Observed Probability Distribution Research Articles

Uncovering dissipation from coarse observables: A case study of a random walk with unobserved internal states.

Inferring underlying microscopic dynamics from low-dimensional experimental signals is a central problem in physics, chemistry, and biology. As a trade-off between molecular complexity and the low-dimensional nature of experimental data, mesoscopic descriptions such as the Markovian master equation are commonly used. The states in such descriptions usually include multiple microscopic states, and the ensuing coarse-grained dynamics are generally non-Markovian. It is frequently assumed that such dynamics can nevertheless be described as a Markov process because of the timescale separation between slow transitions from one observed coarse state to another and the fast interconversion within such states. Here, we use a simple model of a molecular motor with unobserved internal states to highlight that (1) dissipation estimated from the observed coarse dynamics may significantly underestimate microscopic dissipation even in the presence of timescale separation and even when mesoscopic states do not contain dissipative cycles and (2) timescale separation is not necessarily required for the Markov approximation to give the exact entropy production, provided that certain constraints on the microscopic rates are satisfied. When the Markov approximation is inadequate, we discuss whether including memory effects can improve the estimate. Surprisingly, when we do so in a "model-free" way by computing the Kullback-Leibler divergence between the observed probability distributions of forward trajectories and their time reverses, this leads to poorer estimates of entropy production. Finally, we argue that alternative approaches, such as hidden Markov models, may uncover the dissipative nature of the microscopic dynamics even when the observed coarse trajectories are completely time-reversible.

A Generalized Finite Mixture Model for Asymmetric Probability Distribution Observations

Finite mixture model is a useful probabilistic model for representing the probability distributions of observations. Gaussian mixture model (GMM) is a widely used one whose parameters are always estimated by the famous EM algorithm. But when probability distributions contain asymmetric modes, GMM requires much more constituent components to achieve a satisfied accuracy. Therefore, a generalized mixture finite model is proposed. First, it adopts the derivative [Formula: see text]-PDF which can well represent the asymmetric probability distributions as the probability density. However, the differential operation and solving the likelihood equations are scarcely possible in the M step of EM algorithm for the complex expression of the derivative [Formula: see text]-PDF. Second, a pseudo EM method is proposed to avoid the abovementioned difficulty which finds the pseudo maximum likelihood estimates of the parameters by utilizing the moment matching principle. It more easily estimates the parameters by solving a series of moment matching equations. Finally, four examples are presented to verify that the proposed generalized finite mixture model has advantage on representing the asymmetric probability distributions observations.

The most likely common cause

The common cause principle for two random variables A and B is examined in the case of causal insufficiency, when their common cause C is known to exist, but only the joint probability of A and B is observed. As a result, C cannot be uniquely identified (the latent confounder problem). We show that the generalized maximum likelihood method can be applied to this situation and allows identification of C that is consistent with the common cause principle. It closely relates to the maximum entropy principle. Investigation of the two binary symmetric variables reveals a non-analytic behavior of conditional probabilities reminiscent of a second-order phase transition. This occurs during the transition from correlation to anti-correlation in the observed probability distribution. The relation between the generalized likelihood approach and alternative methods, such as predictive likelihood and minimum common entropy, is discussed. The consideration of the common cause for three observed variables (and one hidden cause) uncovers causal structures that defy representation through directed acyclic graphs with the Markov condition.

Interaction of HIV-1 Gag matrix domain with plasma membrane from coarse-grained molecular dynamics simulations

We performed coarse-grained simulations with explicit solvent to analyse the role of myristoyl in the matrix domain of the HIV-1 protein Gag that interacts with three models of the lipid bilayer. These models were chosen to represent:(a) the inner membrane of the mammalian plasma membrane, (b) the lipid-raft domains, and (c) a generic cholesterol-free bilayer. The inner plasma membrane is the target for the Gag protein when forming the HIV-1 capsid. The other two models permit investigations of the effects of bilayer composition on the interaction between the matrix domain and the lipid bilayer. Our coarse-grained simulations allow sampling over several hundreds of microseconds, in contrast to other atomistic-detail simulations that can only cover a few microseconds. We measured the probability distributions associated with position and orientation of the myristoyl interacting with the plasma membrane models and quantified the orientational order of the myristoyl inserted into the membrane. We determined the Gibbs free energies (potentials of mean force and torque) associated with the observed probability distributions and determined binding energy values of the order of 8 kT , equivalent to ∼ 5 kcal mol − 1 at physiological temperature. Our methods can be applied to other acylated proteins interacting with the plasma membrane.

