Back to AI

OK, here we go.

AI is full of many fairly well-developed methods of "propositional" learning: probabilistic, fuzzy, and otherwise fancified versions of stuff we can do with boolean algebra but no quantifiers. In other words, current AI is fairly good at manipulating systems that have a fixed number of variables, but not so good at arbitrary logical structures.

A simple illustration of the two categories:

Propositional Version:
I give the AI data on 3,000 patients, which includes information about which patients had a list of 20 common symptoms, and what of 10 diseases those patients were ultumately determined to be suffering from. The AI learns to predict disease from symptoms.

Predicate Version:
I give the AI data on 3,000 patients, with logical descriptions of symptoms and logical descriptions of diseases. Again, the AI is to learn to predict diseases from symptoms.

In the first case, the AI is learning a mapping from 20 variables (the twenty symptoms) to 10 variables (the diseases). In the second case, the AI is learning a mapping from one infinite space (possible logical descriptions of symptoms) to another (possible logical descriptions of diseases).

There are some borderline cases. For example, if we are learning to map arbitrary real numbers to arbitrary real numbers, the space is infinite, and we *could* treat the real numbers as logical entities rather than merely continuous variables, but this is almost never done; so, it is propositional. (Treating the real numbers as logical entities would mean worrying about interesting exact numbers like 1, 2, 3, pi, e, 1/2, and so on. For propositional methods, 1 is not any more special than 1.002.) It gets fuzzier... suppose we are learning about integers rather than real numbers. We could adapt the same propositional strategies we used in the real-number case, restricting them to integers. It might work well. But with integers there is a greater chance that the exact number matters, and a greater chance that the number should be treated as a logical entity. Perhaps a pattern relies on one of the integers being even. A propositional method will not see this, and will need to learn from the data which integers the pattern applies to and which it doesn't. So, it will only be able to extend the pattern to cases it has seen before. If this sort of pattern is likely, more sophisticated methods become critical. Yet, we're still working with a fixed number of variables. The more sophisticated methods become really critical when a given problem instance can contain a totally new variable, yet one that patterns from old variables might apply to. And then they become really really critical when problem instances can't even be totally separated, because they all fit into one big structure of logical relationships...

The transition from the first type of AI to the second is now taking place, largely under the banner of "probabilistic relational models". Hooray!

(I've labeled the second type "predicate" to distinguish it from "propositional". This way of naming the two types is not uncommon, but many people use "relational" to describe the second class, instead. Another convenient term is "logical".)

Anyway, I'm going to outline one way of carrying over the progress that's been made with propositional models to the higher level... this is an idea I've been sort of thinking about on the back burner, partly as a way of making an inherently parallel learning method. By no means do I think this is the only way of going forward, and in fact I'll probably be working in a different direction myself for at least a little while... maybe I'll write about that later.

So, how can we use propositional algorithms in the wider domain (whatever we choose to call it)?

I described propositional models as models having no quantifiers. Really, though, a universal quantifier over all the variables is taken for granted in the propositional setting. If a propositional method learns that a particular variable is "2 or 3", it doesn't mean that for some one case it is either two or three; it means for any case it is either two or three. This single quantifier gives us some ability to do relational-style learning.

Suppose we want to learn a probabilistic model for pictures. We could treat every pixel as a variable, and learn correlations between them from the dataset. This approach would require a very large dataset before any interesting patterns could be found. It would need to re-learn every object in every possible location, because it would have no way of generalizing from one location to another. A smarter way to apply propositional methods to the problem would be to treat a small square, say 10 by 10, as a single "instance"; we learn correlations between variables in such squares. With this approach, we might be able to get something interesting from even one image. There is a cost, of course: where before it was impossible to generalize from one location in the picture to another, now it is impossible to not do so. For example, the method would not notice that an egg appears in the top right corner of each picture; it will simply notice that an egg is a common object. This can be helped, by adding the location information as an extra variable.

More generally, for any structured data, this approach can be used to learn local structure. For linear data such as text, we can use a fixed number of letters rather than a fixed number of pixels. For tree-structured data (such as computer program code, HTML...), we can use a branching segment. But, we can go even further: there is a totally general mapping that will handle any logical structure. Any data can be represented in the form of predicate statements: a series of entities, each of which can have properties and relationships with other entities. Just as we can select a random 10 by 10 area of a picture and ask what the colors are, we can select 10 entities at random and ask what their properties and relationships are. Let's call this the "universal mapping".

The universal mapping allows us to learn logical structure, but it does have some serious limitations. Suppose once again that we're looking at linearly structured data such as text, but in predicate calculus form, and with the universal mappping. We have a bunch of entities ordered by a next/previous relation, and with a "letter" property that distinguishes each entity as 'a', 'b', 'c', ... 'A', 'B', 'C', ... 'space', 'comma', 'period', et cetera. Now suppose that we sample entities at random. If we've got much text, it will be very unlikely that we'll pick letters that occur next to eachother. We're learning a correct distribution, technically, but we're not looking at the "interesting" part of it (the line). The algorithm could eventually learn what it was supposed to, but it would take too much time, and besides that it would not be in a directly usable form (since the distribution we wnat is embedded in a larger, useless distribution). If we used a special-purpose linear mapping that did not sample so randomly, we'd be much better off.

