Large deviations of continuous regular conditional probabilities

We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate

Integrals for functions with values in a partially ordered vector space

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions

Bochner integrals in ordered vector spaces

We present a natural way to cover an Archimedean directed ordered vector space $E$ by Banach spaces and extend the notion of Bochner integrability to functions with values in $E$. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map

Collagen bundle morphometry in skin & scar tissue: a novel distance mapping method provides superior measurements compared to Fourier analysis

Histopathological evaluations of fibrotic processes require the characterization of collagen morphology in terms of geometrical features such as bundle orientation thickness and spacing. However, there are currently no reliable and valid techniques of measuring bundle thickness and spacing. Hence, two objective methods quantifying the collagen bundle thickness and spacing were tested for their reliability and validity: Fourier first-order maximum analysis and Distance Mapping, with the latter constituting a newly developed morphometric technique. Histological slides were constructed and imaged from 50 scar and 50 healthy human skin biopsies and subsequently analyzed by two observers to determine the interobserver reliability via the intraclass correlation coefficient. An intraclass correlation coefficient larger than 0.7 is considered as representing good reliability. The interobserver reliability for the Fourier first-order maximum and for the Distance Mapping algorithms, respectively, showed an intraclass correlation coefficient above 0.72 and 0.89. Additionally, we performed an assessment of validity in the form of responsiveness, in particular, demonstrating medium to excellent results via a calculation of the effect size, highlighting that both methods are sensitive enough to measure a treatment effect in clinical practice. In summary, two reliable and valid measurement methods were demonstrated for collagen bundle morphometry for the first time. Due to its superior reliability and more useful measures (bundle thickness and bundle spacing), Distance Mapping emerges as the preferred and more practical method. Nevertheless, in the future, both methods can be used for reliable and valid collagen morphometry of skin and scars, whereas further applications evaluating the quantitative microscopy of other fibrotic processes are anticipated

Large Deviations of Continuous Regular Conditional Probabilities

We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate