Space Of Probability Distributions Research Articles

Minimal length effect on the broadening of free wave packets and its physical implications

We study the Generalized Uncertainty Principle (GUP) modified time evolution for the width of wave-packets for a scalar potential. Free particle case is solved exactly where the wave-packet broadening is modified by a coupling between the GUP parameter and higher order moments in the probability distribution in momentum space. We consider two popular forms of deformations widely used in the literature - one of which modifies the commutator with a quadratic term in momentum, while the other modifies it with terms both linear and quadratic in momentum. Unlike the standard case, satisfying Heisenberg uncertainty, here the GUP modified broadening rates, for both deformations, not only depend on the initial size (both in position and momentum space) of the wave-packet, but also on the initial probability distribution and momentum of the particle. The new rates of wave-packet broadening, for both situations, are modified by a handful of new terms - such as the skewness and kurtosis coefficients, as well as the (constant) momentum of the particle. Comparisons with the standard Heisenberg Uncertainty Principle (HUP)-based results show potentially measurable differences in the rates of free wave-packet broadening for physical systems such as the $C_{60}$ and $C_{176}$ molecules, and more so for large organic molecular wave-packets. In doing so, we open a path to scan the GUP parameter space by several orders of magnitude inside the best existing upper bounds for both forms of GUP.

One-dimensional solute transport in open channel flow from a stochastic systematic perspective

Solute transport by river and stream flows in natural environment has significant implication on water quality and the transport process is full of uncertainties. In this study, a stochastic one-dimensional solute transport model under uncertain open-channel flow conditions is developed. The proposed solute transport model is developed by upscaling the stochastic partial differential equations through their one-to-one correspondence to the nonlocal Lagrangian–Eulerian Fokker–Planck equations. The resulting Fokker–Planck equation is a linear and deterministic differential equation, and this equation can provide a comprehensive probabilistic description of the spatiotemporal evolutionary probability distribution of the underlying solute transport process by one single numerical realization, rather than requiring thousands of simulations in the Monte Carlo simulation. Consequently, the ensemble behavior of the solute transport process can also be obtained based on the probability distribution. To illustrate the capabilities of the proposed stochastic solute transport model, various steady and unsteady uncertain flow conditions are applied. The Monte Carlo simulation with stochastic Saint–Venant flow and solute transport model is used to provide the stochastic flow field for the solute transport process, and further to validate the numerical solute transport results provided by the derived Fokker–Planck equations. The comparison of the numerical results by the Monte Carlo simulation and the Fokker–Planck equation approach indicated that the proposed model can adequately characterize the ensemble behavior of the solute transport process under uncertain flow conditions via the evolutionary probability distribution in space and time of the transport process.

Selective Inference for Testing Trees and Edges in Phylogenetics

Selective inference is considered for testing trees and edges in phylogenetic tree selection from molecular sequences. This improves the previously proposed approximately unbiased test by adjusting the selection bias when testing many trees and edges at the same time. The newly proposed selective inference $p$-value is useful for testing selected edges to claim that they are significantly supported if $p>1-\alpha$, whereas the non-selective $p$-value is still useful for testing candidate trees to claim that they are rejected if $p<\alpha$. The selective $p$-value controls the type-I error conditioned on the selection event, whereas the non-selective $p$-value controls it unconditionally. The selective and non-selective approximately unbiased $p$-values are computed from two geometric quantities called signed distance and mean curvature of the region representing tree or edge of interest in the space of probability distributions. These two geometric quantities are estimated by fitting a model of scaling-law to the non-parametric multiscale bootstrap probabilities. Our general method is applicable to a wider class of problems; phylogenetic tree selection is an example of model selection, and it is interpreted as the variable selection of multiple regression, where each edge corresponds to each predictor. Our method is illustrated in a previously controversial phylogenetic analysis of human, rabbit and mouse.

Bounded Rational Decision-Making from Elementary Computations That Reduce Uncertainty.

