Probabilistic Problem Research Articles

Preservice Teachers’ Misconceptions in Solving Probabilistic Problems

Concerns about the influence of misconceptions, culture and social setting on probabilistic reasoning of teacher trainees led to a study in this area to explore the reasoning of preservice teachers from a college of education in Ghana. This study investigates preservice teachers’ misconceptions and difficulties in probabilistic problems solving before they learn Elementary Stochastic as part of their course of studies. The identification of preservice teachers’ misconceptions will serve as a reference point for planning and enactment of lessons for effective training of teachers. The study employed explanatory sequential mixed methods research design. The participants in this study were 181 level 200 female preservice teachers offering a Diploma in Basic Education. Diagnostic test and semi-structured interview were used to collect data. Data collected were analysed descriptively. Also, content analysis was employed to analyse some parts of the data collected. The results indicate that preservice teachers had some probability difficulties such as conceptual difficulties, interpretation difficulties, and procedural difficulties. Similarly, the study found that preservice teachers entered college with the following probability misconceptions: equiprobability bias, representativeness bias, negative and positive recency effects, outcome orientation bias, and belief bias. This study shows the need for teaching and learning activities to focus on addressing probability misconceptions and difficulties in order to developed better probabilistic reasoning of preservice teachers.

On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints

We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.

Universal Equivalence and Majority of Probabilistic Programs over Finite Fields

We study decidability problems for equivalence of probabilistic programs for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication, and conditionals and thus is sufficiently expressive to encode Boolean and arithmetic circuits. We consider two variants of equivalence: The first one considers an interpretation over the finite field F q , while the second one, which we call universal equivalence, verifies equivalence over all extensions F q k of F q . The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F 2 k for any k . While the first problem is obviously decidable, we establish its exact complexity, which lies in the counting hierarchy. To show decidability and a doubly exponential upper bound of the universal variant, we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. We then devise a general way to draw links between the universal probabilistic problems and widely studied problems on linear recurrence sequences. Finally, we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy. We also define and provide some insights about program indistinguishability, proving that it is decidable for programs always returning 0 or 1.

Normal Wiggly Probabilistic Hesitant Fuzzy Information for Environmental Quality Evaluation

As a valuable tool for representing uncertain information, probabilistic hesitant fuzzy sets (PHFS) have gained considerable recognition and in-depth discussion in recent years to increase the flexibility and manifest hesitant information in decision-making problems. However, decision-makers (DMs) cannot express all preferences only through a few probabilistic terms in actual decision-making. Much critical information is hidden behind the original information provided by the DMs. Keeping that in mind, we are interested in mining deeper uncertain information from the original probabilistic hesitant fuzzy evaluation data. To achieve the target, we put forward a novel representation tool called the normal wiggly probabilistic hesitant fuzzy set (NWPHFS) to extract deeper uncertain preferences from original probabilistic information. NWPHFS retains the original evaluation information and carries and assesses the potential uncertain details for increasing the rationality of decision-making outcomes. Herein, we propose some fundamental concepts of NWPHFS, for instance, some elementary operational laws, distance measures between two NWPHFSs, and score function. We also suggest two new aggregation operators, that is, the normal wiggly probabilistic hesitant fuzzy weighted averaging (NWPHFWA) and normal wiggly probabilistic hesitant fuzzy weighted geometric (NWPHFWG). A novel mechanism is proposed here to work out multiattribute decision-making (MADM) in solving normal wiggly probabilistic decision-making problems. Through a practical example of environmental quality assessment, the specific calculation steps of this method are epitomized. Finally, we have demonstrated the feasibility and advancement of the proposed approach via a comprehensive comparative study.

