Probability Space Research Articles

The role of Markov chain in Monte Carlo simulation

The Markov Chain Monte Carlo method has been applied in several fields and has thrived in academic research over the past few decades. The method is utilized to estimate the interest parameters posterior distribution through random sampling in the probability space. Monte Carlo method alone can accomplish random simulations for some complicated continuous integrals or discrete summations. However, the method requires the sample set corresponding to the distribution, which demands the application of Markov chain. Nevertheless, problems occur when the probability function of the samples remain unknown, under which circumstance the Markov Chain Monte Carlo method is no longer available. Therefore, the article introduced Metropolis-Hastings Algorithm as a resolution. The paper is intended to provide specific insights about Metropolis-Hastings Algorithm for Markov Chain Monte Carlo and the corresponding interdisciplinary applications. Specifically, the paper will discuss about using the theory mentioned above to check the relationships among species in a common ecosystem. The paper will cover several prepositive theories, including the Law of Large Numbers for Markov Chains, binary sequences, and Metropolis-Hastings Algorithm. Overall, specific cases and the way to draw correct conclusions will be most concerned.

Uncertain random variables and laws of large numbers under U-C chance space

Uncertainty theory was founded by Baoding Liu for modeling belief degrees about human uncertainty. And chance theory was pioneered by Yuhan Liu based on probability theory and uncertainty theory. In many cases, imprecise risk situation in subjective or objective setting and human uncertainty simultaneously appear in a complex system. To measure uncertain random events in this system, this paper combines uncertainty space with convex non-additive probability space into a new kind of two-dimensional chance space called U-C chance space. Moreover, a new framework for uncertain random variables combining uncertainty theory with convex non-additive probability theory is provided. For applications of this new framework, it can be applied to characterize some kinds of phenomena which possess the characteristics of both imprecise risk situations and belief degrees about human uncertainty in situations of great events, such as financial crisis, major natural disaster, etc. The main contribution of this paper is to derive the types of Kolmogorov and Marcinkiewicz-Zygmund LLNs for uncertain random variables satisfying some conditions under U-C chance space, where the convex non-additive probability is totally monotone. Furthermore, several examples are stated and explained.

From Wald to Schnorr: von Mises’ definition of randomness in the aftermath of Ville’s Theorem

The first formal definition of randomness, seen as a property of sequences of events or experimental outcomes, dates back to Richard von Mises’ work in the foundations of probability and statistics. The randomness notion introduced by von Mises is nowadays widely regarded as being too weak. This is, to a large extent, due to the work of Jean Ville, which is often described as having dealt the death blow to von Mises’ approach, and which was integral to the development of algorithmic randomness—the now-standard theory of randomness for elements of a probability space. The main goal of this article is to trace the history and provide an in-depth appraisal of two lesser-known, yet historically and methodologically notable proposals for how to modify von Mises’ definition so as to avoid Ville’s objection. The first proposal is due to Abraham Wald, while the second one is due to Claus-Peter Schnorr. We show that, once made precise in a natural way using computability theory, Wald’s proposal constitutes a much more radical departure from von Mises’ framework than intended. Schnorr’s proposal, on the other hand, does provide a partial vindication of von Mises’ approach: it demonstrates that it is possible to obtain a satisfactory randomness notion—indeed, a canonical algorithmic randomness notion—by characterizing randomness in terms of the invariance of limiting relative frequencies. More generally, we argue that Schnorr’s proposal, together with a number of little-known related results, reveals that there is more continuity than typically acknowledged between von Mises’ approach and algorithmic randomness. Even though von Mises’ exclusive focus on limiting relative frequencies did not survive the passage to the theory of algorithmic randomness, another crucial aspect of his conception of randomness did endure; namely, the idea that randomness amounts to a certain type of stability or invariance under an appropriate class of transformations.

Infinitesimal operators and the distribution of anticommutators and commutators

In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analyzed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence.

