Standard Probability Research Articles

From quantum carpets to quantum suprematism—the probability representation of qudit states and hidden correlations

We develop the tomographic-probability picture where quantum states of qudits (spin-j systems, N-level atoms) are described by means of sets of fair classical probability distributions associated with properties of sets of dichotomic classical random variables. We formulate the superposition principle of pure quantum states as nonlinear addition rule of standard probabilities. The invertible map of the density matrices of qudit states onto the probability distributions is illustrated by their geometric representation in terms of triangles and squares called Malevich’s squares. We discuss an analogy of the squares with the quantum carpet picture. New entropic inequalities obtained due to the introduced map of the density matrices onto Malevich’s squares are studied. The correlations in systems without subsystems (hidden correlations) are discussed.

A general scheme for finding the static rate–distortion optimized ordering for the bits of the coefficients of all subbands of an [formula omitted]-level dyadic biorthogonal DWT

The expected distortion decrease per bit attained by conveying the significance and refinement information of transform coefficients is derived taking into consideration their Probability Distribution Function (PDF) and their entropy. A general scheme for finding the static rate–distortion optimized ordering for the bits of the coefficients for all subbands of a generic N-level dyadic biorthogonal Discrete Wavelet Transform (DWT) is given by weighting their distortion decrease per bit using the gain of each decomposition subband. Specific formulation for the Exponential Power Distribution (EPD) family is given and closed formulae are derived for the special cases of the Uniform and Laplace distributions. It is shown that under certain circumstances some refinement information of larger magnitude coefficients should be conveyed before the significant information of the current ones. The results can be applied by both conventional context modeling entropy coding algorithms or by set-partitioning algorithms which use multiple lists to keep track of significant and refinement coefficients. A very fast set-partitioning compression algorithm (Depth Embedded Block Tree — DEBT) has been developed based on variable-depth blocks and trees which uses the results presented here and achieves excellent compression ratios along with many other desired properties, e.g., embedded rate–distortion optimized lossy or lossless stream, quality or resolution scalability, and region of interest, among others. Tests show that the rate–distortion curves of most images show a significant improvement when using the weighting derived by the appropriate Exponential Power Distribution fitting in comparison to using a priori standard Uniform or Laplace Probability Distribution Function when using a non orthogonal Discrete Wavelet Transform.

Turbulence statistics in smooth wall oscillatory boundary layer flow

Turbulence characteristics of an asymmetric oscillatory boundary layer flow are analysed through two-component laser-Doppler measurements carried out in a large oscillatory flow tunnel and direct numerical simulation (DNS). Five different Reynolds numbers, $R_{\unicode[STIX]{x1D6FF}}$, in the range 846–2057 have been investigated experimentally, where $R_{\unicode[STIX]{x1D6FF}}=\tilde{u} _{0max}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}$ with $\tilde{u} _{0max}$ the maximum oscillatory velocity in the irrotational region, $\unicode[STIX]{x1D6FF}$ the Stokes length and $\unicode[STIX]{x1D708}$ the fluid kinematic viscosity. DNS has been carried out for the lowest three $R_{\unicode[STIX]{x1D6FF}}$ equal to 846, 1155 and 1475. Both experimental and numerical results show that the flow statistics increase during accelerating phases of the flow and especially at times of transition to turbulent flow. Once turbulence is fully developed, the near-wall statistics remain almost constant until the late half-cycle, with values close to those reported for steady wall-bounded flows. The higher-order statistics reach large values within a normalized wall distance of approximately $y/\unicode[STIX]{x1D6FF}=0.2$ at phases corresponding to the onset of low-speed streak breaking, because of the intermittency of the velocity fluctuations at these times. In particular, the flatness of the streamwise velocity fluctuations reaches values of the order of ten, while the flatness of the wall-normal velocity fluctuations reaches values of several hundreds. Far from the wall, at locations where the vertical gradient of the streamwise velocity is zero, the skewness is approximately zero and the flatness is approximately equal to 3, representative of a normal distribution. At lower elevations the distribution of the fluctuations deviate substantially from a normal distribution, but are found to be well described by other standard theoretical probability distributions.

Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data

Parameter estimation in HEP experiments often involves Monte Carlo simulation to model the experimental response function. A typical application are forward-folding likelihood analyses with re-weighting, or time-consuming minimization schemes with a new simulation set for each parameter value. Problematically, the finite size of such Monte Carlo samples carries intrinsic uncertainty that can lead to a substantial bias in parameter estimation if it is neglected and the sample size is small. We introduce a probabilistic treatment of this problem by replacing the usual likelihood functions with novel generalized probability distributions that incorporate the finite statistics via suitable marginalization. These new PDFs are analytic, and can be used to replace the Poisson, multinomial, and sample-based unbinned likelihoods, which covers many use cases in high-energy physics. In the limit of infinite statistics, they reduce to the respective standard probability distributions. In the general case of arbitrary Monte Carlo weights, the expressions involve the fourth Lauricella function $F_{D}$ , for which we find a new finite-sum representation in a certain parameter setting. The result also represents an exact form for Carlson’s Dirichlet average $R_{n}$ with $n > 0$ , and thereby an efficient way to calculate the probability generating function of the Dirichlet-multinomial distribution, the extended divided difference of a monomial, or arbitrary moments of univariate B-splines. We demonstrate the bias reduction of our approach with a typical toy Monte Carlo problem, estimating the normalization of a peak in a falling energy spectrum, and compare the results with previously published methods from the literature.

Effectiveness of log-logistic distribution to model water-consumption data

Water consumption varies with time of use, season and socio-economic status of consumers, and is defined as a continuous random variable. Incorporating probabilistic nature in water-consumption modelling will lead to more realistic assessments of performance of water distribution systems. Furthermore, fitting water-consumption patterns into a suitable statistical distribution will assist in determining how often peaks will occur, or the probability of exceeding the peaking factor in a system, for incorporation into design calculations. There are few studies in the literature where the random variations of consumption have been considered. The purpose of this study is to evaluate real water-consumption data from the United Kingdom (UK) and North America and to investigate the possibility of establishing a standard probability distribution function to apply in simulating water consumption in developed countries. Daily water-consumption data for five years (2009–2013) were obtained from water companies in the UK and North America and analysed by fitting into normal, log-normal, log-logistic and Weibull distributions. Statistical modelling was performed using MINITAB version 18 statistical package. The Anderson-Darling goodness-of-fit test was used to show how well the selected statistical distribution fits the water-consumption data.

S88. Classification for non-hypoxic ischemic encephalopathy based on standardized critical care EEG terminology: A pilot study

Though encephalopathy is commonly diagnosed on continuous EEG monitoring, no standardized classification exits. Current classification systems are not clinically validated or tested for inter-reader reliability. We prospectively evaluated an easy to use classification of encephalopathy based on standardized critical care EEG terminology (ACNS-2012) in this pilot study. Patients were diagnosed with encephalopathy using continuous EEG monitoring during a two-week study period. Those with hypoxic ischemic encephalopathy/cardiac arrest were excluded. Encephalopathy was classified as mild, moderate and severe based on standard critical care EEG criteria. Patients with suppression, burst suppression, status epilepticus, acute frequent seizures, lack of reactivity and variability were classified as “severe encephalopathy”. Patients with preserved background and transitions for most of the recording without any main term 1 or main term 2 features were classified as “mild encephalopathy”. Patients with all other features including main term 1 or main term 2 features were classified as “moderate encephalopathy”. The patient’s mental status at the time of EEG recording was assessed using the Glasgow Coma Scale and categorized into three categories of increasing severity (GCS less than 5, GCS 6–10, GCS 10–15). The EEG based classification was then clinically validated using standard probability assessment. A total of 26 patients were evaluated. Mild, moderate and severe encephalopathy was diagnosed 12,10 and 4 patients respectively. The pre-and post-classification probabilities (sensitivity, specificity, positive predictive value (PPV), negative predicitive value (NPV) for the classification system are shown in the table below: - TABLE For mild encephalopathy: Sensitivity 85.7%, Specificity 100%, PPV 100%, NPV 85.7% For moderate encephalopathy: Sensitivity 100%, Specificity 94.1%, PPV 90%, NPV 100% For severe encephalopathy: Sensitivity 100%, Specificity 95.6%, PPV 75%, NPV 100% A standardized classification for the severity of non-hypoxic hypoxic ischemic encephalopathy using standardized critical care EEG terminology (ACNS, 2012) was clinically validated in this pilot study. Further large prospective studies are needed to confirm these results.

Two special subgroups of the universal sofic group

We define a subgroup of the universal sofic group, obtained as the normalizer of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each sofic representation can be conjugated inside this subgroup.

