Probabilistic Version Research Articles

Can (should) theories of crowding be unified?

Objects in clutter are difficult to recognize, a phenomenon known as crowding. There is little consensus on the underlying mechanisms of crowding, and a large number of models have been proposed. There have also been attempts at unifying the explanations of crowding under a single model, such as the weighted feature model of Harrison and Bex (2015) and the texture synthesis model of Rosenholtz and colleagues (Balas, Nakano, & Rosenholtz, 2009; Keshvari & Rosenholtz, 2016). The goal of this work was to test various models of crowding and to assess whether a unifying account can be developed. Adopting Harrison and Bex's (2015) experimental paradigm, we asked observers to report the orientation of two concentric C-stimuli. Contrary to the predictions of their model, observers' recognition accuracy was worse for the inner C-stimulus. In addition, we demonstrated that the stimulus paradigm used by Harrison and Bex has a crucial confounding factor, eccentricity, which limits its usage to a very narrow range of stimulus parameters. Nevertheless, reporting the orientations of both C-stimuli in this paradigm proved very useful in pitting different crowding models against each other. Specifically, we tested deterministic and probabilistic versions of averaging, substitution, and attentional resolution models as well as the texture synthesis model. None of the models alone was able to explain the entire set of data. Based on these findings, we discuss whether the explanations of crowding can (should) be unified.

The probabilistic minimum dominating set problem

We present a natural wireless sensor network problem, which we model as a probabilistic version of the min dominating set problem (called probabilistic min dominating set). We first show that calculation of the objective function of this general probabilistic problem is #P-complete. We then introduce a restricted version of probabilistic min dominating set and show that, this time, calculation of its objective function can be performed in polynomial time and that this restricted problem is “just” NP-hard, since it is a generalization of the classical min dominating set. We study the complexity of this restricted version in graphs where min dominating set is polynomial, mainly in trees and paths and then we give some approximation results for it.

Managing risk-adjusted resource allocation for project time-cost tradeoffs

In the traditional project crashing problem (also known as the time-cost tradeoff problem) associated with a project network, the decision variables are assumed to be continuous, representing the amount of reduction in the duration of the tasks in the project, and the objective is to find the least expensive set of time reductions (“crashing” times) to complete the project within a given time limit (to avoid penalties). We will study a variant of this problem with a combinatorial probabilistic version, where the objective is to find the set of “crashing” choices that minimizes the time reduction cost and the expected penalty risk from tardiness. We show this problem is NP-hard, so we propose efficient heuristics that can provide approximate solutions to this problem. We also extend this problem so that resource assignments can be adjusted following project status reviews several times before reaching the project deadline.

A Probabilistic Reformulation of No Free Lunch: Continuous Lunches Are Not Free.

No Free Lunch (NFL) theorems have been developed in many settings over the last two decades. Whereas NFL is known to be possible in any domain based on set-theoretic concepts, probabilistic versions of NFL are presently believed to be impossible in continuous domains. This article develops a new formalization of probabilistic NFL that is sufficiently expressive to prove the existence of NFL in large search domains, such as continuous spaces or function spaces. This formulation is arguably more complicated than its set-theoretic variants, mostly as a result of the numerous technical complications within probability theory itself. However, a probabilistic conceptualization of NFL is important because stochastic optimization methods inherently need to be evaluated probabilistically. Thus the present study fills an important gap in the study of performance of stochastic optimizers.

Piecewise Regression Mixture for Simultaneous Functional Data Clustering and Optimal Segmentation

