers assigned to those events by P.

Can we choose which we're talking about: events or numbers?

What is the probability of a golf ball landing at any exact point? Zero.

Wrong.

I don't know which is more painful: Eliezer's errors, or those of his detractors.

Perhaps you could clarify what exactly is an infinite set atheist in a full post...or maybe it's only worth a comment.

Cumulant, I think the idea behind "infinite set atheism" is not that limits don't exist, but that that infinities are acceptable only as limits approached in a specified way. On this view, limits are not a consequence of infinite sets, as you contend; rather, only the limit exists, and the infinite set or sequence is merely a sloppy way of thinking about the limit.

Eliezer, I'll second Matthew's suggestion above that you write a post on infinite set atheism; it looks as if we don't understand you.

I think I understand the motive for rejecting infinite sets (viz., that whenever you deal with infinites you get all sorts of ridiculously counterintuitive results--sums coming out different when you reärrange the terms, the Banach-Tarski paradox, &c., &c.), but I'm not sure you can give up infinite sets without also giving up the real numbers (as others have touched on above), which seems very wrong.

Caledonian: Not wrong. Take the field you're swinging at to be a plane. There are infinitely many points in that plane; that's just the density of the reals.

Now say there is some probability density of landing spots; and, let's say no one spot is special in that it attracts golf balls more than points immediately nearby (i.e. our pdf is continuous and non-atomic). Right there, you need every point (as a singleton) to have measure 0.

Go pick up Billingsley: measure 0 is not the same as impossible nor does it cause any problems.

Take the field you're swinging at to be a plane. There are infinitely many points in that plane; that's just the density of the reals.

And the location that the ball lands on will also be composed of infinitely many reals. Shall we compare the size of two infinite sets?

I'd say that the ball is a sphere and consider the first point of impact (i.e. the tangency point of the plane to the sphere). Otherwise, you need to know a lot about the ball and the field where it lands.

You can compare infinite sets. Take the sets A and B, A={1,2,3,...} and B={2,3,4,...}. B is, by construction, a subset of A. There's your comparison; yet, both are infinite sets.

What assumptions would you make for the golf ball and the field? (To keep things clear, can we define events and probabilities separately?)

Caledonian, every undergraduate who has ever taken a statistics class knows that the probability of any single point in a continuous distribution is zero. Probabilities in continuous space are measured on intervals. Basic calculus...

I believe according to quantum mechanics the smallest unit of length is Planck length and all distances must be finite multiples of it.

This is what I'm given to understand as well. Doesn't this take the teeth out of Zeno's paradox?

Pragmatically the way this comes out is that "probability 0" doesn't imply impossible.

Janos, would you agree that P=0 is a probability to the same degree that infinity is a number? Apologies for double post.

Caledonian, every undergraduate who has ever taken a statistics class knows that the probability of any single point in a continuous distribution is zero.

Gowder, everyone who's ever given the issue more than three-seconds'-thought knows that no statistical result ever involves a single point.

Usually, if a die lands on edge we say it was a spoiled throw and do it over. Similarly if a Dark Lord writes 37 on the face that lands on top, we complain that the Dark Lord is spoiling our game and we don't count it.

We count 6 possibilities for a 6-sided die, 5 possibilities for a 5-sided die, 2 possibilities for a 2-sided die, and if you have a die with just one face -- a spherical die -- what's the chance that face will come up?

I think it would be interesting to develop probability theory with no boundaries, with no 0 and 1. It works fine to do it the way it's done now, and the alternative might turn up something interesting too.

Ben:

Well, that depends on your number system. For some purposes +infinity is a very useful value to have. For instance if you consider the extended nonnegative reals (i.e. including +infinity) then every measurable nonnegative extended-real-valued function on a measure space actually has a well-defined extended-nonnegative-real-values integral. There are all kinds of mathematical structures where an infinity element (or many) is indispensable. It's a matter of context. The question of what is a "number" is I think very vague given how many interesting number-like notions mathematicians have come up with. But unquestionably "infinity" is not a natural number, or a real number, or a complex number.

Probability theory, on the other hand, would have to change shape if we comfortably wanted to exclude 0 probabilities. What we now call measures would be wrong for the job. I don't know how it would look, but I find the standard description intuitively appealing enough that I don't think it should be changed. It's probably true that for a Bayesian inference engine of some sort, whose purpose is to find likelihoods of propositions given evidence, the "probabilities" it keeps track of shouldn't become 0 or 1. If there's a rich theory there focussing on how to practically do this stuff (and I bet there is, although I know nothing of it beyond Bayes' Theorem, which is a simple result) then ignoring the possibility of 0s and 1s makes sense there: for example you can use the log odds. But in general probability theory? No.

