I'll try to give you the formalist perspective, which is a sort of 'minimal' take on the whole matter.
Everything starts with a set of symbols, usually finite, that can be combined to form strings called formulas.
Logic is then defined as two set of rules: one that tells you which strings of symbols are considered valid (morphology), and one that tells you which valid formulas follows from which valid formulas (syntax).
Then there's the concept of truth: when you have a logic, you notice that sometimes formulas refer to entities or states of some environment, and that syntactic rules somehow reflect processes happening between those entities.
Specifying which environment, which processes and which entities you are considering is the purpose of ontology, while the task of relating ontology and morphology/syntax is the purpose of semantics.
As you can probably imagine, there are a myriad of logics and myriads of ontologies (often called models).
There's Hilbertian ontology, where the environment is a set or a class, the entities are its members and relations and functions between them, and semantics relates Tarskian truths to other Tarskian truths.
There's categorial ontology, where formulas are interpreted as object of a category and syntactic rules as arrow between them.
There's dialogical ontology, where formulas are states of a game and rules are ways for attacking or defending the current state.
There is possible world ontology, which you have described.
And so on and so on.
Hystorically, a specific set of rules and symbols emerged exceedingly often, and seemed to be particularly apt to capture the reasoning that mathematicians intuitively adopted. That is now known as classical logic (CL), and for a very very long time it was believed to be the only "true" logic, that is the logic of the atemporal and universal description of all things.
CL is very useful and can be adapted to a wide array of ontologies: for example, Boolean algebras (which are instances of the Hilbertian ontology) and possible worlds semantics. The latter in particular was ideated for an extension of CL, both in symbols and rules, known as modal logic.
Enter probability.
The determination of which concepts the word refers to has a long and heated history, but now the modern understandig is four-fold. On one side, you have those who refer to a stated property of the world, a not very well specified "long-run randomized frequency" (frequentist). On the other side, you have those who believe that probability does not refer to an objective property of a system, but to a degree of beliefs of an observer (Bayesian). Bayesians themselves are divided among those who thinks that probabilities are entirely subjective (subjectivist), relying on the De Finetti coherence theorem and pragmatic interpretation, and those who thinks that probabilites are the degrees of belief that an idealized rational agent has about a system. These folks we can call objectivist, but since they are the majority here on LessWrong, it's simpler to just call them Bayesian.
Bayesians rely on Cox theorem, that justifies the structure of probability from a small set of minimally rational requirements.
All three of them, frequentists, subjectivists and Bayesians, believe that the structure of probability is correctly described by the mathematical concept of a measure, as formalized by the Kolmogorov axioms.
Enter Jaynes.
He was a physicist, and wrote a book from a Bayesian perspective, showing that probabilities thus intended are an extension of CL. Where CL can be seen as assigning to proposition only two values, 0 and 1, PTEL (probability theory as extended logic) actually relaxes that restriction, assigning values in the [0,1] interval, following two simple rules derived from Cox theorem.
In his work, Jaynes greatly systematized and simplified the field, solved many of its paradox and brought probability theory to previously unexplored areas of application. He also talked about the ideal rational agent as a 'robot', it is thus no surprise that his book is regarded as a cornerstone for LW's understanding of Artificial Intelligence.
Now, on to your questions:
Does the above explanation differ from how other people use probability?
As you can see, you are just using one ontology (possible worlds) to justify one interpretation (Kolmogorov measure), but there are many more. The interpretation of choice here is PTEL, usually coupled with an Hilbertian ontology for the underlying CL or just left unspecified.
why should those estimations follow the laws of probability?
Because of Cox theorem, you can show that with a specified amount of initial information, you can do no better than following the laws of probability.
And if I understand right, they says that Bayes' theorem follows from Boolean logic, which is similar to what I've said above, yes?
Actually no. First, there is no Boolean logic: there is classical logic interpreted in the ontology of Boolean algebra, but that doesn't really matter. Cox theorem is based on CL, but makes additional assumptions (that of a minimal, ideally rational observer). It's not just pure logic.
Also there has been some exploration in the direction of extending Cox with other basal logics.
I'm familiar with predicate logics as well, but I'm not sure what the interaction of any of them is with probability or the use of it
As per above, PTEL is based only on classical logic, which is a particular first order, two-valued predicate logic. AFAIK, no successful extension of Cox has been made to other kind of logic.
Fuzzy logic resembles PTEL in the expansions of the set of truth values, but uses different rules than CL, so the resemblance is only superficial: PTEL and fuzzy logics are two very different beasts.
sets just need to be exhaustive and mutually exclusive for the Kolmogorov axioms to work, right?
No, not really. Kolmogorov axioms are defined on any (sigma)-algebra, wether it is the algebra of subsets of a measurable sets or some other things.
the answer really is out there?
Of course it is ;)
First of all, let me thank you so much, MrMind, for your post. It was really helpful, and I greatly appreciate how much work you put into it!
I'll try to give you the formalist perspective, which is a sort of 'minimal' take on the whole matter.
Much obliged.
Everything starts with a set of symbols, usually finite, that can be combined to form strings called formulas.