OK. Anyway, that is the "state of the art" so to speak. These are all basically a type of markov model. So, what is my new suggestion?

Well, first, let's look at one more relevant detail of the current propositional algorithms. Many of them are hierarchical in form. This means that rather than finding complex patterns directly in the set of variables, the algorithm creates additional variables, and finds simpler relationships. Simpler relationships that include additional entities amount to the more complex relationships that we could have looked for in the first place; but the process is easier to manage by abstracting it in this way. This abstraction is iterated: second and third levels of hidden variables can be created to manage increasingly complex relationships, which treat the lower levels exactly as the first level treats the visible variables.

Two recent sucessful examples are Hinton's deep belief nets and Numenta's Hierarchical Temporal Memory.

So in the propositional domain, we can talk about hidden variables. In the predicate domain, we can also talk about hidden variables, but we can add hidden relations and even hidden entities to the list. For propositional methods, hidden variables are just an efficiency issue. In the predicate domain, it is very different: hidden stuff dramatically increases the representational power (and therefore the model search space).

Starting simple with hidden variables: if each entity is allowed to posses hidden variables, then the modeling power has changed from markov-model to hidden-markov-model.

Adding hidden entities and relations is enough to boost the representational power up to turing-completeness: any computable pattern can be represented.

So, OK, enough background. Now, the question is, how can we best use the power of existing propositional methods to search a turing-complete model space? Here is the idea I've been pondering recently...

For the moment, let's say we're working with linearly structured data, for simplicity. One turing-complete formalism is the cellular automaton. These bear some structural similarity to a hierarchical network of variables. The major difference is that all cells are the same, meaning that all variables are interacting with their neighbors in exactly the same manner. My basic idea is to semi-smoothly integrate deep belief network learning with cellular automaton learning. For the linearly structured data, then, the system would be searching for two-dimensional automata. Intuitively, learning a cellular automaton of hidden variables is similar to learning the physical laws that hold sway equally at any point in space. We can see directly how those physical laws work for the visible space, and we generalize, assuming that there are nearby hidden spaces that partially explain the visible space, and farther-off hidden spaces, and so on, but that all follow the same physical laws.

Well, that was nearly 20 paragraphs of introduction for 1 paragraph of an idea... but, that introductory material was interesting in itself, and it will be useful to refer to in future posts. So, more later...

Bayesian Convergence: What it Will and Won't Do

In my continuing quest for the proper prior, I happened upon a nice result: under very general conditions, the effect that the prior has on the current belief will vanish as evidence accumulates. Nice! This means that a Bayesian learner will be good regardless of the choice of prior-- the learned beliefs will fit the evidence.

Does this mean the search for the correct prior is needless?

To answer that question, I should first give the convergence result in a bit more detail. (To get it in full detail, see this paper.)

The key assumption that is made is that no model of the world has zero probability. Bayesian learning will never increase the probability of such a model, so this makes sense. The convergence result is most easily understood in the language of likelihood ratios. In this version of bayesian learning (which gives the same end results as other versions, just by different intermediate steps) we start out with the "prior odds," rather than the prior probability, of a model. The prior odds of a model is just the probability for that model divided by the probability against. For each new bit of evidence that comes in, we multiply the current odds by the "likelihood ratio". The likelihood ratio is the probability of the evidence given the model, divided by the probability of that evidence given its negation. (The probability of the evidence given the negation is actually the sum of its probability given each of the other possible models.) Now, as we observe more and more evidence, we multiply again and again to update the odds for each model we're considering. Yet the prior odds remain a constant at the beginning of that long line of multiplication. The odds of a model, then, can become as large as they like regardless of prior, and likewise can become as small as they might. The evidence is what matters, not the prior.

In any case, I am not about to drop my concern about which prior a rational entity should choose. The main reason for this is that the convergence result leaves open the question of the class of models to be considered, which is my primary concern. Even if this were settled, however, the convergence theorem would not convince me that ensuring nonzero probability for each model is sufficient. The reason has to do with predictions.

To make the case as extreme as possible, I'm going to ignore probabilistic models, and only consider deterministic ones. This is actually no restriction at all; any prior over probabilistic models could be seen as a fancy way of specifying a prior over totally deterministic ones. A probabilistic model gives a probability to each possible dataset, and a prior over many probabilistic models can be seen as just a weighted sum of these, giving us a new (possibly more complicated) distribution over the possible datasets. This can be used as a prior. In fact, since it gives the same overall probability for each dataset, it is for practical purposes the same prior; yet the models it considers are deterministic.

Now that we're working with completely deterministic models, the data will either fit or not fit with each model. When it doesn't fit, we throw that model out. The convergence theorem still holds, because the set of models we're considering will keep shrinking as we throw more out; whenever this happens, the probability that belonged to the discredited model will be redistributed among the still-valid ones. Thus the probability of the correct model will continue to increase (since it's never thrown out).

However, this is not much comfort! The relative probabilities of the models still in consideration will not be based on the evidence at all; it will still be based purely on the prior. (The probability from a model that gets thrown out is redistributed, but not evenly.) This means that when we make predictions, the prior is (in a loose sense) the only thing that determines our prediction.