Is GN-z11 powered by a super-Eddington massive black hole?

Context. Observations of z ∼ 6 quasars powered by supermassive black holes (SMBHs; MBH ∼ 108 − 10 M⊙) challenge our current understanding of early black hole (BH) formation and evolution. The advent of the James Webb Space Telescope (JWST) has enabled the study of massive BHs (MBHs; MBH ∼ 106 − 7 M⊙) up to z ∼ 11, thus bridging the properties of z ∼ 6 quasars to their ancestors. Aims. The JWST spectroscopic observations of GN-z11, a well-known z = 10.6 star-forming galaxy, have been interpreted with the presence of a super-Eddington (Eddington ratio ≡ λEdd ∼ 5.5) accreting MBH. To test this hypothesis, we used a zoom-in cosmological simulation of galaxy formation and BH co-evolution. Methods. We first tested the simulation results against the observed probability distribution function (PDF) of λEdd found in z ∼ 6 quasars. Then, in the simulation we selected the BHs that satisfy the following criteria: (a) 10 < z < 11, (b) MBH > 106 M⊙. Next, we applied the extreme value statistics to the PDF of λEdd resulting from the simulation. Results. We find that the probability of observing a z ∼ 10 − 11 MBH accreting with λEdd ∼ 5.5 in the volume surveyed by JWST is very low (< 0.2%). We compared our predictions with those in the literature, and discussed the main limitations of our work. Conclusions. Our simulation cannot explain the JWST observations of GN-z11. This might be due to: (i) poor resolution and statistics in simulations, (ii) simplistic sub-grid models (e.g. BH accretion and seeding), (iii) uncertainties in the data analysis and interpretation.

Verification and Enforcement of (ϵ, ξ)-Differential Privacy over Finite Steps in Discrete Event Systems

In the realm of data protection strategies, differential privacy ensures that unauthorized entities cannot reconstruct original data from system outputs. This study explores discrete event systems, specifically through probabilistic automata. Central is the protection of state data, particularly the initial state privacy of multiple starting states. We introduce an evaluation criterion to safeguard initial states. Using advanced algorithms, the proposed method counters the probabilistic identification of any state within this collection by adversaries from observed data points. The efficacy is confirmed when the probability distributions of data observations tied to these states converge. If a system’s architecture does not meet state differential privacy demands, we propose an enhanced supervisory control mechanism. This control upholds state differential privacy across all initial states, maintaining operational flexibility within the probabilistic automaton framework. Concluding, a numerical analysis validates the approach’s strength in probabilistic automata and discrete event systems.

Adjacent initial states-based differential privacy for probabilistic labeled Petri nets

Privacy protection has received widespread attention from the community of discrete event systems to protect the sensitive information of users or organizations from being leaked. The existing privacy protection methods cannot protect the state information of probabilistic discrete event systems via repeated observations, which represents the information pertaining to system resource configurations. This work introduces differential privacy into the framework of probabilistic labeled Petri nets to solve the problems pertaining to the initial state protection. For two initial states that are adjacent under a specified measure, a state differential privacy verification method is proposed by determining whether the probability distributions of observations generated from adjacent initial states are similar. An external attacker is unlikely to infer the initial state via repeated observations if the system satisfies state differential privacy for certain adjacent initial states. For a probabilistic labeled Petri net, which does not satisfy state differential privacy, a supervisory control method is proposed for enforcement. A maximally permissive controller can be constructed based on the control specification proposed in this paper. Experimental studies show that the method proposed in the paper can effectively protect the privacy of given adjacent initial states.

