Probabilistic Version Research Articles

Probabilistic degenerate Dowling polynomials associated with random variables

Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely, the probabilistic degenerate Whitney numbers of the second kind associated with and the probabilistic degenerate Dowling polynomials associated with . We derive some properties, explicit expressions, certain identities, recurrence relations and generating functions for those numbers and polynomials. In addition, we investigate their generalizations, namely, the probabilistic degenerate ‐Whitney numbers of the second kind associated with and the probabilistic degenerate ‐Dowling polynomials associated with , and get similar results to the aforementioned numbers and polynomials.

Deep and shallow conditionals – and three alleged counterexamples

ABSTRACT We analyze three interesting arguments from the literature, where ascribing a probability of 1 to a certain right-nested conditional A→(B→C) leads to strong theses concerning conditionals: they serve as counterexamples to important general claims. The first is the classic and much discussed McGee’s counterexample to Modus Ponens from McGee [“A Counterexample to Modus Ponens.” The Journal of Philosophy 82 (9): 462–471]. The second example was given by Santorio [“Trivializing Informational Consequence.” Philosophy and Phenomenological Research 104:297–320] and is intended to undermine the probabilistic version of Modus Ponens. The third example was presented in Cantwell [“Revisiting McGee’s Probabilistic Analysis of Conditionals.” Journal of Philosophical Logic 51 (5): 973–1017] in order to reject the general version of the Ramsey Test (i.e. PCCP) and Modus Ponens. These examples are based on a specific interpretation of right-nested conditional which is not the only possible one. We provide examples demonstrating that alternative interpretations are feasible. As a result, the arguments presented in the three cited works possess limited efficacy, as they are confined solely to the particular interpretation of the conditional.

Characterizing a probabilistic version of the Boston mechanism under weak priorities

We characterize the probabilistic version of the Boston mechanism from Han (2024). This mechanism uniquely satisfies respect for rank, equal-rank ex-ante stability, and equal-rank equal-priority ordinal fairness. Additionally, it uniquely selects the rank-based leximin-optimal allocation among all equal-rank ex-ante stable allocations.

Fast erasure decoder for hypergraph product codes

We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in O(N2) bit operations where N is the length of the quantum code. A probabilistic version of this decoder can be implemented in O(N1.5) bit operations.

Probabilistic Degenerate Fubini Polynomials Associated with Random Variables

Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Fubini polynomials and the degenerate Fubini polynomials of order r, namely the probabilisitc degenerate Fubini polynomials associated with Y and the probabilistic degenerate Fubini polynomials of order r associated with Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of Y, we treat the gamma random variable with parameters α,β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha ,\\beta > 0$$\\end{document}, the Poisson random variable with parameter α>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha > 0$$\\end{document}, and the Bernoulli random variable with probability of success p.

The impact of state merging on predictive accuracy in probabilistic tree automata: Dietze's conjecture revisited

Dietze's conjecture concerns the problem of equipping a tree automaton M with weights to make it probabilistic, in such a way that the resulting automaton N predicts a given corpus C as accurately as possible. The conjecture states that the accuracy cannot increase if the states in M are merged with respect to an equivalence relation ∼ so that the result is a smaller automaton M∼. Put differently, merging states can never improve predictions. This is under the assumption that both M and M∼ are bottom-up deterministic and accept every tree in C. We prove that the conjecture holds, using a construction that turns any probabilistic version N∼ of M∼ into a probabilistic version N of M, such that N assigns at least as great a weight to each tree in C as N∼ does.