Computation of multi-objective two-stage fuzzy probabilistic programming problem

Computation of multi-objective two-stage fuzzy probabilistic programming problem

Probabilistic Set Covering Location Problem in Congested Networks

This paper focuses on designing a facility network, taking into account that the system may be congested. The objective is to minimize the overall fixed and service capacity costs, subject to the constraints that for any demand the disutility from travel and waiting times (measured as the weighted sum of the travel time from a demand to the facility serving that demand and the average waiting time at the facility) cannot exceed a predefined maximum allowed level (measured in units of time). We develop an analytical framework for the problem that determines the optimal set of facilities and assigns each facility a service rate (service capacity). In our setting, the consumers would like to maximize their utility (minimize their disutility) when choosing which facility to patronize. Therefore, the eventual choice of facilities is a user-equilibrium problem, where at equilibrium, consumers do not have any incentive to change their choices. The problem is formulated as a nonlinear mixed-integer program. We show how to linearize the nonlinear constraints and solve instead a mixed-integer linear problem, which can be solved efficiently.

Machine learning RBF-based surrogate models for uncertainty quantification of age and time-dependent fracture mechanics

This paper proposes a machine learning strategy using radial basis function (RBF) as surrogate models for uncertainty quantification of age and time-dependent fracture mechanics problems. The RBF surrogate models are trained to replace the time-consuming evaluations of the mapping integral of the time-dependent energy release rate. The probabilistic problem considers input random variables of geometry, loading, and material parameters for a concrete plate section with the presence of an initial surface crack. The performance of the RBF surrogates was evaluated through cross-validation and Monte Carlo reference solutions that expensively evaluated the integrals. The results of mean square errors and the Kolmogorov-Smirnov goodness-of-fit tests on the predicted probability and cumulative density functions show that, the RBF surrogates reached good accuracy even for a small training set size, rather than second-order polynomial models which could reasonably perform only for bigger training set sizes. The RBF surrogates accurately predict even long-tail distributions for farther prediction times with relatively small training sets. The proposed RBF surrogate models presented the generalization capabilities to predict thousands of unseen data points from Monte Carlo simulations and successfully assess the uncertainty quantification of the time-dependent energy release rate.

Онтология и практика преподавания математики в высшей и средней школе

Discussed are problems of modern mathematical education: the development of the conceptual apparatus of mathematics, as well as the development of thinking in modal categories when proving mathematical propositions and solving probabilistic problems. The work is based on the results of the explication of the ontological foundations of mathematics obtained earlier. It is shown that the basis of a holistic, deep perception of the mathematical laws of being is categorical intuition with subsequent rationalization of mathematical objects. It is emphasized that such objects are not always possible to describe by means of extensional logic: an important place in the categorical structuring of mathematical knowledge is occupied by ontic modalities of the necessary and possible. The author expresses concern about the tendency of algorithmization of mathematical thinking in the learning process as opposed to historically established methods of creative search for proofs, alternatives, and opportunities. Special attention is paid to the topic of the teacher’s implementation of the ideological and educational function of mathematical science. Taking into account the enormous development of computing and information technologies, it is necessary to remember about the anthropological and humanitarian components of modern science and education. In substantiation of the relevance and discussion of the raised issues, the points of view of specialists from various fields are given: mathematicians, historians, teachers. This article is addressed to a wide range of readers: scientists, teachers, graduate students, freelance researchers.

The bright homunculus in our head: Individual differences in intuitive sensitivity to logical validity

Classic dual process theories of human reasoning attribute explicit reasoning to effortful, deliberative thinking. According to these models, intuitive processes lack any access to the formal rules of logic and probability and hence rely exclusively on superficial problem features to determine a response. However, in recent years, researchers have demonstrated that reasoners are able to solve simple logical or probabilistic problems relatively automatically, a capability which has been called "logical intuition." In four experiments, we instructed participants to judge the validity (Experiments 1 and 4), likeability (Experiments 1, 2, and 3), and physical brightness (Experiments 2, 3, and 4) of the conclusion to several reasoning problems. Brightness judgements were made by evaluating the font shade brightness of the argument's conclusion. Participants were also asked to complete a range of individual differences measures, drawing on cognitive ability and cognitive style, to evaluate the extent to which "logical intuitions" were linked to measures of deliberative reasoning. The results showed that participants judged the conclusion of logically valid statements to be more valid, more likable, and more physically bright than invalid statements. Participants with higher cognitive ability and unlimited processing time showed greater effects of logical validity in their liking judgements (varied across experiments). However, these effects were absent in the brightness tasks, suggesting that logic effects observed under instructions to judge conclusion brightness are a purer measure of "logical intuition." We discuss the implications of our findings for recent dual process theories of human reasoning.