I think it would be interesting to develop probability theory with no boundaries, with no 0 and 1. It works fine to do it the way it's done now, and the alternative might turn up something interesting too.

You might want to check out Kosko's Fuzzy Thinking. I haven't gone any further into fuzzy logic, yet, but that sounds like something he discussed. Also, he claimed probability was a subset of fuzzy logic. I intend to follow that up, but there is only one of me, and I found out a long time ago that they can write it faster than I can read it.

"On some golf courses, the fairway is readily accessible, and the sand traps are not. The green is either."

Haha, very nice CGD. Shows how much those philosophers of language know about golf. :-)

Although... hmm... interesting. I think that gives us a way to think about another probability 1 statement: statements that occupy the entire logical space. Example: "either there are probability 1 statements, or there are not probability 1 statements." That statement seems to be true with probability 1...

Disallowing a symbol for "all events" breaks the definition of a probability space. It's probably easier to allow extended reals and break some field axioms than figure out do rigorous probability without a sigma-algebra.

When re-working this into a book, you need to double check your conversions of log odds into decibels. By definition, decibels are calculated using log base 10, but some of your odds are natural logarithms, which confused the heck out of me when reading those paragraphs.

Probability .0001 = -40 decibels (This is the only correct one in this post, all "decibel" figures afterwards are listed as 10 * the natural logarithm of the odds.) Probability 0.502 = 0.035 decibels Probability 0.503 = 0.052 decibels Probability 0.9999 = 40 decibels Probability 0.99999 = 50 decibels

P.S. It'd be nice if you provided an RSS feed for the comments on a post, in addition to the RSS feed for the posts...

I cannot begin to imagine where those numbers came from. Dangers of "Posted at 1:58 am", I guess. Fixed.

P(A&B)+P(A&~B)+P(~A&B)+P(~A&~B)=1

Isn't the "1" above a probability?

My intution as a mathematician declares that nobody will never develop an elegant mathematical formulation of probability theory that does not allow for statements that are logically impossible or certain, such as statements of the form p AND NOT p. And it is necessary, if the theory is to be isomorphic to the usual one, that these statements have probability 0 (if impossible) or 1 (if certain). However, I believe that it is quite reasonable to declare, as a condition demanded of any prior deemed rational, that only truly impossible or certain statements have those probabilities. I think that this gives you what you want.

It's obvious that you can make this very demand when working with discrete probability distributions. It may not be obvious that you can make this demand when working with continuous probability distributions. Certainly the usual theory of these, based on so-called ‘measure spaces’ and ‘σ-algebras’ (I mention those in case they jog the reader's memory), cannot tolerate this requirement, at least not if anything at all similar to the usual examples of continuous distributions are allowed.

One answer is that only discrete probability distributions apply to the real world, in which one can never make measurements with infinite precision or observe an infinite sequence of events. Even if the world has infinite size or is continuous to infinitesimal scales, you will never observe that, so you don't need to predict anything about that.

However, even if you don't buy this argument, never fear! There is a mathematical theory of probability based on ‘pointless measure spaces’ and ‘abstract σ-algebras’. In this theory, it again makes perfect sense to demand that any prior must assign probability 0 or 1 only to impossible or certain events. The idea is that if something can never be observed, even in principle, then it is effectively impossible, and the abstract pointless theory allows one to treat it as such.

Then I agree that one should require, as a condition on considering a prior to be rational, that it should assign probability 0 only to these impossible events and assign probability 1 only to their certain complements.

As Perplexed points out this is usually known as Cromwell's_rule.

I'm kinda surprised that it's only been mentioned once in the comments (I only just discovered this site, really really great, by the way) and one from 2010 at that, but it seems to me that "a magical symbol to stand for "all possibilities I haven't considered" " does exist: the symbol "~" (i.e. not). Even the commenter who does mention it makes things complicated for himself: P(Q or ~Q)=1 is the simplest example of a proposition with probability 1.

The proposition is of course a tautology. I do think (but I'm not sure) that that is the only sort of statement that receives probability 1. This is in sync with Eliezer's "amount of evidence" interpretation. A bayesian update can only generate 1 if the initial proposition was of probability 1 or if the evidence was tautological (i.e. if Q then Q or, slightly less lame, if "Q or R" and "~R" then Q, where "Q or R" and "~R" are the evidence).