An Efficient Probabilistic Algorithm to Detect Periodic Patterns in Spatio-Temporal Datasets

Deriving insight from data is a challenging task for researchers and practitioners, especially when working on spatio-temporal domains. If pattern searching is involved, the complications introduced by temporal data dimensions create additional obstacles, as traditional data mining techniques are insufficient to address spatio-temporal databases (STDBs). We hereby present a new algorithm, which we refer to as F1/FP, and can be described as a probabilistic version of the Minus-F1 algorithm to look for periodic patterns. To the best of our knowledge, no previous work has compared the most cited algorithms in the literature to look for periodic patterns—namely, Apriori, MS-Apriori, FP-Growth, Max-Subpattern, and PPA. Thus, we have carried out such comparisons and then evaluated our algorithm empirically using two datasets, showcasing its ability to handle different types of periodicity and data distributions. By conducting such a comprehensive comparative analysis, we have demonstrated that our newly proposed algorithm has a smaller complexity than the existing alternatives and speeds up the performance regardless of the size of the dataset. We expect our work to contribute greatly to the mining of astronomical data and the permanently growing online streams derived from social media.

Probabilistic versions of Quantum Private Queries

The no-go theorem regarding unconditionally secure Quantum Bit Commitment protocols is a relevant result in quantum cryptography. Such result has been used to prove the impossibility of unconditional security for other protocols such as Quantum Oblivious Transfer or One-Sided Two Party Computation. In this paper, we formally define two non-deterministic versions of Quantum Private Queries, a protocol addressing the Symmetric-Private Information Retrieval problem. We show that the strongest variant of such scheme is formally equivalent to Quantum Bit Commitment, Quantum Oblivious Transfer and One-Sided Two-Party Computation protocols. This equivalence serves as conclusive evidence of the impracticality of achieving unconditionally secure Strong Probabilistic Quantum Private Queries.

Probabilistic degenerate Stirling polynomials of the second kind and their applications

ABSTRACT The aim of this paper is to further study probabilistic versions of the degenerate Stirling polynomials of the second kind, namely the probabilistic degenerate Stirling polynomials of the second kind associated with Y , which are also degenerate versions of the probabilistic Stirling polynomials of the second kind investigated earlier. Here Y is a random variable whose moment generating function exists in some neighbourhood of the origin. We derive some properties, explicit expressions and certain identities for those polynomials. In addition, we apply our results to the Bernoulli random variable with probability of success p , the normal random variable with parameters μ = 0 , σ 2 = 1 , and to the uniform random variable on ( 0 , 1 ) .

Scoring epidemiological forecasts on transformed scales.

Forecast evaluation is essential for the development of predictive epidemic models and can inform their use for public health decision-making. Common scores to evaluate epidemiological forecasts are the Continuous Ranked Probability Score (CRPS) and the Weighted Interval Score (WIS), which can be seen as measures of the absolute distance between the forecast distribution and the observation. However, applying these scores directly to predicted and observed incidence counts may not be the most appropriate due to the exponential nature of epidemic processes and the varying magnitudes of observed values across space and time. In this paper, we argue that transforming counts before applying scores such as the CRPS or WIS can effectively mitigate these difficulties and yield epidemiologically meaningful and easily interpretable results. Using the CRPS on log-transformed values as an example, we list three attractive properties: Firstly, it can be interpreted as a probabilistic version of a relative error. Secondly, it reflects how well models predicted the time-varying epidemic growth rate. And lastly, using arguments on variance-stabilizing transformations, it can be shown that under the assumption of a quadratic mean-variance relationship, the logarithmic transformation leads to expected CRPS values which are independent of the order of magnitude of the predicted quantity. Applying a transformation of log(x + 1) to data and forecasts from the European COVID-19 Forecast Hub, we find that it changes model rankings regardless of stratification by forecast date, location or target types. Situations in which models missed the beginning of upward swings are more strongly emphasised while failing to predict a downturn following a peak is less severely penalised when scoring transformed forecasts as opposed to untransformed ones. We conclude that appropriate transformations, of which the natural logarithm is only one particularly attractive option, should be considered when assessing the performance of different models in the context of infectious disease incidence.