In its most basic form, decision-making can be viewed as a computational process that progressively eliminates alternatives, thereby reducing uncertainty. Such processes are generally costly, meaning that the amount of uncertainty that can be reduced is limited by the amount of available computational resources. Here, we introduce the notion of elementary computation based on a fundamental principle for probability transfers that reduce uncertainty. Elementary computations can be considered as the inverse of Pigou–Dalton transfers applied to probability distributions, closely related to the concepts of majorization, T-transforms, and generalized entropies that induce a preorder on the space of probability distributions. Consequently, we can define resource cost functions that are order-preserving and therefore monotonic with respect to the uncertainty reduction. This leads to a comprehensive notion of decision-making processes with limited resources. Along the way, we prove several new results on majorization theory, as well as on entropy and divergence measures.

Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis

The Pythagorean fuzzy set (PFS) which is an extension of intuitionistic fuzzy set, is more capable of expressing and handling the uncertainty under uncertain environments, so that it was broadly applied in a variety of fields. Whereas, how to measure PFSs’ distance appropriately is still an open issue. It is well known that the square root of Jensen–Shannon divergence is a true metric in the probability distribution space which is a useful measure of distance. On account of this point, a novel divergence measure between PFSs is proposed by taking advantage of the Jensen–Shannon divergence in this paper, called as PFSJS distance. This is the first work to consider the divergence of PFSs for measuring the discrepancy of data from the perspective of the relative entropy. The new PFSJS distance measure has some desirable merits, in which it meets the distance measurement axiom and can better indicate the discrimination degree of PFSs. Then, numerical examples demonstrate that the PFSJS distance can avoid generating counter-intuitive results which is more feasible, reasonable and superior than existing distance measures. Additionally, a new algorithm based on the PFSJS distance measure is designed to solve the problems of medical diagnosis. By comparing the different methods in the medical diagnosis application, it is found that the new algorithm is as efficient as the other methods. These results prove that the proposed method is practical in dealing with the medical diagnosis problems.

Bounds for the Wasserstein mean with applications to the Lie-Trotter mean

Since barycenters in the Wasserstein space of probability distributions have been introduced, the Wasserstein metric and the Wasserstein mean of positive definite Hermitian matrices have been recently developed. In this paper, we explore some properties of Wasserstein mean of positive definite matrices and find bounds for the Wasserstein mean with respect to operator norm and Loewner order. Using bounds for the Wasserstein mean, we verify that the Wasserstein mean is the multivariate Lie-Trotter mean.

A differential game on Wasserstein space. Application to weak approachability with partial monitoring

Studying continuous time counterpart of some discrete time dynamics is now a standard and fruitful technique, as some properties hold in both setups. In game theory, this is usually done by considering differential games on Euclidean spaces. This allows to infer properties on the convergence of values of a repeated game, to deal with the various concepts of approachability, etc. In this paper, we introduce a specific but quite abstract differential game defined on the Wasserstein space of probability distributions and we prove the existence of its value. Going back to the discrete time dynamics, we derive results on weak approachability with partial monitoring: we prove that any set satisfying a suitable compatibility condition is either weakly approachable or weakly excludable. We also obtain that the value for differential games with nonanticipative strategies is the same that those defined with a new concept of strategies very suitable to make links with repeated games.

Distributed classification based on distances between probability distributions in feature space

We consider a distributed framework where training and test samples drawn from the same distribution are available, with the training instances spread across disjoint nodes. In this setting, a novel learning algorithm based on combining with different weights the outputs of classifiers trained at each node is proposed. The weights depend on the distributional distance between each node and the test set in the feature space. Two different weighting approaches are introduced, which are referred to as per-Node Weighting (pNW) and per-Instance Weighting (pIW). While pNW assigns the same weight to all test instances at each node, pIW allows distinct weights for test instances differently represented at the node. By construction, our approach is particularly useful to deal with unbalanced nodes. Our methods require no communication between nodes, allowing for data privacy, independence of the kind of trained classifier at each node and maximum training speedup. In fact, our methods do not require retraining of the node’s classifiers if available. Although a range of different combination rules are considered to ensemble the single classifiers, theoretical support for the optimality of using the sum rule is provided. Our experiments illustrate all of these properties and show that pIW produces the highest classification accuracies compared with pNW and the standard unweighted approaches.