Guiding Search with Generalized Policies for Probabilistic Planning

We examine techniques for combining generalized policies with search algorithms to exploit the strengths and overcome the weaknesses of each when solving probabilistic planning problems. The Action Schema Network (ASNet) is a recent contribution to planning that uses deep learning and neural networks to learn generalized policies for probabilistic planning problems. ASNets are well suited to problems where local knowledge of the environment can be exploited to improve performance, but may fail to generalize to problems they were not trained on. Monte-Carlo Tree Search (MCTS) is a forward-chaining state space search algorithm for optimal decision making which performs simulations to incrementally build a search tree and estimate the values of each state. Although MCTS can achieve state-of-the-art results when paired with domain-specific knowledge, without this knowledge, MCTS requires a large number of simulations in order to obtain reliable state-value estimates. By combining ASNets with MCTS, we are able to improve the capability of an ASNet to generalize beyond the distribution of problems it was trained on, as well as enhance the navigation of the search space by MCTS.

Isotonic Distributional Regression

AbstractIsotonic distributional regression (IDR) is a powerful non-parametric technique for the estimation of conditional distributions under order restrictions. In a nutshell, IDR learns conditional distributions that are calibrated, and simultaneously optimal relative to comprehensive classes of relevant loss functions, subject to isotonicity constraints in terms of a partial order on the covariate space. Non-parametric isotonic quantile regression and non-parametric isotonic binary regression emerge as special cases. For prediction, we propose an interpolation method that generalizes extant specifications under the pool adjacent violators algorithm. We recommend the use of IDR as a generic benchmark technique in probabilistic forecast problems, as it does not involve any parameter tuning nor implementation choices, except for the selection of a partial order on the covariate space. The method can be combined with subsample aggregation, with the benefits of smoother regression functions and gains in computational efficiency. In a simulation study, we compare methods for distributional regression in terms of the continuous ranked probability score (CRPS) and L2 estimation error, which are closely linked. In a case study on raw and post-processed quantitative precipitation forecasts from a leading numerical weather prediction system, IDR is competitive with state of the art techniques.

Analytical asymptotics for hard diffraction

We show that the cross section for diffractive dissociation of a small onium off a large nucleus at total rapidity $Y$ and requiring a minimum rapidity gap ${Y}_{\mathrm{gap}}$ can be identified, in a well-defined parametric limit, with a simple classical observable on the stochastic process representing the evolution of the state of the onium, as its rapidity increases, in the form of color dipole branchings: it formally coincides with twice the probability that an even number of these dipoles effectively participate in the scattering, when viewed in a frame in which the onium is evolved to the rapidity $Y\ensuremath{-}{Y}_{\mathrm{gap}}$. Consequently, finding asymptotic solutions to the Kovchegov-Levin equation, which rules the $Y$ dependence of the diffractive cross section, boils down to solving a probabilistic problem. Such a formulation authorizes the derivation of a parameter-free analytical expression for the gap distribution. Interestingly enough, events in which many dipoles interact simultaneously play an important role, since the distribution of the number $k$ of dipoles participating in the interaction turns out to be proportional to $1/[k(k\ensuremath{-}1)]$.

Bayesian optimization for robust design of steel frames with joint and individual probabilistic constraints

This work proposes a Bayesian optimization (BO) method for solving multi-objective robust design optimization (RDO) problems of steel frames under aleatory uncertainty in external loads and material properties. Joint and individual probabilistic constrained RDO problems are formulated to consider two different ways the frame reaches its collapse state. Each problem involves three conflicting objective functions, namely, the total mass of the frame, the mean and variance of the maximum inter-story drift. Since the uncertain objective and probabilistic constraint functions of both problems are implicit within a finite element analysis program and the computation of the probabilistic constraints is an NP-hard problem, BO is used to guide the optimization process toward better solutions after it completes an iteration and offers a set of near Pareto-optimal solutions when it terminates. Specifically, Bayesian regression models called Gaussian processes (GPs) serve as surrogates for the structural responses. Two acquisition functions are then developed for the two RDO problems and a maximization problem of these functions is formulated as a mixed-integer nonlinear programming (MINLP) problem. A new random search coupled with simulated annealing is devised to solve the MINLP problem, thereby locating the most promising point in the input variable space at which the current solutions maximize their chance to be improved and the GP models are refined before the BO starts a new iteration. A test problem and two design examples show that exact or good Pareto-optimal solutions to the RDO problems can be found by the proposed method with 20 iterations.