Significance estimation for the Kullback-Leibler divergence: the Poissonian case in seismological studies

The Kullback-Leibler divergence, κ, is a widely used measure of the difference between an observed probability distribution and a reference one; κ=0 when the two distributions are equal, but it has no upper limit to help interpret the significance of any other κ value. Using as an example the problem of distinguishing clustering or gaps in the time occurrence of earthquakes from seismicity uniformly distributed in time, a Monte Carlo method for evaluating the significance of a particular κ value is presented, a method that takes into account the number of classes in the distributions and the length of the sample. Application of this method yields a probability according to which the hypothesis of the observed distribution being a realization of the reference one can be discarded or accepted with a quantitative degree of confidence. This method, and two possible reference values, are presented using the Poisson distribution as an example, but they can be used for other reference distributions.

The growth and size of orogenic gold systems: probability and dynamical behaviour

Every nonlinear system grows by increments, and the final probability distributions for components of that system emerge from an amalgamation of these increments. The resulting probability distribution depends on the constraints imposed on each increment by the physical and chemical processes that produce the system. Hence there is the potential that the observed probability distribution can reveal information on these processes. Complex systems that grow by competition between the supply and consumption of energy and mass have growth laws that are cumulative probability distributions for their component parts that reflect such competition. We show that the type of probability distribution is characteristic of the endowment of orogenic gold deposits with the sequence: Weibull → Fréchet → gamma → log normal representative of increasing endowment. Further, the differential entropy of the probability distribution is indicative of the quality of the deposit, with low-quality deposits represented by high entropy and high-quality deposits represented by low or negative entropy. The type of probability distribution gives an indication of the processes that operated to produce the deposit. These conclusions hold for mineralisation as well as for the associated alteration assemblages. We suggest that the probability distribution for the mineralisation or the alteration assemblage gives a good indication of the endowment and quality of a deposit from a single drill hole. KEY POINTS A single drill hole from a deposit can provide information on endowment and organisation. Weibull → Fréchet → gamma → log normal probability distributions are representative of increasing gold endowment. The differential entropies of these distributions characterise the organisation of the system.

Inflation: a Python library for classical and quantum causal compatibility

We introduce Inflation, a Python library for assessing whether an observed probability distribution is compatible with a causal explanation. This is a central problem in both theoretical and applied sciences, which has recently witnessed significant advances from the area of quantum nonlocality, namely, in the development of inflation techniques. Inflation is an extensible toolkit that is capable of solving pure causal compatibility problems and optimization over (relaxations of) sets of compatible correlations in both the classical and quantum paradigms. The library is designed to be modular and with the ability of being ready-to-use, while keeping an easy access to low-level objects for custom modifications.

Исследование устойчивости к аномальным наблюдениям модификаций метода главных компонент

The paper considers the problem of reducing multidimensional correlated indicators. One of the approaches to solving this problem is based on the method of principal components, which makes it possible to compactly describe the vector with correlated coordinates (components) using the principal components vector with uncorrelated coordinates of much smaller dimension, while retaining most of the information about correlation structure of the original vector. On simulated and real data, several modifications of the principal components method were compared differing in the method of evaluating correlation matrix of the observation vector. The work objective is to demonstrate advantages of the robust modifications of the principal components method in cases, where data contained the abnormal values. To compare the considered modifications on the model data, metric was introduced that measured the difference between estimated and true eigenvalues of the initial data correlation matrix. This metric behavior depending on the probability distribution of observations was studied by computer simulation. As the distributions, multivariate distributions with the off-diagonal correlation matrices simulating a polluted sample were selected. Next, a sample of 13 correlated socioeconomic indicators for 85 countries was considered, where 46 abnormal values were identified. The considered modifications of the principal components method chose the same optimal number of principal components equal to three. However, the real data compression quality, which was defined as the share of the initial indicators total variance described by the first three principal components, turned out to be significantly higher for the robust modifications of the principal components method. Results obtained on these real data are in good agreement with conclusions of the computer simulation

Bayesian probability updates using sampling/importance resampling: Applications in nuclear theory

We review an established Bayesian sampling method called sampling/importance resampling and highlight situations in nuclear theory when it can be particularly useful. To this end we both analyse a toy problem and demonstrate realistic applications of importance resampling to infer the posterior distribution for parameters of ΔNNLO interaction model based on chiral effective field theory and to estimate the posterior probability distribution of target observables. The limitation of the method is also showcased in extreme situations where importance resampling breaks.

Discrete-valued time series based on the exponential family with the multidimensional parameter and their probabilistic and statistical analysis.