Parameter Estimation of KST-IRT Model under Local Dependence

A mantra often repeated in the introductory material to psychometrics and Item Response Theory (IRT) is that a Rasch model is a probabilistic version of a Guttman scale. The idea comes from the observation that a sigmoidal item response function provides a probabilistic version of the characteristic function that models an item response in the Guttman scale. It appears, however, more difficult to reconcile the assumption of local independence, which traditionally accompanies the Rasch model, with the item dependence existing in a Guttman scale. In recent work, an alternative probabilistic version of a Guttman scale was proposed, combining Knowledge Space Theory (KST) with IRT modeling, here referred to as KST-IRT. The present work has, therefore, a two-fold aim. Firstly, the estimation of the parameters involved in KST-IRT models is discussed. More in detail, two estimation methods based on the Expectation Maximization (EM) procedure are suggested, i.e., Marginal Maximum Likelihood (MML) and Gibbs sampling, and are compared on the basis of simulation studies. Secondly, for a Guttman scale, the estimates of the KST-IRT models are compared with those of the traditional combination of the Rasch model plus local independence under the interchange of the data generation processes. Results show that the KST-IRT approach might be more effective in capturing local dependence as it appears to be more robust under misspecification of the data generation process, but it comes with the price of an increased number of parameters.

Almost all Cayley maps are mapical regular representations

Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map  a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.

Single MCMC chain parallelisation on decision trees

AbstractDecision trees (DT) are highly famous in machine learning and usually acquire state-of-the-art performance. Despite that, well-known variants like CART, ID3, random forest, and boosted trees miss a probabilistic version that encodes prior assumptions about tree structures and shares statistical strength between node parameters. Existing work on Bayesian DT depends on Markov Chain Monte Carlo (MCMC), which can be computationally slow, especially on high dimensional data and expensive proposals. In this study, we propose a method to parallelise a single MCMC DT chain on an average laptop or personal computer that enables us to reduce its run-time through multi-core processing while the results are statistically identical to conventional sequential implementation. We also calculate the theoretical and practical reduction in run time, which can be obtained utilising our method on multi-processor architectures. Experiments showed that we could achieve 18 times faster running time provided that the serial and the parallel implementation are statistically identical.

On the Probabilistic Quantum Error Correction

Probabilistic quantum error correction is an error-correcting procedure which uses postselection to determine if the encoded information was successfully restored. In this work, we deeply analyze probabilistic version of the error-correcting procedure for general noise. We generalized the Knill-Laflamme conditions for probabilistically correctable errors. We show that for some noise channels, we should encode the information into a mixed state to maximize the probability of successful error correction. Finally, we investigate an advantage of the probabilistic error-correcting procedure over the deterministic one. Reducing the probability of successful error correction allows for correcting errors generated by a broader class of noise channels. Significantly, if the errors are caused by a unitary interaction with an auxiliary qubit system, we can probabilistically restore a qubit state by using only one additional physical qubit.

A generalization of Bernstein–Vazirani algorithm with multiple secret keys and a probabilistic oracle

A probabilistic version of the Bernstein-Vazirani problem (which is a generalization of the original Bernstein-Vazirani problem) and a quantum algorithm to solve it are proposed. The problem involves finding one or more secret keys from a set of multiple secret keys (encoded in binary form) using a quantum oracle. From a set of multiple unknown keys, the proposed quantum algorithm is capable of (a) obtaining any key (with certainty) using a single query to the probabilistic oracle and (b) finding all keys with a high probability (approaching 1 in the limiting case). In contrast, a classical algorithm will be unable to find even a single bit of a secret key with certainty (in the general case). Owing to the probabilistic nature of the oracle, a classical algorithm can only be useful in obtaining limiting probability distributions of $ 0 $ and $ 1 $ for each bit-position of secret keys (based on multiple oracle calls) and this information can further be used to infer some estimates on the distribution of secret keys based on combinatorial considerations. For comparison, it is worth noting that a classical algorithm can be used to exactly solve the original Bernstein-Vazirani problem (involving a deterministic oracle and a single hidden key containing $n$ bits) with a query complexity of $\mathcal{O}(n)$. An interesting class of problems similar to the probabilistic version of the Bernstein-Vazirani problem can be construed, where quantum algorithms can provide efficient solutions with certainty or with a high degree of confidence and classical algorithms would fail to do so.