Optimized Real-Time Stochastic Model of Power Electronic Converters based on FPGA

Stochastic models can effectively describe the operating characteristics of power electronic converters with stochastic parameters. However, it is difficult to implement the models in FPGA-based real-time simulation (RTS) because their high order leads to a large calculation. This paper proposes an optimized real-time stochastic modeling method for power electronic converters based on generalized polynomial chaos (gPC). Firstly, an orthogonal polynomials construction method is used based on Schmidt orthogonalization to describe stochastic variables with atypical probability distributions and provide conditions for simplifying the system model. Secondly, the method of probability space transformation is adopted to divide the system model into multiple sub-models to suppress the exponential growth of the model order while maintaining the statistical properties. This method has performed over the traditional stochastic modeling method. The proposed model is built on FPGA-based hardware-in-the-loop (HIL) experiment platform with 1us simulation step. The optimized model uses approximately 37% fewer resources than the traditional stochastic model while maintaining the same level of accuracy.

A probabilistic time geographic approach to quantifying seabird‐vessel interactions

AbstractAccounting for uncertainty is essential for precautionary approaches to managing seabird bycatch in commercial fisheries. However, there is no existing mechanism to explicitly quantify the uncertainty of seabird‐vessel interactions (i.e. co‐occurrence in space and time). Here we develop a time geographic method to measure the probability of individual birds encountering (co‐occurring within 30 km) and attending (within 5 km) individual fishing vessels. The approach involves creating voxel‐based probabilistic space–time prisms (PSTPs) to model the movements of individual birds and vessels, with trajectory data from bird‐borne GPS devices and vessel Automatic Identification Systems (AIS). We intersected these PSTPs to quantify the probability of interaction between bird‐vessel pairs over time and space. We demonstrate the approach with a case study of interactions of Endangered Toroa (Antipodean Albatross; Diomedea antipodensis antipodensis) with pelagic longline vessels in part of the South Pacific high seas. We found 15 vessels within 150 km and 3 h of two birds, yet interaction occurred with only two of those vessels. We visualised the probability of encounter and attendance over time and space and determined that interactions lasted several hours each (up to 6.2–14.1 h attendance, 20.8–26.1 h encounter for one bird‐vessel pair). Our time geographic approach adds to existing tools to quantify seabird bycatch risk by providing an explicit measure of uncertainty of seabird‐vessel interactions. We provide a flexible methodological pathway and R scripts, the application of which would allow managers to estimate interaction probability for multiple marine species and fisheries, including those with lower‐resolution positional datasets.

Partial information decomposition for continuous variables based on shared exclusions: Analytical formulation and estimation.

Describing statistical dependencies is foundational to empirical scientific research. For uncovering intricate and possibly nonlinear dependencies between a single target variable and several source variables within a system, a principled and versatile framework can be found in the theory of partial information decomposition (PID). Nevertheless, the majority of existing PID measures are restricted to categorical variables, while many systems of interest in science are continuous. In this paper, we present a novel analytic formulation for continuous redundancy-a generalization of mutual information-drawing inspiration from the concept of shared exclusions in probability space as in the discrete PID definition of I_{∩}^{sx}. Furthermore, we introduce a nearest-neighbor-based estimator for continuous PID and showcase its effectiveness by applying it to a simulated energy management system provided by the Honda Research Institute Europe GmbH. This work bridges the gap between the measure-theoretically postulated existence proofs for a continuous I_{∩}^{sx} and its practical application to real-world scientific problems.

Some Skorohod-type results

AbstractLet S be a metric space, $$g:S\rightarrow \mathbb {R}$$ g : S → R a Borel function, and $$(\mu _n:n\ge 0)$$ ( μ n : n ≥ 0 ) a sequence of tight probability measures on $$\mathcal {B}(S)$$ B ( S ) . If $$\mu _n=\mu _0$$ μ n = μ 0 on $$\sigma (g)$$ σ ( g ) , there are S-valued random variables $$X_n$$ X n , all defined on the same probability space, such that $$X_n\sim \mu _n$$ X n ∼ μ n and $$g(X_n)=g(X_0)$$ g ( X n ) = g ( X 0 ) for all $$n\ge 0$$ n ≥ 0 . Moreover, $$X_n\overset{a.s.}{\longrightarrow }X_0$$ X n ⟶ a . s . X 0 if and only if $$E_{\mu _n}(f\mid g)\,\overset{\mu _0-a.s.}{\longrightarrow }\,E_{\mu _0}(f\mid g)$$ E μ n ( f ∣ g ) ⟶ μ 0 - a . s . E μ 0 ( f ∣ g ) for each $$f\in C_b(S)$$ f ∈ C b ( S ) . This result, proved in Pratelli and Rigo (J Theoret Probab 36:372-389, 2023) , is the starting point of this paper. Three types of contributions are provided. First, $$\sigma (g)$$ σ ( g ) is replaced by an arbitrary sub-$$\sigma $$ σ -field $$\mathcal {G}\subset \mathcal {B}(S)$$ G ⊂ B ( S ) . Second, the result is applied to some specific frameworks, including equivalence couplings, total variation distances, and the decomposition of cadlag processes with finhite activity. Third, following Hansen et al. (Tempered Bayesian analysis, Unpublished manuscript, 2024), the result is extended to models and kernels. This extension has a fairly natural interpretation in terms of decision theory, mass transportation and statistics.