Covariant information theory and emergent gravity

Informational dependence between statistical or quantum subsystems can be described with Fisher information matrix or Fubini-Study metric obtained from variations/shifts of the sample/configuration space coordinates. Using these (noncovariant) objects as macroscopic constraints, we consider statistical ensembles over the space of classical probability distributions (i.e. in statistical space) or quantum wave functions (i.e. in Hilbert space). The ensembles are covariantized using dual field theories with either complex scalar field (identified with complex wave functions) or real scalar field (identified with square roots of probabilities). We construct space–time ensembles for which an approximate Schrodinger dynamics is satisfied by the dual field (which we call infoton due to its informational origin) and argue that a full space–time covariance on the field theory side is dual to local computations on the information theory side. We define a fully covariant information-computation tensor and show that it must satisfy certain conservation equations. Then we switch to a thermodynamic description of the quantum/statistical systems and argue that the (inverse of) space–time metric tensor is a conjugate thermodynamic variable to the ensemble-averaged information-computation tensor. In (local) equilibrium, the entropy production vanishes, and the metric is not dynamical, but away from the equilibrium the entropy production gives rise to an emergent dynamics of the metric. This dynamics can be described approximately by expanding the entropy production into products of generalized forces (derivatives of metric) and conjugate fluxes. Near equilibrium, these fluxes are given by an Onsager tensor contracted with generalized forces and on the grounds of time-reversal symmetry, the Onsager tensor is expected to be symmetric. We show that a particularly simple and highly symmetric form of the Onsager tensor gives rise to the Einstein–Hilbert term. This proves that general relativity is equivalent to a theory of nonequilibrium (thermo)dynamics of the metric, but the theory is expected to break down far away from equilibrium where the symmetries of the Onsager tensor are to be broken.

Uncertainty quantification in the nuclear shell model

The uncertainty quantifications of theoretical results are of great importance to make meaningful comparisons of those results with experimental data and to make predictions in experimentally unknown regions. By quantifying uncertainties, one can make more solid statements about, e.g., origins of discrepancy in some quantities between theory and experiment. We propose a novel method for uncertainty quantification for the effective interactions of nuclear shell-model calculations as an example. The effective interaction is specified by a set of parameters, and its probability distribution in the multi-dimensional parameter space is considered. This enables us to quantify the agreement with experimental data in a statistical manner and the resulting confidence intervals show unexpectedly large variations. Moreover, we point out that a large deviation of the confidence interval for the energy in shell-model calculations from the corresponding experimental data can be used as an indicator of some exotic property, e.g. alpha clustering, etc. Other possible applications and impacts are also discussed.

Dynamical optimal transport on discrete surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows.

Spatiotemporal dynamics of prehistoric human population growth: Radiocarbon ‘dates as data’ and population ecology models

Archaeologists now routinely use summed radiocarbon dates as a measure of past population size, yet few have coupled these measures to theoretical expectations about social organization. To help move the ‘dates as data’ approach from description to explanation, this paper proposes a new integrative theory and method for quantitative analyses of radiocarbon summed probability distributions (SPDs) in space. We present this new approach to ‘SPDs in space’ with a case study of 3571 geo-referenced radiocarbon dates from Wyoming, USA. We develop a SPD for the Holocene in Wyoming, then analyze the spatial distribution of the SPD as a function of time using a standard nearest-neighbor statistic. We compare population growth and decline throughout the Holocene with expectations for different Ideal Distribution Models from population ecology that predict the relationship between habitat quality and population density. Results suggest that populations in Wyoming were initially clustered and then became increasingly dispersed through the course of the Holocene. These results suggest that Allee-like benefits to aggregation, rather than ideal free-driven dispersion patterns, explain settlement decisions in response to growing populations. Our approach is a first step in constructing a method and theory for describing relationships between social organization and population growth trends derived from archaeological radiocarbon time-series.