Approximating the Value of Zero-Sum Differential Games with Linear Payoffs and Dynamics

We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton–Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton–Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes.

Bayesian structural parameter identification from ambient vibration in cultural heritage buildings: The case of the San Jerónimo monastery in Granada, Spain

The deterioration of Cultural Heritage assets caused by the natural hazards is a pressing issue in many countries. Therefore, reliable models based on the large-scale structural response of the assets is key to assess their resilience. However, reliable models such as large and detailed Finite Element (FE) models, require a large number of data and input parameters. This paper proposes a Bayesian learning approach to identify the main parameters of a FE model with quantified uncertainty based on ambient vibration data. As a novelty when compared with other Bayesian structural parameter identification methods from ambient vibration data, here the likelihood function is formulated in a principled way considering information from both frequencies and modes using a probabilistic version of the Modal Assurance Criterion for the modes. This method is embedded into a parameterised computational model to automate the simulation process, and a real case study for a sixteenth century heritage building in Granada (Spain) is presented. The results show the suitability and effectiveness of the proposed Bayesian approach in identifying the most plausible values of the uncertain model parameters in a rigorous probabilistic way, but also in obtaining the modelled frequencies and the modal assurance criterion values with quantified uncertainty.

Lower tails via relative entropy

We show that the naive mean-field approximation correctly predicts the leading term of the logarithmic lower tail probabilities for the number of copies of a given subgraph in G(n,p) and of arithmetic progressions of a given length in random subsets of the integers in the entire range of densities where the mean-field approximation is viable. Our main technical result provides sufficient conditions on the maximum degrees of a uniform hypergraph H that guarantee that the logarithmic lower tail probabilities for the number of edges, induced by a binomial random subset of the vertices of H, can be well approximated by considering only product distributions. This may be interpreted as a weak, probabilistic version of the hypergraph container lemma that is applicable to all sparser-than-average (and not only independent) sets.

Towards a Conceptual Framework for Conspiracy Theory Theories

ABSTRACT I present a conceptual framework for classifying generalist and particularist approaches to conspiracy theories (CTs). Specifically, I exploit a probabilistic version of the hexagon of opposition which allows for systematically visualising the logical relations among basic philosophical positions concerning CTs. The probabilistic interpretation can also account for positions, which make weaker claims about CTs: e.g. instead of claiming ‘every CT is suspicious’ some theorists might prefer to claim ‘most CTs are suspicious’ and then ask about logical consequences of such claims. Finally, I illustrate the proposed conceptual framework by selected claims about CTs drawn from the CT research literature.

Four Approaches to Supposition

Suppositions can be introduced in either the indicative or subjunctive mood. There are also two types of judgments that may be initiated by suppositions of either sort: qualitative (binary) judgments and quantitative (numerical) ones. The former are judgments about whether a given proposition is acceptable and the latter are about how acceptable it is. We systematically explicate the relationships between canonical representatives of each of the four available types of theories. For the qualitative accounts, our representative theories of indicative and subjunctive supposition are based on the belief change operations given by AGM revision and KM update respectively; for the quantitative ones, we consider those given by conditionalization and imaging. This choice is motivated by the familiar approach of understanding supposition as ‘provisional belief revision’ wherein one temporarily treats the supposition as true and forms judgments by making appropriate changes to their other opinions. To compare the numerical judgments recommended by the quantitative theories with the binary ones recommended by the qualitative accounts, we appeal to a suitably adapted version of the Lockean thesis. Ultimately, we establish a number of new results that we interpret as vindicating the often-repeated claim that conditionalization is a probabilistic version of revision, while imaging is a probabilistic version of update.