Sequential tests for monitoring methods to detect elevated incidence \u2013 a simulation study

BackgroundCommon cancer monitoring practice is seldom prospective and rather driven by public requests. This study aims to assess the performance of a recently developed prospective cancer monitoring method and the statistical tools used, in particular the sequential probability ratio test in regard to specificity, sensitivity, observation time and heterogeneity of size of the geographical unit.MethodsA simulation study based on a predefined selection of cancer types, geographical unit and time period was set up. Based on the population structure of Lower Saxony the mean number of cases of three diagnoses were randomly assigned to the geographical units during 2008–2012. A two-stage monitoring procedure was then executed considering the standardized incidence ratio and sequential probability ratio test. Scenarios were constructed differing by the simulation of clusters, significance level and test parameter indicating a risk to be elevated.ResultsPerformance strongly depended on the choice of the test parameter. If the expected numbers of cases were low, the significance level was not fully exhausted. Hence, the number of false positives was lower than the chosen significance level suggested, leading to a high specificity. Sensitivity increased with the expected number of cases and the amount of risk and decreased with the size of the geographical unit.ConclusionsThe procedure showed some desirable properties and is ready to use for a few settings but demands adjustments for others. Future work might consider refinements of the geographical structure.Inhomogeneous unit size could be addressed by a flexible choice of the test parameter related to the observation time.

Measure concentration and the weak Pinsker property

Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of $(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than $\varepsilon$. This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms. This paper proves that it does. The proof actually gives a more general result. Firstly, it gives a relative version: any factor map from one ergodic automorphism to another can be enlarged by arbitrarily little entropy to become relatively Bernoulli. Secondly, using some facts about relative orbit equivalence, the analogous result holds for all free and ergodic measure-preserving actions of a countable amenable group. The key to this work is a new result about measure concentration. Suppose now that $\mu$ is a probability measure on a finite product space $A^n$, and endow this space with its Hamming metric. We prove that $\mu$ may be represented as a mixture of other measures in which (i) most of the weight in the mixture is on measures that exhibit a strong kind of concentration, and (ii) the number of summands is bounded in terms of the difference between the Shannon entropy of $\mu$ and the combined Shannon entropies of its marginals.

Modular discovery of monomeric and dimeric transcription factor binding motifs for large data sets.

In some dimeric cases of transcription factor (TF) binding, the specificity of dimeric motifs has been observed to differ notably from what would be expected were the two factors to bind to DNA independently of each other. Current motif discovery methods are unable to learn monomeric and dimeric motifs in modular fashion such that deviations from the expected motif would become explicit and the noise from dimeric occurrences would not corrupt monomeric models. We propose a novel modeling technique and an expectation maximization algorithm, implemented as software tool MODER, for discovering monomeric TF binding motifs and their dimeric combinations. Given training data and seeds for monomeric motifs, the algorithm learns in the same probabilistic framework a mixture model which represents monomeric motifs as standard position-specific probability matrices (PPMs), and dimeric motifs as pairs of monomeric PPMs, with associated orientation and spacing preferences. For dimers the model represents deviations from pure modular model of two independent monomers, thus making co-operative binding effects explicit. MODER can analyze in reasonable time tens of Mbps of training data. We validated the tool on HT-SELEX and ChIP-seq data. Our findings include some TFs whose expected model has palindromic symmetry but the observed model is directional.

Approximations to the Normal Probability Distribution Function using Operators of Continuous-valued Logic

In this study, novel approximation methods to the standard normal probability distribution function are introduced. The techniques presented are founded on applications of certain operators of continuous-valued logic. It is demonstrated here that application of the averaging Dombi conjunction operator to two symmetric Sigmoid fuzzy membership functions results in a function that is identical with Tocher’s approximation to the standard normal probability distribution function. Next, an approximation connected with a unary fuzzy modifier operator is discussed. Namely, the so-called Kappa function is applied for constructing a novel probability distribution function. It is shown here that the asymptotic Kappa function is just the Sigmoid function and the proposed Quasi Logistic probability distribution function can be utilized to approximate the standard normal probability distribution function. It is also explained how the new probability distribution function is connected with the generator function of Dombi operators. The proposed approximation formula is very simple as it has only one constant parameter. It does not include any exponential term, but has a good approximation accuracy and fulfills certain requirements that only a few of the known approximation formulas do.