We propose herein a new parsimonious Markov model for a discrete-valued time series with conditional probability distributions of observations lying in the exponential family with the multidimensional parameter. A family of explicit consistent asymptotically normal statistical estimators is constructed for the parameters of the proposed model for increasing length of observed time series, and asymptotically effective estimator is found within this constructed family. The obtained results can be used for robust statistical analysis of discrete-valued time series,and for statistical analysis of discrete-valued spatio-temporal data and random fields.

Quantum Bell inequalities from Information Causality – tight for Macroscopic Locality

In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set of quantum correlations in Bell scenarios where the number of settings or outcomes can be arbitrary. We derive these inequalities from the principle of Information Causality, and thus, we do not assume the formalism of quantum mechanics. Moreover, we identify a subspace in the correlation space for which the derived inequalities give the necessary and sufficient conditions for the principle of Macroscopic Locality. As a result, we show that in this subspace, the principle of Information Causality is strictly stronger than the principle of Macroscopic Locality.

Correlation of wind waves and sea level variations on the coast of the seasonally ice-covered Gulf of Finland

Abstract. Both sea level variations and wind-generated waves affect coastal flooding risks. The correlation of these two phenomena complicates the estimates of their joint effect on the exceedance levels for the continuous water mass. In the northern Baltic Sea the seasonal occurrence of sea ice further influences the situation. We analysed this correlation with 28 years (1992–2019) of sea level data, and 4 years (2016–2019) of wave buoy measurements from a coastal location outside the City of Helsinki, Gulf of Finland in the Baltic Sea. The wave observations were complemented by 28 years of simulations with a parametric wave model. The sea levels and significant wave heights at this location show the strongest positive correlation (τ=0.5) for southwesterly winds, while for northeasterly winds the correlation is negative (−0.3). The results were qualitatively similar when only the open water period was considered, or when the ice season was included either with zero wave heights or hypothetical no-ice wave heights. We calculated the observed probability distribution of the sum of the sea level and the highest individual wave crest from the simultaneous time series. Compared to this, a probability distribution of the sum calculated by assuming that the two variables are independent underestimates the exceedance frequencies of high total water levels. We tested nine different copulas for their ability to account for the mutual dependence between the two variables.

Detecting the periodicity of highly irregularly sampled light curves with Gaussian processes: the case of SDSS J025214.67−002813.7

ABSTRACT Based on a 20-yr-long multiband observation of its light curve, it was conjectured that the quasar SDSS J025214.67−002813.7 has a periodicity of ∼4.4 yr. These observations were acquired at a highly irregular sampling rate and feature long intervals of missing data. In this setting, the inference over the light curve’s spectral content requires, in addition to classic Fourier methods, a proper model of the probability distribution of the missing observations. In this article, we address the detection of the periodicity of a light curve from partial and irregularly sampled observations using Gaussian processes, a Bayesian non-parametric model for time series. This methodology allows us to evaluate the veracity of the claimed periodicity of the above-mentioned quasar and also to estimate its power spectral density. Our main contribution is the confirmation that considering periodic component definitely improves the modelling of the data, although being the source originally selected by a large sample of objects, the possibility that this is a chance result cannot be ruled out.

УЧЕТ РИСКОВ ПРИНЯТИЯ РЕШЕНИЙ ПРИ МОДЕЛИРОВАНИИ ВОЗДЕЙСТВИЙ ОКРУЖАЮЩЕЙ СРЕДЫ НА ЭКОНОМИЧЕСКИЕ ОБЪЕКТЫ

Risk analysis of decision-making serves as the basis for ensuring industrial and environmental safety and achieving economic efficiency in the process of economic activity. It is shown that modeling of environmental impacts on the complex systems under study, involving risk analysis in the form of probabilities of decision-making errors, is one of the conditions for achieving economic efficiency and safety. The environmental impact model, which takes into account the probabilities of errors of the first and second kind, is defined by an infinite-dimensional matrix stochastic operator. It is shown that under the influence of the matrix operator, the observed probability distributions of discrete random variables are transformed to a bimodal form. The conformity of the proposed model to the principles of structural harmony is investigated. An example of effective pricing taking into account the environmental impact model is given.