In fact, if the prior assigns nonzero probability to every possible dataset, then the set of models not yet ruled out will contain all possible futures. The only thing that can narrow this down to make a useful prediction is the prior, which may or may not do so in a way dependent on the evidence so far.

Perhaps someone objects: "But then, can't we just require that a prior's predictions do depend on the evidence? Isn't it an obviously silly mistake to construct a prior that violates this?" Unfortunately, simply ruling out these cases doesn't tell us what prior to use. What kind of dependence do we want? I want a prior that can in theory "notice any sort of regularity"; but this includes noticing that the data is just completely random (predictably unpredictable).

In a way, allowing probabilistic models is a very strange move. It's very similar to allowing models that are infinitely large; in a way, a probabilistic model includes information about an infinite number of coin flips, which are used in a well-specified (deterministic) way to decide on predictions. Of course, when we specify a probabilistic model, we don't specify this infinite table of heads and tails; in fact, that's where probability theory gets its power. This is reminiscent of the idea of a "random sequence" being a more fundamental notion then "probability", as discussed in the previous post... but that's enough speculation for today.

Interpretation of Probability, Yet Again

In my previous post, I mentioned that my "personalized "bayesian frequentist" interpretation of probability was struck down by the news that frequentism conflicts with the standard axioms of probability." The issue needs more consideration, so this post will discuss it exclusively.

To that end, I've found a very interesting article on the subject.

The information I had found previously went something like this. The simplistic frequentist position is to say that the probability of an event is the fraction of times it occurs in some set. This is problematic mainly because if we flip a coin five times and get the fractions heads=1/5 and tails=4/5, we can attribute these fractions to chance. We don't automatically believe that the "real" probability of heads is 1/5.

Revision 1 of the frequentist view changes the definition to "limiting frequency": the probability is the fraction we would get if we kept at it. The limiting frequency does not always exist. For example, consider a sequence containing As and Bs, starting with ABBAAAABBBBBBBB... Each time, the number of same letters in a row doubles. The ratio of A to B will wave back and forth forever, never settling. So by definition, probabilities only apply to sequences for which the limiting frequency exists.

This is better, but it still isn't quite right. There are two problems. First, a coin could land on heads every even flip and tails every odd flip. The limiting fraction for both sides would be 1/2, but this is obviously not a random sequence. So the requirement that there is a limiting frequency is not enough to guarantee that the sequence is probabilistic. Second, it is possible to re-order an infinite sequence to make the limiting frequencies different. With the same alternating heads/tails sequence, we could reorder as follows: group heads together in pairs by moving them backwards, but keep tails isolated. This makes the limiting frequency of heads 2/3. It seems odd that the probability would change when we're just counting in a different order.

To fix this, von Mises defined something called a "collective". Before reading the paper, I knew that a collective had the additional property that any subsequence chosen without knowledge of where the heads were and where the tails were would have the same limit frequencies. I had also read that the resulting theory was inconsistant with the standard axioms of probability. I wondered: if this definition is inconsistant with the standard axiomization, what sort of alternative probability theory does it yield?

What the paper immediately revealed was that the collective-based definition of probability was a competitor to the now-standard axiomization, Kolmogorov's axiomization. It is not surprising, then, that the two are inconsistant with eachother. Where the two differ, von Mises preferred the collection-based account. It is not hard to see why; his account is grounded in a mathematical concept of a random sequence, while Kolmogorov's axioms simply tell us how probabilities must be calculated, without any explanation of what a probability means.

Mainly, the notion of a collective is weaker; it does not allow us to prove as much. For example, it is quite possible that a coin being flipped approaches the correct frequency "from above": with only a finite number of exceptions, the ratio of heads at any finite time can be greater than 1/2, although it gets to 1/2 as we keep going. Perhaps a stonger notion of random sequence is needed, then. But I do think the von Mises approach of defining random sequences first, and then probabilities, seems like a better foundation. I wonder: is there any definition of randomness from which the Kolmogorov axioms automatically follow?

Modal Probability Stuff

My personalized "bayesian frequentist" interpretation of probability was struck down by the news that frequentism conflicts with the standard axioms of probability. In the wake of this disaster, I am forced to develop some new ideas about how to interpret probabilities.

I've been thinking of using modal logic to support probability theory. I looked around the internet somewhat, and it seems like there is some action in the opposite direction (using probability theory as a foundation for modal logic), but not so much in the direction I want to go. (I assume of course that my direction isn't totally unique... it seems natural enough, so someone's probably tried it. But anyway it is in the minority.)

Modal logic is essentially the logic of possibility and necessity. Because these words are ambiguous, there are many possible modal logics. We have a number of worlds, some of which can "see" eachother. Different facts may be true and false in each world. In a given world, a fact is "necessary" if it is true in all worlds that world can see. It is "possible" if it is true in at least one of those worlds. Modal logic lets us assign definite meaning to larger stacks such as "necessarily necessary" (it is true in all worlds that the worlds we can see are able to see), "possibly necessary" (in some world we can see, it is true in all worlds that world can see), and "possibly possibly possible" (in some world we can see, in some world it can see, in some world it can see, the fact holds).