Model aggregation using optimal transport and applications in wind speed forecasting

AbstractIn several environmental affected socio‐economic activities, including renewable energy site assessment, search and rescue operations, and local microclimate modeling, the need of very local wind speed prediction is critical and not completely covered by the use of numerical weather prediction models. In meteorology and, particularly, in wind speed prediction, the spatial location of the prediction does not coincide with the spatial locations where numerical models provide estimates of the relevant quantity (which are typically grid points used for the numerical resolution of the wind transport equations). Hence, the important problem of constructing a predictive model for the wind speed at the required location using a combination of actual measurements and model predictions arises. This problem is far from trivial on account of the fact that measurements and predictions do not refer to the same quantity for the reason that typical grid points for the numerical scheme that provide model predictions and the location of the meteorological stations that provide measurements do not coincide. In this work, a new approach is proposed based on optimal transportation theory for the aggregation of model predictions and measurements for the construction of an optimal predictor for wind speed at the location of interest. Our model provides a linear predictive model in the space of probability distributions of the predictors (Wasserstein space), which is then mapped into observation space using a generalized quantile regression technique. Importantly, the proposed scheme allows also for the construction of zone monitoring the extremes, which when applied to real data, provides superior results with respect to other existing methods.

Unsupervised Domain Adaptation for Face Anti-Spoofing

Face anti-spoofing (a.k.a. presentation attack detection) has recently emerged as an active topic with great significance for both academia and industry due to the rapidly increasing demand in user authentication on mobile phones, PCs, tablets, and so on. Recently, numerous face spoofing detection schemes have been proposed based on the assumption that training and testing samples are in the same domain in terms of the feature space and marginal probability distribution. However, due to unlimited variations of the dominant conditions (illumination, facial appearance, camera quality, and so on) in face acquisition, such single domain methods lack generalization capability, which further prevents them from being applied in practical applications. In light of this, we introduce an unsupervised domain adaptation face anti-spoofing scheme to address the real-world scenario that learns the classifier for the target domain based on training samples in a different source domain. In particular, an embedding function is first imposed based on source and target domain data, which maps the data to a new space where the distribution similarity can be measured. Subsequently, the Maximum Mean Discrepancy between the latent features in source and target domains is minimized such that a more generalized classifier can be learned. State-of-the-art representations including both hand-crafted and deep neural network learned features are further adopted into the framework to quest the capability of them in domain adaptation. Moreover, we introduce a new database for face spoofing detection, which contains more than 4000 face samples with a large variety of spoofing types, capture devices, illuminations, and so on. Extensive experiments on existing benchmark databases and the new database verify that the proposed approach can gain significantly better generalization capability in cross-domain scenarios by providing consistently better anti-spoofing performance.

Simplicial Palatini action

We consider the piecewise flat spacetime and a simplicial analog of the Palatini form of the general relativity (GR) action where the discrete Christoffel symbols are given on the tetrahedra as variables that are independent of the metric. Excluding these variables with the help of the equations of motion gives exactly the Regge action. This paper continues our previous work. Now, we include the parity violation term and the analog of the Barbero–Immirzi parameter introduced in the orthogonal connection form of GR. We consider the path integral and the functional integration over the connection. The result of the latter (for certain limiting cases of some parameters) is compared with the earlier found result of the functional integration over the connection for the analogous orthogonal connection representation of Regge action. These results, mainly as some measures on the lengths/areas, are discussed for the possibility of the diagram technique where the perturbative diagrams for the Regge action calculated using the measure obtained are finite. This finiteness is due to these measures providing elementary lengths being mostly bounded and separated from zero, just as the finiteness of a theory on a lattice with an analogous probability distribution of spacings.

The Visualization of the Space Probability Distribution for a Particle Moving in a Double Ring-Shaped Coulomb Potential

The analytical solutions to a double ring-shaped Coulomb potential (RSCP) are presented. The visualizations of the space probability distribution (SPD) are illustrated for the two- (contour) and three-dimensional (isosurface) cases. The quantum numbers (n,l,m) are mainly relevant for those quasi-quantum numbers (n′,l′,m′) via the double RSCP parameter c. The SPDs are of circular ring shape in spherical coordinates. The properties for the relative probability values (RPVs) P are also discussed. For example, when we consider the special case (n,l,m)=(6,5,0), the SPD moves towards two poles of z-axis when P increases. Finally, we discuss the different cases for the potential parameter b, which is taken as negative and positive values for c&gt;0. Compared with the particular case b=0, the SPDs are shrunk for b=-0.5, while they are spread out for b=0.5.