This paper introduces a novel mixture model-based approach to the simultaneous clustering and optimal segmentation of functional data, which are curves presenting regime changes. The proposed model consists of a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associated with a cluster, and within each cluster, each piecewise polynomial component is associated with a regime (i.e., a segment). We derive two approaches to learning the model parameters: the first is an estimation approach which maximizes the observed-data likelihood via a dedicated expectation-maximization (EM) algorithm, then yielding a fuzzy partition of the curves into K clusters obtained at convergence by maximizing the posterior cluster probabilities. The second is a classification approach and optimizes a specific classification likelihood criterion through a dedicated classification expectation-maximization (CEM) algorithm. The optimal curve segmentation is performed by using dynamic programming. In the classification approach, both the curve clustering and the optimal segmentation are performed simultaneously as the CEM learning proceeds. We show that the classification approach is a probabilistic version generalizing the deterministic K-means-like algorithm proposed in Hebrail, Hugueney, Lechevallier, and Rossi (2010). The proposed approach is evaluated using simulated curves and real-world curves. Comparisons with alternatives including regression mixture models and the K-means-like algorithm for piecewise regression demonstrate the effectiveness of the proposed approach.

Maximum entropy discrimination factor analyzers

Devising generative models that allow for inferring low dimensional latent feature representations of high-dimensional observations is a significant problem in statistical machine learning. Factor analysis (FA) is a well-established linear latent variable scheme addressing this problem by modeling the covariances between the elements of multivariate observations under a set of linear assumptions. FA is closely related to principal components analysis (PCA), and might be considered as a generalization of both PCA and its probabilistic version, PPCA. Recently, the invention of Gaussian process latent variable models (GP-LVMs) has given rise to a whole new family of latent variable modeling schemes that generalize FA under a nonparametric Bayesian inference framework. In this work, we examine generalization of FA models under a different Bayesian inference perspective. Specifically, we propose a large-margin formulation of FA under the maximum entropy discrimination (MED) framework. The MED framework integrates the large-margin principle with Bayesian posterior inference in an elegant and computationally efficient fashion, allowing to leverage existing high-performance solvers for convex optimization problems. We devise efficient mean-field inference algorithms for our model, and exhibit its advantages by evaluating it in a number of diverse application scenarios, dealing with high-dimensional data classification and reconstruction.

Szemerédi's Regularity Lemma via Martingales

We prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the model, we design a two-step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the Stiefel manifold, i.e., the manifold of matrices with orthogonal columns. The additional benefit of our probabilistic formulation, is that it allows us to select the dimensionality of the AS via the Bayesian information criterion. We validate our approach by showing that it can discover the right AS in synthetic examples without gradient information using both noiseless and noisy observations. We demonstrate that our method is able to discover the same AS as the classical approach in a challenging one-hundred-dimensional problem involving an elliptic stochastic partial differential equation with random conductivity. Finally, we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system.

The Eigenvector Moment Flow and Local Quantum Unique Ergodicity

We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.

Isosurface Visualization of Data with Nonparametric Models for Uncertainty.

The problem of isosurface extraction in uncertain data is an important research problem and may be approached in two ways. One can extract statistics (e.g., mean) from uncertain data points and visualize the extracted field. Alternatively, data uncertainty, characterized by probability distributions, can be propagated through the isosurface extraction process. We analyze the impact of data uncertainty on topology and geometry extraction algorithms. A novel, edge-crossing probability based approach is proposed to predict underlying isosurface topology for uncertain data. We derive a probabilistic version of the midpoint decider that resolves ambiguities that arise in identifying topological configurations. Moreover, the probability density function characterizing positional uncertainty in isosurfaces is derived analytically for a broad class of nonparametric distributions. This analytic characterization can be used for efficient closed-form computation of the expected value and variation in geometry. Our experiments show the computational advantages of our analytic approach over Monte-Carlo sampling for characterizing positional uncertainty. We also show the advantage of modeling underlying error densities in a nonparametric statistical framework as opposed to a parametric statistical framework through our experiments on ensemble datasets and uncertain scalar fields.

Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation

That superpositions of states can be useful for performing tasks in quantum systems has been known since the early days of quantum information, but only recently has quantitative theory of quantum coherence been proposed. Here we apply that theory to an analysis of the Deutsch-Jozsa algorithm, which depends on quantum coherence for its operation. The Deutsch-Jozsa algorithm solves a decision problem, and we focus on a probabilistic version of that problem, comparing probability of being correct for both classical and quantum procedures. In addition, we study a related decision problem in which the quantum procedure has one-sided error while the classical procedure has two-sided error. The role of coherence on the quantum success probabilities in both of these problems is examined.