Skimming the comments, I saw two other proposals for "sure bets", the runner who clocked a negative time and the golf ball landing in a particular spot. That last one degenerated pretty quickly into a discussion about how many points there are in a field and on a ball. I think that's typical of such arguments: it depends on your model. Once you have your model specified the probability becomes 1 (or not) if the statement is (or isn't) tautological in the model. If the model isn't specified, then neither is the statement (what is a precise point?) and hence the probability. Ask the next man what the probability is of a runner clocking a negative time and he'll rightly respond: "Huh?" (unless he is a particularly obfuscatory know-it-all, in which case he might start blabbering about the speed of light. But then too, he makes a claim because he can ascribe meaning to the question, that is, he picks his model). So these are also tautological examples.

I think Eliezer's hold up pretty well for proposition that aren't tautological and hence empirical in nature: they require evidence and only tautological evidence will suffice for certainty.

About the problem of inserting 0's in certain standard theorems: I don't see a problem with Bayes' theorem (I'm curious about other examples). Dividing by 0 is not defined, so the probability of it raining when hell freezes over is not defined. That seems like a satisfactory arrangement.

For any state of information X, we have P(A or not A | X) = 1 and P(A and not A | X) = 0. We have to have 0 and 1 as probabilities for probability theory even to work. I think you're taking a reasonable idea -- that P(A | X) should be neither 0 nor 1 when A is a statement about the concrete physical world -- and trying to apply it beyond its applicable domain.

Consider the set of all possible hypotheses. This is a countable set, assuming I express hypotheses in natural language. It is potentially infinite as well, though in practice a finite mind cannot accomodate infintely-long hypotheses. To each hypothesis, I can try to assign a probability, on the basis of available evidence. These probabilities will be between zero and one. What is the probability that a rational mind will assign at least one hypothesis the status of absolute certainty? Either this is one (there is definitely such a hypothesis), or zero (there is definitely not such a hypothesis, which cannot be, because the hypothesis "there is definitely not such a hypothesis" is then a counterexample), or somewhere in between (there may be, somewhere, a hypothesis that a rational mind would regard as being absolutely certain). So I cannot accept your hypothesis that there does not exist, anywhere, ever, a hypothesis that I should regard as being absolutely certain.

Yes 0 and 1 are not probabilities. They're truth or falseness values. it's necessary to make a third 'truth value' for things that are unprovable, and possibly a fourth for things that are untestable.

Digging up an old thread here, but an interesting point I want to bring up: a friend of mine claims that he internally assigns probability 1 (i.e. an undisprovable belief) only to one statement: that the universe is coherent. Because if not, then mnergarblewtf. Is it reasonable to say that even though no statement can actually have probability 1 if you're a true Bayesian, it's reasonable to internally establish an axiom which, if negated, would just make the universe completely stupid and not worth living in any more?

The ("Bayesian") framework explored in these essays replaces the two Cartesian options, affirmation and denial, by a continuum of judgmental probabilities in the interval from 0 to 1, endpoints included, or -- what comes to the same thing -- a continuum of judgmental odds in the interval from 0 to infinity, endpoints included. Zero and 1 are probabilities no less than 1/2 and 99/100 are. Probability 1 corresponds to infinite odds, 1:0. That's a reason for thinking in terms of odds: to remember how momentous it may be to assign probability 1 to a hypothesis."

Richard Jeffrey, "Probability and the art of judgement".

I leave it as an exercise to correctly state the relationships between Eliezer's article, the Jeffrey quote, and the value of P(A|A).

(Note: Jeffrey is not to be confused with Jeffreys, although both were Bayesian probability theorists.)

Interesting Log-Odds paper by Brian Lee and Jacob Sanders, November 2011.

"When you work in log odds, the distance between any two degrees of uncertainty equals the amount of evidence you would need to go from one to the other. That is, the log odds gives us a natural measure of spacing among degrees of confidence."

That observation is so useful and intuition friendly it probably deserves it's own blog post, and a prominent place in your book.

Forgive me if this sounds condescending, but isn't saying "0 and 1 are not probabilities because they won't let you update your knowledge" basically the same as saying "you can't know something because knowing makes you unable to learn"? If we assign tautologies as having probability 1, then anything reducible to a tautology should have probability 1 (and similarly, all contradictions and things reducible to contradictions should have probability 0). For any arbitrarily large N, if you put 2 apples next to 2 apples and repeat the test N times, you'll get 4 apples N out of N times, no less (discounting molecular breakdowns in the apples or other possible interferences).

