Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part I

Abstract

This paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower-dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 Ä m < n be fixed positive integers, ; ¤ D Â R n some open domain, and hW D ! R m a function yielding a full partitioning of D into a family, denoted M.h/, of lower-dimensional surfaces/manifolds via inverseh/º noting each h 1 .t / can also be considered the solution set of all X in D of the simultaneous equations h.X/ D t . Let X be a random vector (rv) over D having a probability density function (pdf) f . Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh.X /=dX over D, etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M.h/ that satisfies a list of natural desirable properties? More generally, for a fixed positive integer r, we can pose a similar question concerning the rv .X /, when W D ! R r is some bounded a.e. continuous function, not necessarily admitting a pdf. Problem considered hereOriginally, over thirty years ago, Higgins and Saw in a series of papers considered special lower-dimensional cases of the problem considered in the above abstract and came to the conclusion, in effect, but without formally recognizing it, that the Implicit Function Theorem (IFT) could be used to address this problem. In this paper, using a stronger global form of IFT (GIFT), it is shown that a succinct theory of inducing distributions on families, denoted M.h/ in the abstract, of lowerdimensional surfaces can be developed, extending the ideas of Higgins and Saw, using just the tools of elementary analysis, yet the proposed approach possesses a number of natural properties. More specifically, under the GIFT assumption for h, we take as the definition of E. .X / j X in h 1 .t//, for each t in range.h/, the limit of the standard conditional expectation with non-zero antecedent probability, E. .X/ j X in h 1 .a//, for any sequence or net of open neighborhoods of t in range(h) with positive probability nesting down to singleton ¹tº in a way equivalent to a nested family of open m-spheres. It is shown that not only all of Higgins and Saw's results hold concerning compatibility and connections with the rv's .X/ and h.x/, but also that: the definition obeys the Radon-Nikodym constraint in many various ways, it has a strong invariance property, satisfies an optimal L 2approximation criterion, as propounded, e.g., by Tjur, has natural connections with surface measures and surface densities, and in particular for h affine and the rv X distributed as multivariate gaussian, is completely compatible with the standard theory of sing...

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