Re: Reigniting Probability theory debate

On Wednesday, 11 April 2012 14:19:59 UTC-7, Tegiri Nenashi wrote:

> Propositional Logic is essentially Boolean Algebra which is rather
> simplistic algebraic structure, known to be friendly to
> generalizations (e.g. algebra of binary relations). Introductory
> chapter of Jone's textbook focused on Propositional calculus and is
> very convincing that it can be generalized to Calculus of
> Probabilities. However, step up to Predicate Calculus, and it is not
> at all evident that there is any connection to Probability Theory.

Yes, and predicate logic is exactly Codd's (named attribute) relational algebra.

A (named attribute) relation can be seen as a set of or mapping on multi(-named-)dimensional points. Functional dependencies are properties of relation values and expressions.

Database relational operators are designed so that there is a correspondence between relation expressions and predicates (and predicate expressions aka wffs). The value of a relation expression is the extension of a corresponding predicate (and wff) where the relation value's attributes are the predicate's parameters (and the wff's free variables). A relation expression has an associated predicate (and wff) built from it in a certain way according to its operators and its variables' given predicates (and wffs). The fundamental theorem of the relational model is that IF the body of each relation variable's value is the set of tuples that make a given predicate (or wff) true THEN the body of each relation expression's value is the set of tuples that make that expression's predicate (or wff) true. Eg if the predicate of relation R is R(X,Y) "person X loves person Y" and the predicate of relation variable S is S(Y,Z) "person Y loves food Z" then the expression for (R JOIN S) PROJECT_AWAY Z is EXISTS Z [S(X,Y) AND R(Y,Z)] "there exists a Z such that person X loves person Y and person Y loves food Z" ie "person X loves person Y who loves some food".

Probability operators for treating relations as probability distributions will do different things (in general) than database operators. They will satisfy different theorems.

A relation with a functional dependency can represent a function. Composition and images are relevant in databases when a relation expression corresponds to a function's (or wff term's) value. To the extent that distributions are used as (relational or functional) mappings such representation-independent mapping-oriented operators will appear in that system.

So what we can expect is that what the two systems have in common is... they both somehow involve a relation as a set of or mapping on multi(-named-)dimensional points. Correspondences between operators other than ones that are oriented to mappings would be coincidental. I don't call that much of an analogy/parallel.

I doubt that relations are an appropriate abstraction for distributions per se. I expect that a relation is just one constituent of a proper distribution representation (which would include notions of dependent and independent coordinates/variables/attributes and calculating and renormalizing probabilities to sum to 1 and multiplying and summing for conditional and marginal probabilities and binary mappings in particular) and that any relevant operators on relations (which generally won't be database ones) are in turn used to define other operators on distribution representations per se. The paper's abstract says that conditional distributions and marginal distributions are given by selection and projection respectively. However I expect that what is actually the case is that for their relational representation of (some of) a distribution some notion of removing rows and columns happens as part of complex relation operators implementing distribution operations. That is human vague reminiscence, not semantic correspondence.

philip Received on Thu Apr 12 2012 - 00:02:41 CEST