Perhaps the idea of "worlds we can see" seems ambiguous. My choice of wording is essentially arbitrary; a common choice is "accessibility". As with necessity and possibility, the exact meaning depends on the modal logic we're using. For example, we might want to talk about immediate future necessity and possibility. The worlds we can see are the possible next moments. "Possibly possible" refers to possibility two moments ahead, "possibly possibly possible" is three moments ahead, and so on. Another modal logic corresponds to possibility and necessity in the entire future. We can "see" any possible world we can evetually reach. Additionally, we might also say that we can see the present moment. (We could choose not to do this, but it is convenient.) In this modal logic, "possibly possible" amounts to the same thing as "possible"; "necessarily necessary" amounts to "necessary". However, "necessarily possible" does not collapse (since it means that a fact remains possible no matter which path we go down in the future), and neither does "possibly necessary" (which means we can take some path to make a fact necessary). Which strings of modal operators collapse in a given modal logic is one of the basic questions to ask.

Other modal logics might represent possibility given current knowledge (so we can only access worlds not ruled out by what we already know), moral necessity and possibility ("must" vs "may"), and so on. Each has a different "seeability" relationship between worlds, which dictates a different set of logical rules governing necessity and possibility.

Just to give an example of using probability theory as a foundation for modal logic, we might equate "possible" with "has probability greater than 0," and "necessary" with "has probability 1". I doubt this simple approach is very interesting, but I know very little about this. I'm more interested in the opposite possibility.

The idea would be something like this: we attach a weight to each world, and at any particular world, we talk about the probability of an event in the worlds we can see. The probability of a fact is the sum of the weights of each seeable world in which it is true, divided by the sum of the weight of all seeable worlds.

The idea here is that probability does not make sense without a context of possible worlds. Probability within a single world doesn't make sense; each thing is either true or false (or perhaps undefined or meaningless). Furthermore, there is no privileged scope of possibilities; we might at one moment talk about probabilities assuming the context of physical possibility (in which case the flip is nearly deterministic, we just don't have enough information to calculate the result), or epistemic possibility (in which case it is just fine to assign a probability to the coin flip without sufficient knowledge of the physical conditions).

One advantage of this approach is that we get an automatic meaning for the troublesome notion of a "probability of a probability". We need this concept, for example, if we want to use bayesian learning to learn the bias of a possibly-biased coin. We assign some probability to each possible ratio of heads to tails, and then start flipping the coin and updating our belief based on the outcomes. This particular example is a bit complicated to treat with the framework I've outlined (which is perhaps a point against): we need to mix modal logics, taking one probability to be based in worlds in which the coin has different properties, and the other to only range over changes in the suurounding conditions of the coin flip. The probability-of-a-probability, then, is the sum over seeable worlds in which,in their seeable worlds, the sum of an event divided by the total sum equals a particular ratio. In the case of the coin, the two instances of "seeable" have different definitions: we jump from the current world to worlds in which the coin has different properties, but from them we jump to worlds in which the flip has different starting conditions. (Note: we know neither the coin's actual properties nor the actual exact starting conditions, so for us itis not much of a "jump". But sometimes we reason with posible worlds which are quite distinctly not our own.)

Perhaps this is not especially intuitive. One problem I can think of is the lack of justification for picking particular ranges of worlds to reason in. But it is an interesting idea.

In partial answer to the questions posed in the last post, I found an interesting paper that shows how, given any logic (within certain broad boundaries), it is possible to construct a corresponding probability theory. In particular, this paper uses the technique to construct an intuitionistic probability theory.

The paper also argues against abandoning what it calls the "principle of addition". The principle is simple: given non-overlapping states, A and B, the probability of A or B should be the probability of A plus the probability of B. (Formally: if P(A and B) = 0, then P(A or B) = P(A) + P(B). The paper gives a slightly different version.) This may not sound very worrisome, but it actually causes severe problems for the frequency interpretation of probabilities. (I was very surprised to learn this. The idea that probabilities are frequencies is logically incompatible with the standard rules of probability theory! I have not yet found a detailed explanation of the incompatibility; it is often mentioned, but rarely explained...) So although a large class of alternative probability theories are covered by the definition (as many as there are alternative logics), the definition is very conservative in some ways.

I do not know much about what happens when we drop the principle of addition (also commonly called "finite additivity"). But in some sense, fuzzy logics are among this field of possible notions of "probability". This idea (together with the fact that frequencies are actually incompatible with normal probability) makes my slightly less antagonistic towards the fuzzy logic approach... but not much. I was previously of the mindset "Probabilities are the mathematically correct way to go, and anything else is just silly." Now I am of the mindset "Probabilities have a fair mathematical motivation, but there is room for improvement." So another theory may be workable, but I'm asking a very strong foundation before I'll stop clinging to standard probability (or if I finally settle on an alternative logic, some conservative generalization of probability such as the one above).

In other news, I've found some very interesting material concerning intuitionistic logic. Sergei N. Artemov has done some amazing-looking work. In particular, on this page, he explicitly claims to have a logic that "circumvents the Incompleteness Theorem". I don't yet know exactly what he means, but the approach is described in this paper, which I will read ASAP. (Actually, I'm reading this one first, because it is also very interesting...)