A note about almost uniform convergence on D-poset of intuitionistic fuzzy sets

The aim of this contribution is studying the almost uniform convergence on D-poset of intuitionistic fuzzy sets. We prove the connection between almost everywhere convergence of random variables in Kolmogorov probability space and almost uniform convergence of observables in the mentioned D-poset. We define a product operation on D-poset of intuitionistic fuzzy sets and prove the existence of a joint observable, too.

The normality of L-coarse spaces

For L being a completely distributive lattice, we introduce a notion of L-coarse spaces, which can be regarded as a large-scale analogue of probabilistic uniform spaces. The main goal of this paper is to investigate the properties of L-coarse spaces that can induce L-valued coarse proximity spaces. First of all, we introduce a notion of L-valued coarse neighborhood systems, and show that the resulting category is isomorphic to that of L-valued coarse proximity spaces. Then we study the normality of coarsely connected L-coarse spaces and of asymptotically connected L-valued asymptotic resemblance spaces, and show that these two normality properties are coincident. More importantly, we show that a coarsely connected L-coarse space induces an L-valued coarse proximity if and only if it is coarsely normal. We conclude with noting that the previous result are also valid for asymptotically connected L-valued asymptotic resemblance spaces.

Complete convergence theorems for weighted sums of extended negatively dependent random variables under sub-linear expectations

. This article aims to study and establish the Lai law for the weighted sum of extended negatively dependent random variables under sub-linear expectations. Using inequalities under sub-linear expectation spaces, we extend the complete convergence of weighted sums from classical probability space to sub-linear expectation space.

Path integration solutions for stochastic systems with Markovian jumps

A path integration method for solving Markovian jump stochastic dynamical systems is presented. The Markovian jump process and the state vector of the system are combined into a new augmented state vector. The randomness of the Markovian jump is modeled by a stochastic process, which is merged with the stochastic perturbations to form the stochastic source affecting the evolution of the augmented system. Then a probability mapping is constructed, mapping the probability space of the stochastic source to the short-time transition probability density function of the augmented system. By solving this probability mapping, the short-time transition probability density function of the path integration method is obtained, thus a path integration method is customized for Markovian jump stochastic systems. Finally, a hydraulic relief valve system is used as an application example to demonstrate the availability of the path integration algorithm. The results suggest that the pre-compression parameters and coefficient of restitution of the plug can induce stochastic P-bifurcation phenomena. When the above two parameters are small, the system becomes unstable with the parameters taken in this example. Monte Carlo simulations prove that the proposed method effectively captures the transient and stationary responses of Markovian jump systems.

Self-supervised Gaussian Restricted Boltzmann Machine via joint contrastive representation and contrastive divergence

In this paper, we propose a novel self-supervised Gaussian Restricted Boltzmann Machine with contrastive learning (CL-GRBM), which fuses contrastive representation learning and contrastive divergence to optimize and enhance the representation of GRBM. Built upon the concept of contrastive representation learning, CL-GRBM aims to enhance the representation capacity of the GRBM. Firstly, a pair of positive samples are constructed by one times Gibbs sampling with the original data. Then, the contrastive loss is used to pull the positive samples closer and push other samples further apart, making similar representations closer and different representations farther apart. In summary, during the training process of CL-GRBM using the CD algorithm, the objective is to align the sampling probability distribution of the visible layer in CL-GRBM as closely as possible with the empirical distribution of the original data. In the feature space of the sampling probabilities in the hidden layer, the distance between positive samples is minimized to capture the intrinsic structure of the data. According to the experimental verification, the proposed CL-GRBM shows better performance than contrastive models. As a shallow self-supervised model, it has even better performance than some excellent deep self-supervised models.

