Are there nonequivalent randomnesses?

Question

There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean http://www.formalontology.it/suszkor.htm), even nonequivalent logicians;-)

Are there nonequivalent randomnesses?

The main two theories we know dealing with randomness and probability is Kolmogorov Probability Theory ( by means of measure theory and Borel $\sigma$-algebras), and Bayesian a priori approach. Are they equivalent enough to say that they are the same in some deeper meaning?

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@ Johannes Hahn - no, I am not asking about non isomorphic probability spaces as it would be trivial. rather I as about possible probability theories as different as different are geometries euclidean and noneuclidean. The obvious generalization is to change linearity in the second axiom of probability ($P(A U B) = P(A) + P(B)$ when A,B are independent.

In fact I mention about it after reading probability overview by Terence Tao http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/ He wrotes:

probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space.

Which in my opinion is something very deep ( but I am only a hobbyist;-). So probably if You have initial probability space, and You need to extent it to describe some additional phenomena, You have have done some kind of morphisms between structures of first space and another wider one. Is there a unique, canonical or any defined way of doing this? May we perform this kind of extension always in the same way or there are different ways of doing that? Gives it any predictable and interesting structure?

@sheldon-cooper Bayesian approach to probability is sometimes seen as alternative ( not very well defined) to Kolmogorov axiomatic probability system, because it do not require given a priori probability space. For example in this approach we may say that probability that tomorrow will be could day ( say temp<-10) is defined, whilst in Kolmogorov approach You probably cannot define proper space ( because You cannot have equivalent population of Wednesdays which are tomorrow days, with different temperatures). Agree - when You have possibility to use properly defined probability space this two approach coincides. From Wikipedia: http://en.wikipedia.org/wiki/Bayesian_probability

Bayesian probability interprets the concept of probability as "a measure of a state of knowledge",[1] in contrast to interpreting it as a frequency or a physical property of a system.

@Qiaochu Yuan - of course randomness here is colloquialism. Yes You have right: maybe I just should ask about different probability theories, but note that non-euclidean geometries in analogy are just geometries but in different spaces, with some special properties. So in fact they share the same meaning of geometric set, figure, space, even so complicated objects as coordinate systems, and angle. But they have different relations between them. So I ask about something similar: different kinds of randomness which are in scope of probability theory but describes different relations between for example different classes of ways of extensions of probability spaces. If the last procedure changes anything in resultants;-) Agree - maybe this is not very interesting question. Maybe it would be more interesting in scope of algorithmic information theory and its randomness concept?

Answer 1

A different answer from the ones so far: Quantum randomness is another kind of randomness that is a generalization of traditional randomness, i.e. classical or non-quantum probability. I think that it fits the question because you could likewise say that non-Euclidean geometry, interpreted as not-necessarily-Euclidean geometry, is a generalization of Euclidean geometry.

A classical probability space is usually defined as a $\sigma$-algebra $\Omega$ with a normalized measure. From the Bayesian viewpoint the measure could equally well be called a "state". Now, a $\sigma$-algebra is the algebra of Boolean random variables with a certain set of axioms. But you can just as well write down axioms for $L^\infty(\Omega)$, the algebra of bounded complex random variables. In favorable cases, it is a commutative von Neumann algebra. In quantum probability you instead allow a non-commutative von Neumann algebra $\mathcal{M}$. Also, in standard quantum probability you keep the usual completed tensor product $\mathcal{M} \otimes \mathcal{N}$ as the model of a joint system. (Free probability theory is still quantum probability, but with a certain free product instead of a tensor product.) You also still have states, conditional states, joint states, correlations, generalized stochastic maps, etc.

Some of the variant models mentioned so far lead to different theorems, but generally give the same answers in combinatorial probability, questions like the birthday paradox or modeling games of chance. Quantum probability leads to a significantly different picture of combinatorial probability, generalizing the old one, but also allowing new answers such as violation of Bell's inequalities, covariance matrices that are Hermitian rather than real symmetric, new complexity classes such as BQP, etc.

Other variant models mentioned so far no longer give any answers for combinatorial probability, for instance models of forcing. But, part of the interest in probability is that it models real life. Amazingly, so does quantum probability; that was the central discovery of quantum mechanics when it was defined in the 1920s and 1930s.

Answer 2

Probably you are interested in an answer coming from measure theory or probability, and I would be interested in such an answer also, but let me give you an answer from set theory, which I believe does in fact offer a precise answer, with something like the sense that I believe you intend in your question.

In set theory, the method of forcing is fundamentally about different kinds of randomness or genericity. Specifically, in order to carry out a forcing argument, we first must specify the notion of forcing or genericity that will be used. This is done by specifying a certain partial order or Boolean algebra, whose natural topology will be used to determine a notion of dense sets. Generic objects will be those objects that are in many dense sets. In the extreme case in which forcing is applied, one has a model of set theory V, and considers adjoining an object G that is V-generic, in the sense that G is a filter containing elements from every dense set in V. The generic extension V[G] is a new model of set theory containing this new generic object G. The overall construction is something like a field extension, because V sits inside V[G], which is the smallest model of ZFC containing V and G, and everything in V[G] is constructible algebraically from G and objects in V. Forcing was first used to prove the consistency of ZFC with the negation of the Continuum Hypothesis, and the notion of forcing used had the effect of adding ω₂ many new real numbers. The generic object was a list of ω₂ many new reals, forcing CH to fail.

The point is that in order to control the nature of the forcing extension V[G], one must carefully control the forcing notion, that is, the notion of randomness that will be used to build V[G].