Probabilistic spatiotemporal solar irradiation forecasting using deep ensembles convolutional shared weight long short-term memory network

High-quality and reliable solar radiation spatiotemporal probabilistic prediction is very significant for the planning and management of solar energy. In this paper, convolutional shared weight long short-term memory network is proposed to deal with solar radiation spatiotemporal prediction problems. The model can minimize the training time of model without significantly reducing prediction accuracy. Furthermore, a spatiotemporal probabilistic prediction model that combines convolutional shared weight long short-term memory network and deep ensemble method is proposed to deal with the solar radiation probabilistic prediction problem. The model obtains the uncertainty estimation of predictands by adjusting the network structure and optimizes the uncertainty by employing a proper scoring rule. In addition, a combination of ensembles are used in the model to improve the robustness of probabilistic prediction. The new spatiotemporal probabilistic prediction model is applied to predict solar radiation in a real area of the United States. Five state-of-the-art models and seven evaluation indicators are used for comparation. The comparation results show the proposed model is able to provide accurate point prediction, reasonable prediction interval and reliable probabilistic prediction results for a whole area.

Probabilistic Thinking Profile of Mathematics Teacher Candidates in Problem Solving based on Self-Regulated Learning

<p style="text-align:justify">Many problems in real life are full of uncertainty. Probabilistic thinking is one way to solve this uncertainty problem. The personality abilities that are interesting to examine include how independent the students are in facing probabilistic solutions. For this reason, it is necessary to study the probabilistic thinking profile of prospective mathematics teachers in solving probabilistic problems in terms of learning the self-regulation. The research method used in this research is qualitative research. In this study, the researcher conducted an analysis of students' probabilistic thinking with the scope of the probability material related to the sample space, the probability of events, and the probability of the random variable space. The profile of the level of probabilistic thinking is influenced by cognitive factors in the form of probabilistic knowledge obtained by students and personality in the form of self-regulated learning. Student learning self-regulated influences the mental activity in decision making. By studying the profile of probabilistic thinking, students are better prepare to face uncertainty with various aspects that influence it.</p>

Reliability-Based Design of an Aircraft Wing Using a Fuzzy-Based Metaheuristic

The purpose of this paper is to design aircraft wing using reliability-based design optimization concerned to fuzzy uncertainty variables. A possibilistic safety index-based design optimization (PSIBDO) with fuzzy uncertainties is proposed to overcome difficult tasks from the original probabilistic problem. The design problem is to minimize mass of a composite aircraft wing subject to aeroelastic and structural constraints through consideration of the material properties are the uncertainties. The design variables include aircraft wing structure dimensions. The reliability-based design approach is needed to alleviate such a problem. Due to the complexity of the aircraft wing structures design and aeroelastic analysis, nonprobability-based design is an alternative choice to increase computational efficiency in the design process. The optimum results show the efficiency of our proposed approach.

A CRITERION UTILITY CONVERSION TECHNIQUE FOR PROBABILISTIC LINGUISTIC MULTIPLE CRITERIA ANALYSIS IN EMERGENCY MANAGEMENT

In multiple criteria decision making (MCDM), the even swaps method uses the relationships of criteria to make trade-offs but the burdens of experts are heavy; the linear programming technique for multidimensional analysis of preference (LINMAP) method cannot deal with the inter-dependencies among criteria but the cognitive burdens of experts are low. Taking the advantages of both these methods, this study proposes a criterion utility conversion (CUC) technique to solve probabilistic linguistic MCDM problems given that the probabilistic linguistic term set (PLTS) can reflect the psychology of experts when making evaluations. The utility conversion process is first proposed based on the marginal utilities of criteria. Then, the criterion preference ratios of experts are refined from the utility conversion process. Based on the criterion preference ratios and the operations of PLTSs, the adjusted probabilistic linguistic expected values of alternatives are calculated. The consistency and inconsistency indexes of alternatives and criteria are defined to set up the linear programming used to work out the criterion preference ratios. An illustration about the selection of emergency logistics supplier is given to validate the proposed method. The comparative analysis indicates the low cognitive burden, high stability, and strong applicability of the proposed method.