So you are saying that statement "0 and 1 are not probabilities" has probability of 1?

O = (P / (1 - P))

probabilities and odds are isomorphic

This is undefined for P = 1. If you claim that that function is a real-valued bijection between probabilities and odds then P = 1 doesn't work so you're begging the question. Always take care to not divide by zero.

Whether or not real-world events can have a probability of 0 or 1 is a different question than "are 0 and 1 probabilities?". They most certainly are.

If I roll a die, then one of the events that can happen will happen. That's just saying that if S is my sample space, then P(S) = 1. Similarly, P(~S) = 0, which is just saying that impossible things won't happen. The former statement is an axiom in the standard mathematical treatments of the subject. These statements may be trivial, but I distrust any mathematics that can't handle trivial cases.

Rejecting 1 as a probability would be catastrophic when you're dealing with discrete spaces. If you're the sort to reject infinity, then it would follow that all probability spaces are discrete. At that point probability loses its rigor. Preference for odds or log odds just means that you have to live with using the extended reals with special conventions for the infinities.

This article is largely incoherent. The main justification is the abuse of an invalid transformations: y=x/(1-x) is not the bijection that he asserts it is, because it's not a function that maps [0,1] onto R. It's a function that maps [0,1] onto [1,\intfy] as a subset of the topological closure of R. And that's okay, but you can't say "well I don't like the topological closure of R, so I'll just use R and claim that 1 is where the problem is."

Additionally, his discussion of log odds and such is perfectly fine, but ignores the fact that there are places where you do need to have an odds of 0:1, or a log odds of negative infinity. Probability theory stops working when you throw out 0 and 1, it's as simple as that.

Even if you don't want to handle tautologies or contradictions, there are other ways to get P(X)=0 or 1. The probability that a real number chosen uniformly from the real interval [0,1] is 0. It has to be. It's a provable fact under ZFC and to decide otherwise is to say that you're more attached to the idea of 0 and 1 not being probabilities than you are to the fact that mathematics is consistent and if you really believe that, well, there's absolutely nothing I have to say to you.

This is one of those situations where EY just demonstrates he knows very little mathematics.

A real mathematician got in a debate with EY over this post, and made some really good points: https://np.reddit.com/r/badmathematics/comments/2bazyc/0_and_1_are_not_probabilities_any_more_than/cj43y8k

Maybe this doesn't stand up mathematically, but I really like the intuition of log odds instead of probability. And this post explained it quite well. And the main point that you shouldn't believe in absolute certainties is still true. An ideal AI using probability theory would probably use log odds, and not have a 0 or 1.

What are the odds that the face showing is 1? Well, the prior odds are 1:5 (corresponding to the real number 1/5 = 0.20)

I'm years late to this party, and probably missing something obvious. But I'm confused by Yudkowsky's math here. Wouldn't it be more correct to say that the prior odds of rolling a 1 are 1:5, which corresponds to a probability of 1/6 or 0.1666...? If odds of 1:5 correspond to a probability of 1/5 = 0.20, that makes me think there are 5 sides to this six-sided die, each side having equal probability.

Put differently: when I think of how to convert odds back into a probability number, the formula my brain settles on is not P = o / (1 + o) as stated above, but rather P = L / (L + R), if the odds are expressed as L:R. Am I missing something important about common probability practice / jargon here?

It's a nice analogy, but it all rests on whether infinite evidence is a thing or not, and there aren't arguments one way or the other here. (Sure, infinite evidence would mean "whatever log odds you come up with, this is even stronger", but that doesn't rule out it is a thing). 

Like, how much evidence for the hypothesis "I'll perceive the die to come up a 4" does the event "Ok, die was thrown and I am perceiving it to be a 3" provide? Or how much evidence do I have of being conscious right now when I am feeling like something? I think any answer different from infinity is just playing a word game.

Wouldn't the prior odds in the bell example be 1:4 when the chance is 0.2? But written is 1:5.

When you work in log odds, the distance between any two degrees of uncertainty equals the amount of evidence you would need to go from one to the other.

What does "amount of evidence" in this sentence is supposed to mean? Is it the same idea that "bits of evidence" mentioned in these posts previously?

The only way I can interpret this sentence as a definition of "amount of evidence", but then I don't understand what's the point of highlighting the sentence as if it's saying something more significant.