Use of directed quasi-metric distances for quantifying the information of gene families

A large hindrance to analyzing information in genetic or protein sequence data has been a lack of a mathematical framework for doing so. In this paper, we present a multinomial probability space X as a general foundation for multicategory discrete data, where categories refer to variants/alleles of biosequences. The external information that is infused in order to generate a sample of such data is quantified as a distance on X between the prior distribution of data and the empirical distribution of the sample. A number of distances on X are treated. All of them have an information theoretic interpretation, reflecting the information that the sampling mechanism provides about which variants that have a selective advantage and therefore appear more frequently compared to prior expectations. This includes distances on X based on mutual information, conditional mutual information, active information, and functional information. The functional information distance is singled out as particularly useful. It is simple and has intuitive interpretations in terms of 1) a rejection sampling mechanism, where functional entities are retained, whereas non-functional categories are censored, and 2) evolutionary waiting times. The functional information is also a quasi-metric on X, with information being measured in an asymmetric, mountainous landscape. This quasi-metric property is also retained for a robustified version of the functional information distance that allows for mutations in the sampling mechanism. The functional information quasi-metric has been applied with success on bioinformatics data sets, for proteins and sequence alignment of protein families.

The Hierarchical Discrete Pursuit Learning Automaton: A Novel Scheme With Fast Convergence and Epsilon-Optimality.

Since the early 1960s, the paradigm of learning automata (LA) has experienced abundant interest. Arguably, it has also served as the foundation for the phenomenon and field of reinforcement learning (RL). Over the decades, new concepts and fundamental principles have been introduced to increase the LA's speed and accuracy. These include using probability updating functions, discretizing the probability space, and using the "Pursuit" concept. Very recently, the concept of incorporating "structure" into the ordering of the LA's actions has improved both the speed and accuracy of the corresponding hierarchical machines, when the number of actions is large. This has led to the ϵ -optimal hierarchical continuous pursuit LA (HCPA). This article pioneers the inclusion of all the above-mentioned phenomena into a new single LA, leading to the novel hierarchical discretized pursuit LA (HDPA). Indeed, although the previously proposed HCPA is powerful, its speed has an impediment when any action probability is close to unity, because the updates of the components of the probability vector are correspondingly smaller when any action probability becomes closer to unity. We propose here, the novel HDPA, where we infuse the phenomenon of discretization into the action probability vector's updating functionality, and which is invoked recursively at every stage of the machine's hierarchical structure. This discretized functionality does not possess the same impediment, because discretization prohibits it. We demonstrate the HDPA's robustness and validity by formally proving the ϵ -optimality by utilizing the moderation property. We also invoke the submartingale characteristic at every level, to prove that the action probability of the optimal action converges to unity as time goes to infinity. Apart from the new machine being ϵ -optimal, the numerical results demonstrate that the number of iterations required for convergence is significantly reduced for the HDPA, when compared to the state-of-the-art HCPA scheme.

UBlade: Efficient Batch Processing for Uncertainty Graph Queries

The study of uncertain graphs is crucial in diverse fields, including but not limited to protein interaction analysis, viral marketing, and network reliability. Processing queries on uncertain graphs presents formidable challenges due to the vast probabilistic space they encapsulate. While existing systems employ batch processing to address these challenges, their performance is often compromised by the suboptimal selection of parallel graph traversal methods, the excessive costs in random number generation, and additional workloads intrinsic to batch processing. In this paper, we introduce uBlade, an efficient batch-processing framework for uncertain graph queries on multi-core CPUs. uBlade utilizes the work-efficient graph traversal, achieving superior parallelism in the batch processing model. Additionally, our Quasi-Sampling technique reduces the random number generation cost by a factor of B, with O(B) denoting the batch size. We further examine the extra workload resulting from batch processing and introduce an efficient strategy to reorder possible worlds, minimizing this associated overhead. Through comprehensive evaluations, we showcase that uBlade achieves up to two orders of magnitude speedups against the state-of-the-art CPU and GPU-based solutions.