Question. I'm making my way through George Lakoff's works on metaphor and embodied thought; are familiar with the theory at all? (I know lukeprog did a blog post about them, but it's not n...
Hello, everyone.
I'm relatively new here as a user rather than as a lurker, but even after trying to read ever tutorial on Bayes' Theorem I could get my hands on, I'm still not sure I understand it. So I was hoping that I could explain Bayesianism as I understand it, and some more experienced Bayesians could tell me where I'm going wrong (or maybe if I'm not going wrong and it's a confidence issue rather than an actual knowledge issue). If this doesn't interest you at all, then feel free to tap out now, because here we go!
Abstraction
Bayes' Theorem is an application of probability. Probability is an abstraction based on logic, which is in turn based on possible worlds. By this I mean that they are both maps that refer to multiple territories: whereas a map of Cincinatti (or a "map" of what my brother is like, for instance), abstractions are good for more than one thing. Trigonometry is a map of not just this triangle here, but of all triangles everywhere, to the extent that they are triangular. Because of this it is useful even for triangular objects that one has never encountered before, but only tells you about it partially (e.g. it won't tell you the lengths of the sides, because that wouldn't be part of the definition of a triangle; also, it only works at scales at which the object in question approximates a triangle (i.e. the "triangle" map is probably useful at macroscopic scales, but breaks down as you get smaller).
Logic and Possible Worlds
Logic is an attempt to construct a map that covers as much territory as possible, ideally all of it. Thus when people say that logic is true at all times, at all places, and with all things, they aren't really telling you about the territory, they're telling you about the purpose of logic (in the same way that the "triangle" map is ideally useful for triangles at all times, at all places).
One form of logic is Propositional Logic. In propositional logic, all the possible worlds are imagined as points. Each point is exactly one possible world: a logically-possible arrangement that gives a value to all the different variables in the universe. Ergo no two possible universes are exactly the same (though they will share elements).
These possible universes are then joined together in sets called "propositions". These "sets" are Venn diagrams, or what George Lakoff refers to as "container schemas"). Thus, for any given set, every possible universe is either inside or outside of it, with no middle ground (see "questions" below). Thus if the set I'm referring to is the proposition "The Snow is White", that set would include all possible universes in which the snow is white. The rules of propositional logic follow from the container schema.
Bayesian Probability
If propositional logic is about what's inside a set or outside of a set, probability is about the size of the sets themselves. Probability is a measurement of how many possible worlds are inside a set, and conditional probability is about the size of the intersections of sets.
Take the example of the dragon in your garage. To start with, there either is or isn't a dragon in your garage. Both sets of possible worlds have elements in them. But if we look in your garage and don't see a dragon, then that eliminates all the possibilities of there being a *visible* dragon in your garage, and thus eliminates those possible universes from the 'there is a dragon in your garage' set. In other words, the probability of that being true goes down. And because not seeing a dragon in your garage would be what you would expect if there in fact isn't a dragon in your garage, that set remains intact. Then if we look at the ratio of the remaining possible worlds, we see that the probability of the no-dragon-in-your-garage set has gone up, not because in absolute terms (because the set of all possible worlds is what we started with; there isn't any more!) but relative to the alternate hypothesis (in the same way that if the denominator of a fraction goes down, the size of the fraction goes up.)
This is what Bayes' Theorem is about: the use of process of elimination to eliminate *part* of the set of a proposition, thus providing evidence against it without it being a full refutation.
Naturally, this all takes place in ones mind: the world doesn't shift around you just because you've encountered new information. Probability is in this way subjective (it has to do with maps, not territories per se), but it's not arbitrary: as long as you accept that possible worlds/logic metaphor, it necessarily follows
Questions/trouble points that I'm not sure of:
*I keep seeing probability referred to as an estimation of how certain you are in a belief. And while I guess it could be argued that you should be certain of a belief relative to the number of possible worlds left or whatever, that doesn't necessarily follow. Does the above explanation differ from how other people use probability?
*Also, if probability is defined as an arbitrary estimation of how sure you are, why should those estimations follow the laws of probability? I've heard the Dutch book argument, so I get why there might be practical reasons for obeying them, but unless you accept a pragmatist epistemology, that doesn't provide reasons why your beliefs are more likely to be true if you follow them. (I've also heard of Cox's rules, but I haven't been able to find a copy. And if I understand right, they says that Bayes' theorem follows from Boolean logic, which is similar to what I've said above, yes?)
*Another question: above I used propositional logic, which is okay, but it's not exactly the creme de la creme of logics. I understand that fuzzy logics work better for a lot of things, and I'm familiar with predicate logics as well, but I'm not sure what the interaction of any of them is with probability or the use of it, although I know that technically probability doesn't have to be binary (sets just need to be exhaustive and mutually exclusive for the Kolmogorov axioms to work, right?). I don't know, maybe it's just something that I haven't learned yet, but the answer really is out there?
Those are the only questions that are coming to mind right now (if I think of any more, I can probably ask them in comments). So anyone? Am I doing something wrong? Or do I feel more confused than I really am?