A New Chaotic Artificial Bee Colony for the Risk-Constrained Economic Emission Dispatch Problem Incorporating Wind Power

Due to the rapid increase in the consumption of electrical energy and the instability of fossil fuel prices, renewable energy, such as wind power (WP), has become increasingly economically competitive compared to other conventional energy production methods. However, the intermittent nature of wind energy creates certain challenges to the power network operation. The combined economic environmental dispatch (CEED) including WP is one of the most fundamental challenges in power system operation. Within this context, this paper presents a new attempt to solve the probabilistic CEED problem with WP penetration. The optimal WP to be incorporated in the grid is determined in such a way that the system security is within acceptable limits. The system security is described by various fuzzy membership functions in terms of the probability that power balance cannot be met. These membership functions are formulated based on the dispatcher’s attitude. This probabilistic and non-convex CEED problem is solved using a new technique combining chaos theory and artificial bee colony (ABC) technique. In this improved version of ABC (IABC), chaotic maps are used to generate initial solutions, and the random numbers involved in the standard ABC are substituted by chaotic sequences. The effectiveness of IABC is tested on two groups of benchmark functions and practical cases. The impacts of dispatcher’s attitude and risk level are investigated in the simulation section.

Dynamic Pricing and Matching for Two-Sided Markets with Strategic Servers

Motivated by applications in online marketplaces such as ridesharing, we study dynamic pricing and matching in two-sided queues with strategic servers. We consider a discrete-time process in which, heterogeneous customers and servers arrive. Each customer joins their type's queue, while servers might join a different type's queue depending on the prices posted by the system operator and an inconvenience cost. Then the system operator, constrained by a compatibility graph, decides the matching. The objective is to maximize the profit minus the expected waiting times. We develop a general framework that enables us to analyze a broad range of strategic behaviors. In particular, we encode servers' behavior in a properly defined cost function that can be tailored to various settings. Using this general cost function, we introduce a novel probabilistic fluid problem as an infinite dimensional optimization program. The probabilistic fluid model provides an upper bound on the achievable profit. We then study the system under a large market regime in which the arrival rates are scaled by η and present a probabilistic two-price policy and a max-weight matching policy which results in $O(η^1/3 )$ profit-loss. In addition, under a broad class of customer pricing policies, we show that any matching policy has profit-loss Ω(η1/3). Conditional on a given expected waiting time, we also establish scale-free lower and upper bounds for the profit. Our asymptotic analysis provides insight into near-optimal pricing and matching decisions, and our scale-free bounds provide insights into how different service levels impact the firm's profit.

Indirect illusory inferences from disjunction: a new bridge between deductive inference and representativeness

We provide a new link between deductive and probabilistic reasoning fallacies. Illusory inferences from disjunction are a broad class of deductive fallacies traditionally explained by recourse to a matching procedure that looks for content overlap between premises. In two behavioral experiments, we show that this phenomenon is instead sensitive to real-world causal dependencies and not to exact content overlap. A group of participants rated the strength of the causal dependence between pairs of sentences. This measure is a near perfect predictor of fallacious reasoning by an independent group of participants in illusory inference tasks with the same materials. In light of these results, we argue that all extant accounts of these deductive fallacies require non-trivial adjustments. Crucially, these novel indirect illusory inferences from disjunction bear a structural similarity to seemingly unrelated probabilistic reasoning problems, in particular the conjunction fallacy from the heuristics and biases literature. This structural connection was entirely obscure in previous work on these deductive problems, due to the theoretical and empirical focus on content overlap. We argue that this structural parallelism provides arguments against the need for rich descriptions and individuating information in the conjunction fallacy, and we outline a unified theory of deductive illusory inferences from disjunction and the conjunction fallacy, in terms of Bayesian confirmation theory.