I have also been running into some amazing math blogs lately. Good Math has a nice introduction to lambda calculus and intuitionistic logic (as well as pleny of other introductions to important areas). (Note: the blog moved from Blogspot to ScienceBlogs, but the old posts I'm referring to are the ones on Blogspot.) Also, Mathematics and Computation looks like a good read. There is a very interesting-looking post on seemingly impossible functional programs, which gives an algorithm for exhaustively searching through the space of infinite sequences of 1s and 0s. Unfortunately, I did not understand very well on my first read, because the algorithms are written in Haskell, which I am unfamiliar with... I found this blog through Machine Learning (Theory), another good blog.

My next post was *going* to be an essay showing how the halting problem, Godel's theorem, Russel's paradox, and several other things can all be demonstrated using one method, called the diagonal method. Perhaps this post is still coming, but I have not felt inspired to write the whole thing out yet. For the moment:

Some Important/Crazy Questions

1. OK, so there's classical logic. In classical logic, everything is either true or false: not neither, not both. Classical logic has some oddities, which have led people to propose alternative logics. Two such logics are: intuitionistic logic, and paraconsistent logics. In some sense, these are opposite solutions to the same problem. (See here.) I like to think of it this way: both logics admit that there are true cases, false cases, and border cases.
They differ in how they deal with the border cases. Intuitionistic logic calls them "neither true nor false". Paraconsistant logic calls them "both true and false". This choice results in very different treatments. This notion of a "border case" has surprisingly been formalized using topology, and that is where I got the metaphor. (See here.) But If there is such a nice relationship between the two, shouldn't they be usefully equivalent, or perhaps part of some larger system? A movie I recently watched made a random reference to the idea of a four-valued logic, one with "true", "false", "both", and "neither". How would this work? (Googling the idea seems to bring up random unrelated stuff.)

2. Another alternative logic I have heard of is called "quantum logic". This is literally a logical system to work with the oddness of quantum mechanics. Strange! So my second question is: how does this logic work? How does it relate to paraconsistant and intuitionistic logics?

3. For applications in artificial intelligence, it is very useful to attempt a seamless integration of probability and logic, rather than simply using logic to reason about probabilities. (This is of course the goal of the Alchemy system I recently mentioned, as well as many other AI systems.) The natural question is how these alternative logics generalize to alternative theories of probability. For intuitionistic logic and paraconsistant logic, possible probabilitic versions readily present themselves: intuitionistic probability theory would let probabilities sum to less than one, and paraconsistant probability theory would let them sum to more than one. I was amazed to hear that the quantum logic has a simple probabilitic version: allow probabilities to have both real-number and imaginary-number components. (I heard this on the AGIRI mailing list. See here, second post in the thread.) How should all these things be interpreted??

Googling all of this stuff comes up with many results, so perhaps I will have a post that answers some of these questions soon. Meanwhile, I am still convinced that some form of nonmonotonic logic that can reason about incomputable models of the world is the important direction to go.

Well, I think I've come up with something of an answer. I want a prior to be interpreted as a frequency estimate on possible worlds. This sounds funny, because we can't possibly estimate such a frequency: we only live in one world. But this is actually just fine: we shouldn't be estimating it, because it's our prior. It's what we use to estimate.

Anything more we learn, we learn using our prior. So can't improve upon our prior. If you've got a bad prior, tough luck.

A prior is an estimate of the frequency of alternative worlds. The perfect prior would contain all knowledge we ever needed; it would give our actual world-of-birth a probability of 1, and all other worlds, 0. But no two people are born to the same world, so evolution couldn't find this prior. (By the way, we could also view evolution as using a prior-- this prior is given to it by the very nature of chemistry and physics, and is not very good, but far better then it might have been.) So a slightly weaker and more useful notion of the perfect prior would be one that would do a fair job if all humans had it. Forcing all humans to have the same prior (which is close to being true) causes the perfect prior to have far more interesting structure (ie some learning occurs), although it still would have freakish foresight for things common to all humans (it would know the one true physics, for example).

Since what I'm interested in is learning, I want some way of ruling out this freakish foresight: I want to talk about a universal prior, one that will learn well no matter what the true physics turns out to be (and so on). I'm rejecting the Solomonoff prior because I think computability is too strict a requirement, but I also know that some restrictions are needed (otherwise there is no structure for the prior to take advantage of). What kind of a universal prior is this? And once I've figured that out, is it really of any use?

More on the Interpretation of Probability

In my previous discussion, I have failed to distinguish between logical necessity and physical necessity. This distinction is critical for my analysis requirement 2: a probability is a statement of uncertainty that could always be turned into certainty given more information. In my first post on the interpretation of probability, I talked about alternatives (A) and (B):

(A): The universe has some set of uncaused events.

(B): The universe does not have uncaused events.

(A) implies that there are actual alternatives: the universe could have happened differently, if these uncaused events happened differently.

(B) implies that there are no such alternatives. However, this comes at the cost of implying an infinite chain of causes. Worse, as I reasoned before, we should still desire a cause even of this infinite chain; an infinite past history is not enough, because we also need an explanation of why that particular infinite past is the one we have. Furthermore, we need a cause for this cause, and so on.