Deterministic and probabilistic liquefaction potential evaluation of Guwahati city

Northeastern region of India is one of the six most seismically active regions of the world. Guwahati, the major city in the region is growing rapidly in every aspect. The 1897 great Assam earthquake and the 1950 Assam earthquake both of magnitude 8.7 are two of the many great earthquakes that had occurred in this region. There have been reports of extensive liquefaction occurring during these two earthquakes. These instances of liquefaction necessitate the need to evaluate liquefaction potential of the area. In the present study, an attempt has been made to predict the liquefaction susceptibility of Guwahati city based on corrected SPT values using deterministic as well as probabilistic performance based approach. The deterministic and probabilistic version of Toprak et al., Youd et.al, Idriss and Boulanger and Cetin, standard penetration test (SPT)- based liquefaction triggering correlations have been used. The results obtained in the probabilistic approach are then compared with their deterministic approach. Standard Penetration Test (SPT) N values, engineering properties of the soils and depth of water table were taken from a data base of 200 boreholes upto 30 meter depth covering an area of 262 km2 in Guwahati city. Liquefaction susceptibility from the methods is presented as maps showing zones of levels of risk of liquefaction. Comparison of the three methods in the deterministic approach have shown that there is a difference in the factors of safety in the same depth and comparison in the probabilistic approach have also shown a difference in the values of probabilities in the same depth. The SPT- based liquefaction evaluation procedures have been found to yield significantly different predictions. This study feels that the issues and uncertainties regarding the use of both the deterministic and probabilistic liquefaction triggering relationships in practice must be resolved so as to arrive at a common consensus to determine the same.

A probabilistic version of the game of Zombies and Survivors on graphs

We consider a new probabilistic graph searching game played on graphs, inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighboring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighboring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is at least 1/2. We present asymptotic results for the zombie numbers of several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and Cartesian and toroidal grids.

Korean n-insertion: a mismatch between data and learning

This paper explores the variable application of n-insertion in Korean. Several tendencies emerge from the distribution of n-insertion in existing Korean words, using data drawn from a dictionary and from surveys and experimental studies. Most, but not all, of these tendencies are mirrored in the results of experiments involving novel words, suggesting that Korean speakers are aware of the differential influence of certain phonological factors on the probability of the application of n-insertion. The observed patterns of variation are analysed within the framework of a probabilistic version of Optimality Theory. In addition, the paper shows that the main aspects of n-insertion are motivated by the requirement of perceptually minimal modification, employing an analysis based on Steriade's P-map theory. The observed data vs. learning mismatch is also attributed to the lack of a perceptual basis for the tendency in question.

Stochastic parameterizations of biogeochemical uncertainties in a 1/4° NEMO/PISCES model for probabilistic comparisons with ocean color data

In spite of recent advances, biogeochemical models are still unable to represent the full complexity of natural ecosystems. Their formulations are mainly based on empirical laws involving many parameters. Improving biogeochemical models therefore requires to properly characterize model uncertainties and their consequences. Subsequently, this paper investigates the potential of using random processes to simulate some uncertainties of the 1/4° coupled Physical–Biogeochemical NEMO/PISCES model of the North Atlantic ocean.Starting from a deterministic simulation performed with the original PISCES formulation, we propose a generic method based on AR(1) random processes to generate perturbations with temporal and spatial correlations. These perturbations are introduced into the model formulations to simulate 2 classes of uncertainties: the uncertainties on biogeochemical parameters and the uncertainties induced by unresolved scales in the presence of non-linear processes. Using these stochastic parameterizations, a probabilistic version of PISCES is designed and a 60-member ensemble simulation is performed.With respect to the simulation of chlorophyll, the relevance of the probabilistic configuration and the impacts of these stochastic parameterizations are assessed. In particular, it is shown that the ensemble simulation is in good agreement with the SeaWIFS ocean color data. Using these observations, the statistical consistency (reliability) of the ensemble is evaluated with rank histograms. Finally, the benefits expected from the probabilistic description of uncertainties (model error) are discussed in the context of future ocean color data assimilation.