Uniform syndeticity in multiple recurrence

Abstract The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers $d,l\geq 1$ and any $\varepsilon> 0$ , we prove the existence of $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that the following holds: Consider a solvable group $\Gamma $ of derived length l, a probability space $(X, \mu )$ , and d pairwise commuting measure-preserving $\Gamma $ -actions $T_1, \ldots , T_d$ on $(X, \mu )$ . Let E be a measurable set in X with $\mu (E) \geq \varepsilon $ . Then, K many (left) translates of $$ \begin{align*} \big\{\gamma\in\Gamma\colon \mu(T_1^{\gamma^{-1}}(E)\cap T_2^{\gamma^{-1}} \circ T^{\gamma^{-1}}_1(E)\cap \cdots \cap T^{\gamma^{-1}}_d\circ T^{\gamma^{-1}}_{d-1}\circ \cdots \circ T^{\gamma^{-1}}_1(E))\geq \delta \big\} \end{align*} $$ cover $\Gamma $ . This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers $d,l\geq 1$ and any $\varepsilon> 0$ , there are $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that for all finite solvable groups G of derived length l and any subset $E\subset G^d$ with $m^{\otimes d}(E)\geq \varepsilon $ (where m is the uniform measure on G), we have that K-many (left) translates of $$ \begin{align*} \{g\in G\colon &m^{\otimes d}(\{(a_1,\ldots,a_n)\in G^d\colon \\ & (a_1,\ldots,a_n),(ga_1,a_2,\ldots,a_n),\ldots,(ga_1,ga_2,\ldots, ga_n)\in E\})\geq \delta \} \end{align*} $$ cover G. The proof of our main result is a consequence of an ultralimit version of Austin’s amenable ergodic Szeméredi theorem.

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

Synthesis and Properties of the Ba2PrWO6 Double Perovskite.

We report details on the synthesis and properties of barium praseodymium tungstate, Ba2PrWO6, a double perovskite that has not been synthesized before. Room-temperature (RT) powder X-ray diffraction identified the most probable space group (SG) as monoclinic I2/m, but it was only slightly distorted from the cubic structure. X-ray photoelectron spectroscopy confirmed that the initial (postsynthesis) material contained praseodymium in both 3+ and 4+ charge states. The former (Pr3+) disappeared after exposure to UV light at RT. Photoluminescence studies of Pr3+ revealed that Ba2PrWO6 is an insulator with a band gap exceeding 4.93 eV. Pressure-dependent Raman spectroscopy excluded the possibility of a phase transition up to 20 GPa; however, measurements between 8 and 873 K signified that there might be a change toward the lower symmetry SG below 200 K. Electron paramagnetic resonance spectra revealed the presence of interstitial oxygen which acts as a deep electron trap.

Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games

Abstract We propose a new sequential Efficient Pseudo-Likelihood (k-EPL) estimator for dynamic discrete choice games of incomplete information. k-EPL considers the joint behaviour of multiple players simultaneously, as opposed to individual responses to other agents’ equilibrium play. This, in addition to reframing the problem from conditional choice probability (CCP) space to value function space, yields a computationally tractable, stable, and efficient estimator. We show that each iteration in the k-EPL sequence is consistent and asymptotically efficient, so the first-order asymptotic properties do not vary across iterations. Furthermore, we show the sequence achieves higher-order equivalence to the finite-sample maximum-likelihood estimator with iteration and that the sequence of estimators converges almost surely to the maximum-likelihood estimator at a nearly superlinear rate when the data are generated by any regular Markov perfect equilibrium, including equilibria that lead to inconsistency of other sequential estimators. When utility is linear in parameters, k-EPL iterations are computationally simple, only requiring that the researcher solve linear systems of equations to generate pseudo-regressors which are used in a static logit/probit regression. Monte Carlo simulations demonstrate the theoretical results and show k-EPL’s good performance in finite samples in both small- and large-scale games, even when the game admits spurious equilibria in addition to one that generated the data. We apply the estimator to analyse competition in the U.S. wholesale club industry.