Now comes the new stuff. When I stated this, I had in mind that the explanation for the infinite past was physics. This isn't quite sufficient: physics needs to be supplemented by a single completely-specified time-slice. Then a deterministic physics can specify the infinite past and future for us. So allowing this, we further ask: why does the universe contain this particular physics? To answer this, we create a meta-physics that specifies for us an explanation of why physics is the way it is. (My impression is that there is work in theoretical physics corresponding to this desire.) Again, in addition to a meta-physics, we need something like a physics-slice sufficient to specify the rest of physics (that is, a minimal number of variables that determine the rest via the meta-physical laws).

(I should note that my concept of "meta-physics" is not the typical concept of "metaphysics": metaphysics is a philosophical pursuit, but meta-physics is essentially very abstract physics, which worries about things like how we might predict the existence of electrons from base principles.)

The chain continues to meta-meta-physics, meta-meta-meta-physics, et cetera ad infinitum.

All of these things are akin to physical necessity. Each fact is determined by some law of a corresponding physics (except for the determining slices). However, as the sets of rules get more abstract, it seems as if we will hit the ceiling of mathematical necessity. This may have happened already: the physical theory of A. Garrett Lisi (which from what I know looks like what I called a meta-physics) describes our (meta-) physics as an algebra, so that the obvious alternatives are different algebras. (As I understand it: our physics obeys the algebra E8, but E8 does not fully determine our physics; in addition, we need information about symmetry-breaking. So E8 is the meta-physics and symmetry breaking information is the minimal set of variables needed to determine everything else about the physics. But don't take my word for fact.) The algebraic alternatives are governed by mathematical laws. The mathematical laws are governed by logic. So logic may be as little as 3 metas away! (E8 is meta-physics, so math is meta-meta, so logic is meta-meta-meta-physics.) This seems to stop the questionable infinite regress: it seems at least plausible for logic to be uncaused and unexplained.

But even this doesn't tie us down to one possible world, namely because of all those minimal slices that we specify along the way. These do the real work of specifying everything; logic's being "at the top" is only a convenient trick. Presumably we could forever play the game of navigating plausible infinite regresses of explanation and plausible places for the regress to stop, but it seems that we could never find a real end to it. (This is particularly chronic if we start to question which logic is at the top, i.e. classical vs intuitionistic vs many other possibilities, and why that logic is at the top, i.e. do we need a meta-logical theory?)

Therefore we are forced into (A)! There must be fundamentally unexplained things, and therefore actual alternatives.

By the way, I don't think that exempts humanity from the investigation of the hierarchy I've described ultimately topped by logic: this hierarchy seems very important despite its ultimate futility (we cannot in principle explain everything, but we must still explain as much as possible).

Concluding (A) does not force me to give up (1) or (2). In particular, I had previously assumed that it went against (2), because I took "more information" to mean causal (or explanatory) information. This is unnecessary; all I need to say is that all meaningful statements are either true or false. Thus the "more information" may merely be the fact itself. As an example, suppose that some physical events really are random: physical law dictates probabilities but not the definite outcome. What I'm saying is that there still is a definite fact of the matter, although not determined by physical law; if we knew everything there is to know, we would know the outcome.

There are still some clarifications needed. I have dealt with issues arising from my first post, but there are also difficulties with what I said in my second post. Basically, in that post I trapped myself into a fundamental confusion that always will arise for those that try to do away with the concept of a bayesian prior. This is hinted at in the infinite regress I get into when I try to take into account uncertainty about relevant information. At some point, if the probability estimate given is to be coherent, the person must invoke a prior belief concerning each probability. So to revise the conclusion I reached there:

A probability (used by a person) is (1) a belief concerning a frequency, (2) a statement of uncertainty given limited information which can always be turned into certainty given more information, and (3) based on some prior belief (updated by the limited information available).

The image here is that our limited information narrows down the space of possible worlds somewhat, and that we hold beliefs about the frequencies of events in the remaining possible worlds. So, for example, if we assert a 50% probability that our friend will buy a pink car, we mean that (given our prior) our information narrows us down to a space of possible worlds such that in about half our friend will get a pink car.

I want to interpret the prior probability in a way consistent with the way I interpret the beliefs formed using the prior plus evidence. Otherwise, it doesn't seem like a full interpretation. I think the best way of doing this, in line with the idea that a probability is a frequency estimate, is to say that the prior is the person's estimate of the frequency of all possible worlds.

I admit this sounds strange. Accepting (A) forces me to say that there are actual alternatives, meaning possible worlds. It seems somewhat reasonable to attach probabilities to these (by attaching probabilities to the uncaused facts). But to go a step further and call these frequencies? Does this make sense?

I suppose I'm forced to leave this question open for now. I think the answer is yes, but my only reason for thinking so is the way it simplifies the whole scheme.

Let's take another look at the two statements I wanted to make about probability.

1. A probability asserted by a human is really a believed frequency.

2. A probability is a statement of uncertainty that could always be turned into certainty given more information.

If the second statement is true, then any probabilistic model is inherently incomplete. This means that there are no single-event probabilities; any "believed frequency" I assert is wrong unless it's 0 or 1, because a single event can only happen or not, and it's meaningless to give a probability.

What is meaningful however, is giving a probability based on the limited information at hand. In doing so, we give our belief about the frequency that would result from looking at all situations that have the same givens; situations that would look the same from our perspective, but might turn out differently. I'd argue that this is basically what people intend when they give such probabilities.