It turns out that many of the most fruitful forcing notions involve a notion of genericity or randomness on the reals. For example, with Cohen forcing, a V-generic real is one that is a member of every co-meager set in V. A random real is a member of every Lebesgue measure one set in V. There are dozens of different forcing notions (see the list of forcing notions), and these have been proved to be fundamentally different, in the sense that a generic filter for one of these is never generic with respect to another. With forcing, we can add Cohen reals, random reals, Mathias reals, Laver reals, dominating reals and so on, and these notions have no generic instances in common.

There isn't any probability space here to speak of, and being V-generic for a particular notion of forcing is not equivalent to any probabilistic property. Rather, one is tailoring the notion of forcing to describe a certain kind of randomness or genericity that is then used to build the forcing extension. Most of the detailed care in a forcing argument is about choosing the forcing notion and making sure that it works as desired.

Thus, since each notion of forcing corresponds fundamentally to a notion of genericity or randomness, I take the proofs that the different notions of forcing are different as an answer to your question.

Answer 3

Short answer to the question in your title: yes, indeed, and a number of recursion theorists computability theorists have been busy investigating various notions of randomness for the past few years. For them, it seems that ``randomness'' is a notion that you apply to infinite sequences (say, of 0's and 1's), like Kolmogorov randomness -- but there is also ''Martin-Löf randomness'' and some other related notions.

This is far from my expertise, but here is a reasonable-looking survey article, and you can follow the references to find many other similar papers. My impression is that this has been a hot subfield of computability theory lately.

Answer 4

Cox's theorem states that if you accept certain `common sense' rules (e.g. consistency with logic), then standard probability laws are the only valid inference laws. I.e. you cannot arbitrarily change sums to products and such. Wikipedia page is not bad, and a longer derivation appears in Jaynes's book here.

If you don't accept some (or all) of these rules as 'common sense', then it seems you can derive other laws, which I guess could be called 'nonequivalent probabilities' in the sense that they are inference laws not corresponding to any standard probability. One example is given in [3] (also listed on the wiki page).

[3]: J. Halpern, "A counterexample to theorems of Cox and Fine," JAIR 1999

Answer 5

First, there's a difference between Kolmogorov probability and Bayesian probability theory. The Kolmogorov axioms require countable additivity, which Bayesians have difficulty justifying, for a variety of reasons.

Cox's theorem is an example of how "objective Bayesians" justify the laws of probability theory. They believe that probability theory should be an extension of logic which can deal with uncertainty. His theorem only justifies finite additivity, and not countable additivity. (Objective Bayesians are "objective" because they believe there all rational agents should begin with noninformative, or maximum entropy, priors. The difficulty/art of objective Bayesianism arises from the fact that uniform distributions over infinite sets don't necessarily exist.)

Subjective Bayesians don't agree that probability theory is an extension of logic -- instead, they believe that probabilities reflect subjective states of belief, and the laws of probability should be derivable as the decision-theoretically optimal way of updating beliefs to reflect evidence. The usual way of formalizing this is via "Dutch book" arguments -- given an agent with a set of beliefs, it should not be possible to come up with a set of acceptable bets that in expectation causes the agent to lose money. Here, the difficulties with countable additivity arise from the fact that it's not clear that agents can talk about infinite sets of bets.

Kolmogorov introduced the axiom of countable additivity in order to apply measure theory to probability. This has been enormously fruitful, and Terry Tao has a wonderful blog post about how measure theorists think about probability as a formal mathematical theory.

In addition to Bayesian concepts of probability, there are frequentist concepts of probability as well. Here, the idea is to formalize the idea of an infinite random sequence of binary digits, as a sequence in which you can't determine the $n+1$-th element from the first $n$, but which in the limit satisfy statistical properties like the law of large numbers (e.e., we will get approximately equal numbers of 1s and 0s). This was introduced by von Mises in the early twentieth century, and there were a lot of difficulties in formalizing if -- it turns out that the set of random sequences with the straightforward definition is empty(!), and this is what motivated recursion-theoretic notions of random sequences, as sequences without computably detectable regularities.

There are also wilder variants of probability theory. For example, if you take a logical view of probability, it follows that disagreements about logic (e.g., intuitionistic versus classical) will yield different probability theories. AI researchers have also looked at qualitative probability theories, in which probability is not a real number, but rather discrete (e.g., "likely" versus "not likely") -- this is to have a theory of what they learn when they elicit preferences from experts when building (e.g.) expert systems. They have also looked at theories in which probability is a set of primitive evidence, so that they can calculate what facts explain a given logical assertion.

Answer 6

Although as the OP admits it's not clear what the question is here, there is a sense in which no, there do not exist nonequivalent randomnesses (for a fixed probability distribution $\mathbb P$), rather there exist different degrees of randomness.

If we fix a ground model of set theory $M$ then we can declare:

a real number $r\in [0,1]$ is random if for each set $S\subseteq [0,1]$ with $\mathbb P(S)=0$ and $S\in M$, we have $r\not\in S$.

So $r$ looks random as far as $M$ can tell; such an $r$ is extremely random! Then one can consider more down-to-earth notions such as

as far as any polynomial-time-bounded Turing machine can tell, $r$ looks random.

This would mean that an impatient clerk, looking at the bits of $r$, cannot detect more and more evidence that $r$ exhibits an unusual pattern (i.e. that $r\in S$).

An intermediate degree is

as far as any Turing machine can tell, $r_1$ looks random

There are several distinct notions that attempt to fit this latter description (Martin-Löf randomness is a greater or equal degree of randomness than Kolmogorov-Loveland randomness, which is greater than computable randomness, which in turn is greater than Schnorr randomness...).