The Evolutionary Game of Electronic Seal Usage Behaviour Supervision From the Perspective of Credit and Penalty

The illegal use of electronic seals has the potential to harm both the contracting parties and the country’s economy. This paper studies the phylogenetic relationship between government regulatory bodies and enterprise behaviors through the payoff matrix and an evolutionary game model. According to both the equilibrium point and stability of the game’s dynamic replication relationship phase diagram, this paper analyzes the incentive effects of the legal use of management costs and the illegal use of economic penalties, government regulatory costs, and credit ratings on the use of electronic seals. The results show that the steady state does not exist as a time extension of government supervision and standardize the behaviors of electronic seal enterprises. In addition, the selection specification evolutionary path using electronic seals is different based on the different initial conditions. The use of the restricted speed and the evolutionary result not only depends on the choice of the enterprise’s standard initial probability of usage behavior but also on the initial probability of government departmental supervision. This occurs without using any means of supervision and penalty measures under the conditions that all enterprises tend to choose not to regulate the use of the electronic seal. Due to regulation with a low probability of no significant effect of the standard, the need to reach a certain level of supervision to regulate the use of electronic seals was improved. The government simply increasing the penalty is not an effective solution since the electronic seal does not regulate the problematic use of credit rating factors. On the one hand, it reduces the probability of government regulations, and on the other hand, it greatly improves the standard probability of the use of electronic seals. Finally, conclusions are based on the governmental point of view to promote the use of electronic seal specification recommendations.

Probability Theory Plus Noise: Descriptive Estimation and Inferential Judgment.

We describe a computational model of two central aspects of people's probabilistic reasoning: descriptive probability estimation and inferential probability judgment. This model assumes that people's reasoning follows standard frequentist probability theory, but it is subject to random noise. This random noise has a regressive effect in descriptive probability estimation, moving probability estimates away from normative probabilities and toward the center of the probability scale. This random noise has an anti-regressive effect in inferential judgement, however. These regressive and anti-regressive effects explain various reliable and systematic biases seen in people's descriptive probability estimation and inferential probability judgment. This model predicts that these contrary effects will tend to cancel out in tasks that involve both descriptive estimation and inferential judgement, leading to unbiased responses in those tasks. We test this model by applying it to one such task, described by Gallistel etal. ). Participants' median responses in this task were unbiased, agreeing with normative probability theory over the full range of responses. Our model captures the pattern of unbiased responses in this task, while simultaneously explaining systematic biases away from normatively correct probabilities seen in other tasks.

The Snow White problem

The Snow White problem is introduced to demonstrate how learning something of which one could not have learnt the opposite (due to observer selection bias) can change an agent’s probability assignment. This helps us to analyse the Sleeping Beauty problem, which is deconstructed as a combinatorial engine and a subjective wrapper. The combinatorial engine of the problem is analogous to Bertrand’s boxes paradox and can be solved with standard probability theory. The subjective wrapper is clarified using the Snow White problem. Sample spaces for all three problems are presented. The conclusion is that subjectivity plays no irreducible role in solving the Sleeping Beauty problem and that no reference to centered worlds is required to provide the answer.

Cosine Similarity Entropy: Self-Correlation-Based Complexity Analysis of Dynamical Systems

The nonparametric Sample Entropy (SE) estimator has become a standard for the quantification of structural complexity of nonstationary time series, even in critical cases of unfavorable noise levels. The SE has proven very successful for signals that exhibit a certain degree of the underlying structure, but do not obey standard probability distributions, a typical case in real-world scenarios such as with physiological signals. However, the SE estimates structural complexity based on uncertainty rather than on (self) correlation, so that, for reliable estimation, the SE requires long data segments, is sensitive to spikes and erratic peaks in data, and owing to its amplitude dependence it exhibits lack of precision for signals with long-term correlations. To this end, we propose a class of new entropy estimators based on the similarity of embedding vectors, evaluated through the angular distance, the Shannon entropy and the coarse-grained scale. Analysis of the effects of embedding dimension, sample size and tolerance shows that the so introduced Cosine Similarity Entropy (CSE) and the enhanced Multiscale Cosine Similarity Entropy (MCSE) are amplitude-independent and therefore superior to the SE when applied to short time series. Unlike the SE, the CSE is shown to yield valid entropy values over a broad range of embedding dimensions. By evaluating the CSE and the MCSE over a variety of benchmark synthetic signals as well as for real-world data (heart rate variability of three different cardiovascular pathologies), the proposed algorithms are demonstrated to be able to quantify degrees of structural complexity in the context of self-correlation over small to large temporal scales, thus offering physically meaningful interpretations and rigor in the understanding the intrinsic properties of the structural complexity of a system, such as the number of its degrees of freedom.

Quantum Information: What Is It All About?

