3,756 research outputs found
Convex Hulls under Uncertainty
We study the convex-hull problem in a probabilistic setting, motivated by the
need to handle data uncertainty inherent in many applications, including sensor
databases, location-based services and computer vision. In our framework, the
uncertainty of each input site is described by a probability distribution over
a finite number of possible locations including a \emph{null} location to
account for non-existence of the point. Our results include both exact and
approximation algorithms for computing the probability of a query point lying
inside the convex hull of the input, time-space tradeoffs for the membership
queries, a connection between Tukey depth and membership queries, as well as a
new notion of \some-hull that may be a useful representation of uncertain
hulls
Polyhedral Predictive Regions For Power System Applications
Despite substantial improvement in the development of forecasting approaches,
conditional and dynamic uncertainty estimates ought to be accommodated in
decision-making in power system operation and market, in order to yield either
cost-optimal decisions in expectation, or decision with probabilistic
guarantees. The representation of uncertainty serves as an interface between
forecasting and decision-making problems, with different approaches handling
various objects and their parameterization as input. Following substantial
developments based on scenario-based stochastic methods, robust and
chance-constrained optimization approaches have gained increasing attention.
These often rely on polyhedra as a representation of the convex envelope of
uncertainty. In the work, we aim to bridge the gap between the probabilistic
forecasting literature and such optimization approaches by generating forecasts
in the form of polyhedra with probabilistic guarantees. For that, we see
polyhedra as parameterized objects under alternative definitions (under
and norms), the parameters of which may be modelled and predicted.
We additionally discuss assessing the predictive skill of such multivariate
probabilistic forecasts. An application and related empirical investigation
results allow us to verify probabilistic calibration and predictive skills of
our polyhedra.Comment: 8 page
Graphics for relatedness research
Studies of relatedness have been crucial in molecular ecology over the last decades. Good evidence of this is the fact that studies of population structure, evolution of social behaviours, genetic diversity and quantitative genetics all involve relatedness research. The main aim of this article is to review the most
common graphical methods used in allele sharing studies for detecting and identifying family relationships. Both IBS and IBD based allele sharing studies are considered. Furthermore, we propose two additional graphical methods from the field of compositional data analysis: the ternary diagram and scatterplots of isometric log-ratios of IBS and IBD probabilities. We illustrate all graphical tools with genetic data from the HGDP-CEPH diversity panel, using mainly 377 microsatellites genotyped for 25 individuals from the Maya population of this panel. We enhance all graphics with convex hulls obtained by simulation and use these to confirm the documented relationships. The proposed compositional graphics are shown to be useful in relatedness research, as they also single out the most prominent related pairs. The ternary diagram is advocated for its ability to display all three allele sharing probabilities simultaneously. The log-ratio plots are advocated as an attempt to overcome the problems with the Euclidean distance interpretation in the
classical graphics.Peer ReviewedPostprint (published version
Believing Probabilistic Contents: On the Expressive Power and Coherence of Sets of Sets of Probabilities
Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions but as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Moss’ account of probabilistic knowledge or her semantics for epistemic modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower
probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rational agents have consistent probabilistic beliefs? We show that an important subclass of Mossean believers avoid Dutch
bookability iff they have consistent probabilistic beliefs
The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
This article investigates matrix convex sets and introduces their tracial
analogs which we call contractively tracial convex sets. In both contexts
completely positive (cp) maps play a central role: unital cp maps in the case
of matrix convex sets and trace preserving cp (CPTP) maps in the case of
contractively tracial convex sets. CPTP maps, also known as quantum channels,
are fundamental objects in quantum information theory.
Free convexity is intimately connected with Linear Matrix Inequalities (LMIs)
L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets
{ X : L(X) is positive semidefinite }, called free spectrahedra. The
Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states
that matrix convex sets are solution sets of LMIs with operator coefficients.
Motivated in part by cp interpolation problems, we develop the foundations of
convex analysis and duality in the tracial setting, including tracial analogs
of the Effros-Winkler Theorem.
The projection of a free spectrahedron in g+h variables to g variables is a
matrix convex set called a free spectrahedrop. As a class, free spectrahedrops
are more general than free spectrahedra, but at the same time more tractable
than general matrix convex sets. Moreover, many matrix convex sets can be
approximated from above by free spectrahedrops. Here a number of fundamental
results for spectrahedrops and their polar duals are established. For example,
the free polar dual of a free spectrahedrop is again a free spectrahedrop. We
also give a Positivstellensatz for free polynomials that are positive on a free
spectrahedrop.Comment: v2: 56 pages, reworked abstract and intro to emphasize the convex
duality aspects; v1: 60 pages; includes an index and table of content
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