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Poisson process partition calculus with an application to Bayesian . . .
, 2005
"... This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
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Cited by 31 (10 self)
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This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 26 (9 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Semipullbacks and Bisimulation in Categories of Markov Processes
, 1999
"... this paper, we show that the answer to the above question is positive. More specifically, we give a canonical construction for semipullbacks in the category whose objects are families of Markov processes, with given transition kernels, on Polish spaces and whose morphisms are transition probability ..."
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Cited by 5 (1 self)
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this paper, we show that the answer to the above question is positive. More specifically, we give a canonical construction for semipullbacks in the category whose objects are families of Markov processes, with given transition kernels, on Polish spaces and whose morphisms are transition probability preserving surjective continuous maps. One immediate consequence is that the category of probability measures on Polish spaces with measurepreserving continuous maps has semipullbacks. Our construction gives semipullbacks for various full subcategories, including that of Markov processes on locally compact second countable spaces and also in the larger category where the objects are Markov processes on analytic spaces (i.e. continuous images of Polish spaces) and morphisms are transition probability preserving surjective Borel maps. It also applies to the corresponding categories of ultrametric spaces. Finally, our result also holds in the larger categories with Markov processes which are given by subprobability distributions, i.e. the total probability of transition from a state can be strictly less than one. We now explain the relevance of our result in computer science. The consequences of Semipullbacks and Bisimulation 3 our mathematical result in the theory of probabilistic bisimulation has been investigated in (Blute et al., 1997; Desharnais et al., 1998). We will briefly review this here. Following the work of Joyal, Nielsen and Winskel (Joyal et al., 1996) on the notion of bisimulation using open maps, define two objects A and B in a category to be bisimular if there exists an object C and morphisms f : C ! A and g : C ! B, i.e.,
The disintegration of the Lebesgue measure on the faces of a convex function
, 2009
"... We consider the disintegration of the Lebesgue measure on the graph of a convex function f: R n → R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the kdimensional Hausdorff ..."
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Cited by 3 (1 self)
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We consider the disintegration of the Lebesgue measure on the graph of a convex function f: R n → R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the kdimensional Hausdorff measure of the kdimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we
SPLITTING OF LIFTINGS IN PRODUCTS OF PROBABILITY SPACES 1
, 2002
"... We prove that if (X,A,P) is an arbitrary probability space with countably generated σalgebra A, (Y,B,Q) is an arbitrary complete probability space with a lifting ρ and ̂R is a complete probability measure on A ̂⊗R B determined by a regular conditional probability {Sy:y ∈ Y} on A with respect to B, ..."
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We prove that if (X,A,P) is an arbitrary probability space with countably generated σalgebra A, (Y,B,Q) is an arbitrary complete probability space with a lifting ρ and ̂R is a complete probability measure on A ̂⊗R B determined by a regular conditional probability {Sy:y ∈ Y} on A with respect to B, then there exist a lifting π on (X × Y,A ̂⊗R B, ̂R) and liftings σy on (X,Ây, ̂Sy), y ∈ Y, such that, for every E ∈ A ̂⊗R B and every y ∈ Y, [π(E)] y = σy([π(E)] y). Assuming the absolute continuity of R with respect to P ⊗ Q, we prove the existence of a regular conditional probability {Ty:y ∈ Y} and liftings ̟ on (X × Y,A ̂⊗R B, ̂R), ρ ′ on (Y,B, ̂Q) and σy on (X,Ây, ̂Sy), y ∈ Y, such that, for every E ∈ A ̂⊗R B and every y ∈ Y, [̟(E)] y = σy([̟(E)] y) and ̟(A × B) = y∈ρ ′ (B) σy(A) × {y} if A × B ∈ A × B. Both results are generalizations of Musia̷l, Strauss and Macheras [Fund. Math. 166 (2000) 281–303] to the case of measures which are not necessarily products of marginal measures. We prove also that liftings obtained in this paper always convert ̂Rmeasurable stochastic processes into their ̂Rmeasurable modifications. 1. Preliminaries. If (Z,Z,S) is a probability space, then we denote by ̂Z the completion of Z with respect to S and by ̂ S the completion of S. We write L ∞ (S): = L ∞ (Z,Z,S) for the space of bounded Zmeasurable realvalued functions. Functions equal a.e. are not identified. We use the notion of lower density and lifting in the sense of [7]. It is known (cf. [7]) that there is a 1–1 correspondence among liftings on Z with
Preprint Ser. No. 26, 1991, Math. Inst. Aarhus Perfect Measures and Maps
"... This paper is designed to provide a solid introduction to the theory of nonmeasurable calculus. In this context basic concepts are reviewed and fundamental facts are presented. Perfect measures and maps are shown to of a vital importance to support the theory, and therefore these concepts are treate ..."
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This paper is designed to provide a solid introduction to the theory of nonmeasurable calculus. In this context basic concepts are reviewed and fundamental facts are presented. Perfect measures and maps are shown to of a vital importance to support the theory, and therefore these concepts are treated separately as well. In particular, many equivalent statements to the property of being perfect are given. The last part of the paper deals with a setting arising in the investigation on extension of measures in the case of perturbations of theiralgebras. 1.
Disintegration of cylindrical measures
, 2002
"... We show that the existence of disintegration for cylindrical measures follows from a general disintegration theorem for countably additive measures. ..."
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We show that the existence of disintegration for cylindrical measures follows from a general disintegration theorem for countably additive measures.