The stochastic order of probability measures on ordered metric spaces

The general notion of a stochastic ordering is that one probability distribution is smaller than a second one if the second attaches more probability to higher values than the first. Motivated by recent work on barycentric maps on spaces of probability measures on ordered Banach spaces, we introduce and study a stochastic order on the space of probability measures $\mathcal{P}(X)$, where $X$ is a metric space equipped with a closed partial order, and derive several useful equivalent versions of the definition. We establish the antisymmetry and closedness of the stochastic order (and hence that it is a closed partial order) for the case of a partial order on a Banach space induced by a closed normal cone with interior. We also consider order-completeness of the stochastic order for a cone of a finite-dimensional Banach space and derive a version of the arithmetic-geometric-harmonic mean inequalities in the setting of the associated probability space on positive matrices.Comment: 25 page

all_data

A .zip file containing all data required to repeat analyses presented in Miller et al. "Vitis phylogenomics: hybridization intensities from a SNP array outperform genotype calls" published in PLoS ONE. Includes a ReadMe file

PCA plot of six genotyped grapevines projected on to principal components calculated from 33 accessions of East-Asian wild <i>Vitis</i> species and 33 cultivars of <i>V</i>. <i>vinifera</i>.

<p>Numbers in parentheses are estimated <i>V</i>. <i>vinifera</i> ancestry proportions based on PC1.</p

Differences in partial sequence of chloroplast DNA between cultivars with <i>V</i>. <i>vinifera</i> type chloroplasts and cultivars with non-<i>V</i>. <i>vinifera</i> type chloroplasts.

<p>Nucleotide positions of Maxxa (NC_007957.1) are shown as the reference.</p

Principal coordinates analysis plot based on partial chloroplast sequences from diverse <i>Vitis</i> accessions.

<p>The proportion of variation explained by each PC is found in parantheses along each axis.</p

Additional file 6 of LinkImputeR: user-guided genotype calling and imputation for non-model organisms

Full cannabis results. (DOCX 13 kb

Multiple species alignment of the death domain of EDAR.

<p>The derived 370A allele is shaded in dark grey. Variable positions in the alignment are shaded in light grey. Sites at which nonsynonymous substitutions are known to result in hypohidrotic ectodermal dysplasia are boxed in. An asterisk indicates the site at which an Arg-His substitution causes decreased activation of NF-κB <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0002209#pone.0002209-Shimomura1" target="_blank">[28]</a>.</p

The derived 370A allele of EDAR results in enhanced activation of NF-κB in vitro.

<p>Transfection of a HEK293 cell line with ancestral (370V) and derived (370A) versions of EDAR activate NF-κB to a different extent on both a normal genetic background (comparison on the left) and on a background containing a known disease mutation (375H; comparison on the right). Luciferase activity is driven by EDAR-activated NF-κB. Data was averaged from two independent experiments with reads from at least 9 wells total for each clone. Error bars are standard error of the mean (* p<0.05, ** p<0.01).</p

Density plot of the posterior distribution of estimates of the time since fixation of the 370A allele.

<p>The estimated time since fixation of the 370A allele is the mode of the distribution (10740 years before present). The 2.5% and 97.5% confidence intervals are boxed in and are indicated by the hashed vertical lines.</p

Worldwide allele frequencies and population differentiation for V370A.

<p>The vertical bar chart displays the frequency of the 370A allele in each of the populations represented in the CEPH-HGDP panel with sample sizes (number of individuals) on the left. The shaded boxes in the 53×53 and 7×7 matrices show which pairwise Fst values are significant compared to the empirical distribution at three <i>P</i> value thresholds (see the boxed-in <i>P</i> value legend).</p