However, this also has some snags: I can't seriously assert that people believe such parallel situations always literally exist. People could give probability estimates in unique situations that never occurred before and may never occur again.

To get past this, I reformulate my statement:

People give probability estimates (1) based on the limited information at hand, (2) only using relevant information, and (3) ignoring potentially relevant information if it doesn't match any previous experience and so doesn't help predict.

The first addition, that only information thought to be relevant is used, helps somewhat by reducing the previously crippling number of unique situations. Now, situations can be considered the same if they vary only in ways irrelevant to the event being predicted. The other addition, however, that potentially relevant information be ignored if it turns the situation into a unique situation, is the real clincher. It guarantees that the probability estimate is meaningful.

But there are still problems.

Clause 3 above may fix everything, but it's pretty problematic from the point of view of machine learning. A prediction made by ignoring some evidence should be given lower certainty. The math there is straightforward; we have some probability of the variable being relevant, and we have no idea how it effects things if it is. We therefore weight the two possibilities, adding our prediction of what happens if the variable isn't relevant to an even wash if it is. (This is an inexact statement, but whatever.) So the result is weaker for each ignored item.

The probabilities-of-relevance here are necessary, but to fit them in to the interpretation, must be given the same sort of interpretation; in other words, they've got to be estimates based on the limited amount of relevant information and so on. The "so on" includes a potential infinite regress because we need to again weaken our statements based on any potentially relevant but new variables, and again this involves a probability of relevance, which again must be estimated in the same way, and so on. However, I'm not sure this is a problem. The reason I say this is that I see it as a series of progressively better estimates; in practice, we can cut it off at some point if we need to, and just use the coarsest possible estimates of the next-down level of probabilities. This could be reflected in a further modification:

People give probability estimates based on as much of the relevant information at hand as they can quickly decide the consequences of.

In other words, we don't compute instantaneously, so we may not use all the relevant information at our disposal, or may use some only partially (using it in the most important estimates, but making less important estimates more quickly by ignoring more information).


This basically seems to reconcile the statements I began with, (1) and (2) at the top of the page. However, I'm still not completely sure about (2). I still may want to assert that there are actual alternatives in the world.

An Interpretation of Probability


You know that something is seriously wrong when, in a science, there is as serious of talk about correctly interpreting a theory as there is work attempting to extend the theory or propose new theories. I personally think it is ridiculous that so much effort is put into "interpreting" quantum mechanics. There's the many-worlds interpretation, the ideas about what counts as an "observer", questions of whether or not a particle actually has a definite speed and location if they can't even in principle be observed simultaneously (what can't be observed doesn't exist), arguments about what it means for an event to be probabilistic, and so on. I'm frustrated both because these don't seem to be real questions (or where they are real, we should experimentally test them rather than argue), and because I want to claim obvious answers myself.

But enough of criticizing a science I know little about. Another field, which I know somewhat more about, is experiencing a similar problem: probability theory. The argument between Bayesianism and Frequentism, two separate schools of thought that have their own very different mathematics for statistics and machine learning, is essentially an argument about the meaning of probability.

Frequentists interpret probability based on the idea of random variables and physical random processes. Probability is defined as a statistical frequency; a probability for an event is the ratio at which that event will occur if we sample for a sufficiently long amount of time. For example, if we flip a coin enough times, we can determine to any desired degree of accuracy the actual ratio of heads to tails for that coin. This seems intuitive, right? This notion turns probability into a solid, objective quantity. In fact, frequentist probabilities are often called "objective probabilities".

Bayesians, however, disagree. For a bayesian, probability can also represent a degree of belief. This distinction is particularly important when an event only occurs once, or when we're talking about the probability of a statement, which can only be actually true or actually false. Frequentist probability cannot be used here, because there is no frequency to measure. For example, suppose we're back before the time of Magellan. You ask me my opinion on the shape of the Earth. I, being well-versed in the popular philosophy of the time, suppose that the earth is round; but I'm not sure. So I give it a 95% chance. From a frequentist view, this is nonsense. The earth is either flat or round; it isn't a physical random process. The frequency of a flat earth is either 1 or 0.

At this point, you're probably siding with the frequentists. The earth does not have a 5% chance of being flat-- sticking an exact number on it sounds silly, even if I want to say that there's a possibility. But let's consider an example you might sympathize with a little better. Suppose you have a friend who is purchasing a vehicle for the first time. You know that your friend (for some reason) refuses to own any car that is not either pink or purple. However, you don't know of any bias between the two colors, so before you see the car, you can do no better than assign a 50% probability to each color. Upon showing you the car, which is pink, your friend explains that the decision was made based on alphabetical order; pink comes before purple.

Now notice-- you had no way of knowing which of the two colors your friend would choose. It seems very reasonable to assign equal probabilities to each. However, such a probability does not seem to be a frequency-- if we rewound history to "try again", the friend would always reason the same way, and the new car would always be pink. Probability can only be interpreted as degree of belief here; and that's exactly what bayesians want to allow. In contrast to the frequentist objective probabilities, bayesian probabilities are called "subjective probabilities". (It is fairly common practice to admit both kinds of probability, so that one might say "The coin has an objective probability of landing on heads [since it could land on either heads or tails], but a subjective probability of being counterfeit [it either is or it isn't]".)

The battle between bayes and frequency has been a long one, but that's not exactly what I'm going to talk about. I'd like to talk about my own grapplings with the interpretation of probability. While I am (at the moment) firmly bayesian in terms of the mathematics, I actually agree with both interpretations of probability; I think that, all probabilities represent a belief about a frequency. But I also want to be able to say a few other things about probability, which I'm not sure are entirely consistent with that view.

Here's what I want to say about the meaning of probability:

(1) All probabilities are a belief about a frequency.

-Even the most obviously frequentist example of probability, coin flipping, can be thought of in this way. We never know the exact distribution of heads to tails; but we use the 50/50 estimate regularly, and it causes no problems.

-Many probabilities that seem to be fundamentally bayesian can be explained with the concept of possible universes. The frequency of a flat earth in our universe is 0. But the frequency of a flat earth in all possible universes may be higher. Even if it's not, because we can never know probabilities for certain, but only estimate them, it's entirely reasonable for someone who does not have the benefit of modern science to estimate something like 5% of all alternatives universes have a flat earth. The estimate can be improved later. (Because we can only ever visit one universe, our estimate of the distribution of possible universes will always be quite crude; so probabilities based on such estimates will have a "subjective" feel to them. But I claim that they are not really a different kind of probability.)

-When counting the frequencies, we only consider universes that match ours sufficiently; in the flat earth example, we consider alternative universes that match everything we *know* about earth, and estimate the frequency of something we *don't* know: whether the earth is flat or round. Similarly, in the example of the pink car, we consider universes that match things we know, but (being only human) are unable to use facts we don't know (such as the fact that our friend loves using alphabetical order to resolve difficult decisions). (This is called a "conditional probability"; if you think it doesn't sound very well-defined, I assure you that it is a very mathematical notion, which has been rigorously founded in logic.) This explains another reason that bayesian probabilities seem "subjective": different people are often considering different evidence (pinning down different facts) when giving probabilities.

(2) All probabilities are really just statements about uncertainty.

-I'm claiming here that the concept of a "physical random process" is philosophically unnecessary. When an event like a coin flip seems random, it is actually quite deterministic; we just don't know all the relevant factors involved. Even if we do know all the relevant factors, we aren't necessarily able to calculate their influence on the final result (at least not quickly).

-Whenever we assign something a probability, then, it's simply because we don't know all the relevant facts (or haven't calculated their influence). Quantum mechanics, for example, gives us probabilistic laws for the behavior of particles; I'm claiming that the probabilistic nature of these laws shows that they aren't the fundamental laws, and that our predictions of quantum events could in principle be improved with more information (or perhaps with a better method of calculating).


To be perfectly honest, I'm less certain about the second claim. I think both claims seem to be what's intuitively correct; but the two seem to contradict eachother.

If probabilities are statements of uncertainty, can they also be frequency estimates?

It seems possible in at least some situations. For example, when flipping a coin, we can create a frequency estimate for both heads and tails, but still claim that if we knew more, we would be able to calculate ahead of time which it would be. In this case, the frequency is based on common known variables from toss to toss (mainly the coin involved), whereas the uncertainty is caused by the unknown variables (the physics of the toss). But this doesn't necessarily solve everything.

The first conflict that I see is that the idea that there are other possible worlds, necessary to my argument for (1), seems to be ruled out by (2). If anything can in principle be narrowed down to one possibility by knowing the relevant facts, then there can be no actual alternatives! Alternatives are illusions, which can be eliminated by the diligent scientist.

Never mind the conflict with (1)! Is this view even self-consistant? (2) asserts that any question can be answered by looking at the relevant information. But this implicitly assumes that every event has a cause. The result is an infinite regress of causes. (While this isn't a direct inconsistency, it is reason to worry.)

So, I'm willing to back down on (2), instead stating two possibilities:

(A) The universe has some set of first causes. Therefore, there exist real alternatives; the universe actually could have been different. The probability of these different universes is unknown and unknowable, because we can only observe on universe, but it isn't an unintelligible concept. (1) holds but (2) does not hold if the event we're examining happens to be a first cause.

(B) The universe has no uncaused events. There is an infinite chain of causes leading to each event. (2) holds, but (1) does not always hold: the universe is deterministic, so probability cannot always be interpreted as a belief about frequency. Specifically, (1) doesn't work when we would resort to alternative universes because we don't have multiple instances of the given event in our universe.

I'm a bit torn between the two, but I prefer (A). I don't have anything against the infinite past implicit in (B), except that the entire timeline has an uncaused feel to it. Why this timeline rather than another? If (B) is correct, then there is some reason. OK. Sure. But (B) also states that this reason has a cause, and it's cause has a cause, and so on. Another infinite causal chain, over and above time. But what caused that infinite chain? And so on. The idea of infinite causes is a bit unsettling.

So is the idea of an uncaused event, to be sure. But I like being able to say that there are real alternatives, and to say that, it seems I've got to admit uncaused events.

Notice that these problems melt away if we restrict our talk to manageable islands of reality rather than entire universes. Both (1) and (2) hold just fine if we don't run into an uncaused event or an event so singular that none like it have ever occurred before or will occur again.

Can we ever know what's really true? Can we experimentally differentiate between (A) and (B)? Probably not. Then why does it matter?

Maybe it doesn't.