Theory and practice of electron diffraction from single atoms and extended objects using an EMPAD.

What does the diffraction pattern from a single atom look like? How does it differ from the scattering from long-range potential? With the development of new high-dynamic range pixel array detectors to measure the complete momentum distribution, these questions have immediate relevance for designing and understanding momentum-resolved imaging modes. We explore the asymptotic limits of long-range and short-range potentials. We use a simple quantum mechanical model to explain the general and asymptotic limits for the probability distribution in both real and reciprocal space. Features in the scattering potential much larger than the probe size cause the bright field (BF) disk to deflect uniformly, while features much smaller than the probe size, instead of a deflection, cause a redistribution of intensity within the BF disk. Because long-range and short-range features are encoded differently in the diffraction pattern, it is possible to separate their contributions in differential phase-contrast (DPC) or center-of-mass (CoM) imaging. The shape profiles for atomic resolution CoM imaging are dominated by the shape of the probe gradient and not the highly singular atomic potentials or their local fields. Instead, only the peak height shows an atomic number sensitivity, whose precise dependence is determined by the convergence angle. At lower convergence angles, the contrast oscillates with increasing atomic number, similar to BF imaging. The range of collection angles impacts DPC and CoM imaging differently, with CoM being more sensitive to the upper cutoff limit, while DPC is more sensitive to the lower cutoff.

Embedding optimal transports in statistical manifolds

We consider Monge-Kantorovich optimal transport problems on ℝ d , d ≥ 1, with a convex cost function given by the cumulant generating function of a probability measure. Examples include theWasserstein-2 transport whose cost function is the square of the Euclidean distance and corresponds to the cumulant generating function of the multivariate standard normal distribution. The optimal coupling is usually described via an extended notion of convex/concave functions and their gradient maps. These extended notions are nonintuitive and do not satisfy useful inequalities such as Jensen’s inequality. Under mild regularity conditions, we show that all such extended gradient maps can be recovered as the usual supergradients of a nonnegative concave function on the space of probability distributions. This embedding provides a universal geometry for all such optimal transports and an unexpected connection with information geometry of exponential families of distributions.

Geometrical structures for classical and quantum probability spaces

On the affine space containing the space [Formula: see text] of quantum states of finite-dimensional systems, there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding the relevant geometrical properties of [Formula: see text]. Guided by Dirac’s analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyze their geometrical properties. Most of our considerations will be dealt with for the simple example of a three-level system.

Chempy: A flexible chemical evolution model for abundance fitting

Elemental abundances of stars are the result of the complex enrichment history of their galaxy. Interpretation of observed abundances requires flexible modeling tools to explore and quantify the information about Galactic chemical evolution (GCE) stored in such data. Here we present Chempy, a newly developed code for GCE modeling, representing a parametrized open one-zone model within a Bayesian framework. A Chempy model is specified by a set of 5-10 parameters that describe the effective galaxy evolution along with the stellar and star-formation physics: e.g. the star-formation history, the feedback efficiency, the stellar initial mass function (IMF) and the incidence of supernova type Ia (SN Ia). Unlike established approaches, Chempy can sample the posterior probability distribution in the full model parameter space and test data-model matches for different nucleosynthetic yield sets. We extend Chempy to a multi-zone scheme. As an illustrative application, we show that interesting parameter constraints result from only the ages and elemental abundances of Sun, Arcturus and the present-day interstellar medium (ISM). For the first time, we use such information to infer IMF parameter via GCE modeling, where we properly marginalize over nuisance parameters and account for different yield sets. We find that of the IMF $11.6_{-1.6}^{+2.1}$ % explodes as core-collapse SN, compatible with Salpeter 1955. We also constrain the incidence of SN Ia per 10^3 Msun to 0.5-1.4. At the same time, this Chempy application shows persistent discrepancies between predicted and observed abundances for some elements, irrespective of the chosen yield set. These cannot be remedied by any variations of Chempy's parameters and could be an indication for missing nucleosynthetic channels. Chempy should be a powerful tool to confront predictions from stellar nucleosynthesis with far more complex abundance data sets.