Probabilistic Type Theory and Natural Language Semantics

Type theory has played an important role in specifying the formal connection between syntactic structure and semantic interpretation within the history of formal semantics. In recent years rich type theories developed for the semantics of programming languages have become influential in the semantics of natural language. The use of probabilistic reasoning to model human learning and cognition has become an increasingly important part of cognitive science. In this paper we offer a probabilistic formulation of a rich type theory, Type Theory with Records (TTR), and we illustrate how this framework can be used to approach the problem of semantic learning. Our probabilistic version of TTR is intended to provide an interface between the cognitive process of classifying situations according to the types that they instantiate, and the compositional semantics of natural language.

Transitivity in coherence-based probability logic

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study selected probabilistic versions of classical categorical syllogisms and construct a new version of the square of opposition in terms of defaults and negated defaults.

Role of Information in Decision Making of Social Agents

The influence of additional information on the decision making of agents, who are the interacting members of a society, is analyzed within the mathematical framework based on the use of quantum probabilities. The introduction of social interactions, which influence the decisions of individual agents, leads to a generalization of the quantum decision theory (QDT) developed earlier by the authors for separate individuals. The generalized approach is free of the standard paradoxes of classical decision theory. This approach also explains the error-attenuation effects observed for the paradoxes occurring when decision makers, who are members of a society, consult with each other, increasing in this way the available mutual information. A precise correspondence between QDT and classical utility theory is formulated via the introduction of an intermediate probabilistic version of utility theory of a novel form, which obeys the requirement that zero-utility prospects should have zero probability weights.

New approach for robust multi-objective optimization of turning parameters using probabilistic genetic algorithm

In this paper, a contribution to the determination of reliable cutting parameters is presented, which is minimizing the expected machining cost and maximizing the expected production rate, with taking into account the uncertainties of uncontrollable factors. The concept of failure probability of stochastic production limitations is integrated into constrained and unconstrained formulations of multi-objective optimization problems. New probabilistic version of the nondominated sorting genetic algorithm P-NSGA-II, which incorporates the Monte Carlo simulations for accurate assessment of cumulative distribution functions, was developed and applied in two numerical examples based on similar and anterior work. In the first case, it is a question of the search space that is completely ‘closed’ by high natural variability related to the multi-pass roughing operation: in this case, the failure risk of technological limitations are considered as objectives to minimize with economic objectives. The second case is related to deformed search space due to the uncertainties specific to finishing operation; therefore, the economic objectives are minimized under imposed maximum probabilities of failure. In both situations, the efficiency and robustness of optimal solutions generated by the P-NSGA-II algorithm are analysed, discussed and compared with sequence of unconstrained minimization technique (SUMT) method.

Calibration of a CRR model based on an expanded SPT-based database for assessing soil liquefaction potential

A new calibrated empirical liquefaction triggering correlation equation with five parameters determined by trial and error is derived from the expanded SPT-based database of the Idriss and Boulanger (2010) and Xie (1984) case histories. The new calibrated correlation equation is reasonably conservative for the expanded SPT-based database and is insensitive to reasonable variations in the components of the liquefaction analysis framework adopted in this study. In addition, a probabilistic version of the new calibrated correlation equation expressed in the form of a mapping function that relates the liquefaction probability to a nominal safety factor obtained using the weighted maximum likelihood technique is verified using a weighted empirical probability approach and Monte Carlo simulations. Liquefaction, transition and non-liquefaction zones for clean sands are defined based on different probability contours, i.e., empirical correlations with different liquefaction probabilities. Liquefaction triggering correlations for liquefaction probabilities of 4%, 15% and 25% represent the very low, low, and moderate probability events, respectively, of a liquefiable soil being mistaken for a non-liquefiable soil in the liquefaction potential evaluation.