This paper answers Bell’s question: What does quantum information refer to? It is about quantum properties represented by subspaces of the quantum Hilbert space, or their projectors, to which standard (Kolmogorov) probabilities can be assigned by using a projective decomposition of the identity (PDI or framework) as a quantum sample space. The single framework rule of consistent histories prevents paradoxes or contradictions. When only one framework is employed, classical (Shannon) information theory can be imported unchanged into the quantum domain. A particular case is the macroscopic world of classical physics whose quantum description needs only a single quasiclassical framework. Nontrivial issues unique to quantum information, those with no classical analog, arise when aspects of two or more incompatible frameworks are compared.

Surprising rationality in probability judgment: Assessing two competing models

We describe 4 experiments testing contrasting predictions of two recent models of probability judgment: the quantum probability model (Busemeyer, Pothos, Franco, & Trueblood, 2011) and the probability theory plus noise model (Costello & Watts, 2014, 2016a). Both models assume that people estimate probability using formal processes that follow or subsume standard probability theory. One set of predictions concerned agreement between people’s probability estimates and standard probability theory identities. The quantum probability model predicts people’s estimates should agree with one set of identities, while the probability theory plus noise model predicts a specific pattern of violation of those identities. Experimental results show the specific pattern of violation predicted by the probability theory plus noise model. Another set of predictions concerned the conjunction fallacy, which occurs when people judge the probability of a conjunction P(A∧B) to be greater than one or other constituent probabilities P(A) or P(B), contrary to the requirements of probability theory. In cases where A causes B, the quantum probability model predicts that the conjunction fallacy should only occur for constituent B and not for constituent A: the noise model predicts that the fallacy should occur for both A and B. Experimental results show that the fallacy occurs equally for both, contrary to the quantum probability prediction. These results suggest that people’s probability estimates do not follow quantum probability theory. These results support the idea that people estimate probabilities using mechanisms that follow standard probability theory but are subject to random noise.

A standard PHD filter for joint tracking and classification of maneuvering extended targets using random matrix

This paper presents a novel filter for jointly tracking and classification (JTC) of maneuvering extended targets using the standard probability hypothesis density (PHD) framework. For an extended target, the extended state that describes the target size, shape and orientation is also estimated, in addition to the kinematic state. Assuming that the target size information is known in advance, the presented filter can classify the extended target based on different sizes, instead of based on different kinematic motion modes in point target tracking. By utilizing the known target size information, the presented filter can contribute to a better extended state estimation while classifying, and how a good classification result can improve the estimation is mathematically analyzed. Simulation results show that the presented filter simultaneously provides a superior tracking performance and the correct classification of multiple extended targets, compared to the gamma Gaussian inverse Wishart PHD (GGIW-PHD) filter.

Predictive model for site specific simulation of ground motions based on earthquake scenarios

SummaryA predictive stochastic model is developed based on regression relations that inputs a given earthquake scenario description and outputs seismic ground acceleration time histories at a site of interest. A bimodal parametric non‐stationary Kanai‐Tajimi (K‐T) ground motion model lies at the core of the proposed predictive model. The functional forms that describe the temporal evolution of the K‐T model parameters can effectively represent strong non‐stationarities of the ground motion. Fully non‐stationary ground motion time histories can be generated through the powerful Spectral Representation Method. A Californian subset of the available NGA‐West2 database is used to develop and calibrate the predictive model. Samples of the model parameters are obtained by fitting the K‐T model to the database records, and the resulting marginal distributions of the model parameters are efficiently described by standard probability models. The samples are translated to the standard normal space and linear random‐effect regression models are established relating the transformed normal parameters to the commonly used earthquake scenario defining predictors: moment magnitude Mw, closest‐to‐site distance Rrup, and average shear‐wave velocity at a site of interest. The random‐effect terms in the developed regression models can effectively model the correlation among ground motions of the same earthquake event, in parallel to taking into account the location‐dependent effects of each site. For validation purposes, simulated acceleration time histories based on the proposed predictive model are compared with recorded ground motions. In addition, the median and median plus/minus one standard deviation elastic response spectra of synthetic ground motions, pertaining to a variety of different earthquake scenarios, are compared to the associated response spectra computed by the NGA‐West2 ground motion prediction equations and found to be in excellent agreement.

On the classification problem of free ergodic actions of nonamenable groups

We show that, for any countable discrete nonamenable group Γ, the relations of conjugacy, orbit equivalence, stable orbit equivalence, von Neumann equivalence, and stable von Neumann equivalence of free ergodic pmp actions of Γ on the standard atomless probability space are not Borel. This answers a question of Kechris.