Learning Quantities of Interest from dynamical systems for observation-consistent inversion

Dynamical systems arise in a wide variety of mathematical models from the physical, engineering, life, and social sciences. A common challenge is to quantify uncertainties on model inputs (i.e., parameters) that correspond to a quantitative characterization of uncertainties on observable Quantities of Interest (QoI). To this end, we consider a stochastic inverse problem (SIP) with a solution described by a pullback probability measure. This is referred to as an observation-consistent solution since its subsequent push-forward through the QoI map matches the observed probability distribution on model outputs. A distinction is made between QoI useful for solving the SIP and arbitrary model output data. In dynamical systems, model output data are often given as a series of state variable responses recorded over a particular time window. Consequently, the dimension of output data can easily exceed O(1E4) or more due to the frequency of observations, and the correct choice or construction of a QoI from this data is not self-evident. We present a new framework, Learning Uncertain Quantities (LUQ), that facilitates the tractable solution of SIPs for dynamical systems. Given ensembles of predicted (simulated) time series and (noisy) observed data, LUQ provides routines for filtering data, learning the underlying dynamics in an unsupervised manner, classifying the observations, and performing feature extraction to learn the QoI map. Subsequently, time series data are transformed into samples coming from the underlying predicted and observed distributions associated with the QoI so that solutions to the SIP are computable. Following the introduction and demonstration of LUQ, numerical results from several SIPs are presented for a variety of dynamical systems arising in the life and physical sciences. In the interest of scientific reproducibility, we provide links to our Python implementation of LUQ, as well as all data and scripts required to reproduce the results in this manuscript.

Evaluation of Weather Research and Forecasting model downscaled rainfall and its variability over India

AbstractLong‐term variations in the frequencies of different rainfall events during summer monsoon season (June through September) are analysed over a period of 38‐years (1980–2017) using the simulations of weather research and forecasting (WRF) model evaluated against the India Meteorological Department (IMD). Further, we have divided the study period into two sub‐periods such as before (1980–1998), and after the mega ENSO event (1999–2017). Our analysis suggests the WRF exhibits good skill in capturing the increased trends in the frequency of heavy and very heavy (extreme) rainfall events during the recent epoch (1999–2017).We have assessed the skill of WRF in capturing the observed probability distributions of daily rain rates over different sub‐regions of India. Our results depict that the downscaled products of WRF captured well the extreme rainfall as observed in IMD, signifying that this dynamically downscaled model with continuous 36‐hr re‐initialization makes it suitable to simulate extreme rainfall events compared to light rainfall event. However, our analysis reports that there is a slight overestimation in WRF rainfall and fails in capturing the light rainfall events compared to IMD. The overestimation of rainfall by WRF possibly be due to increased pressure gradients and enhances the convection in the upper streams and associated rainfall. Our study emphasizes that the variability in summer monsoon rainfall during the recent period is due to the changes in the monsoon circulations, particularly monsoon low‐level jet that leads to alter the moisture budget in terms of moisture convergence and transport. Moreover, the variability in frequencies of rainfall events are analysed for strong La Niña/El Niño years and observed an increase in extremes over major part of India during strong La Niña except over parts of Central India and NEI where strong El Niño dominates; which is well captured by the WRF model.

Supercooled Liquid Water Detection Capabilities from Ka-Band Doppler Profiling Radars: Moment-Based Algorithm Formulation and Assessment

The occurrence of supercooled liquid water in mixed-phase cloud (MPC) affects their cloud microphysical and radiative properties. The prevalence of MPCs in the mid- and high latitudes translates these effects to significant contributions to Earth’s radiative balance and hydrological cycle. The current study develops and assesses a radar-only, moment-based phase partition technique for the demarcation of supercooled liquid water volumes in arctic, MPC conditions. The study utilizes observations from the Ka band profiling radar, the collocated high spectral resolution lidar, and ambient temperature profiles from radio sounding deployments following a statistical analysis of 5.5 years of data (January 2014–May 2019) from the Atmospheric Radiation Measurement observatory at the North Slope of Alaska. The ice/liquid phase partition occurs via a per-pixel, neighborhood-dependent algorithm based on the premise that the partitioning can be deduced by examining the mean values of locally sampled probability distributions of radar-based observables and then compare those against the means of climatologically derived, per-phase probability distributions. Analyzed radar observables include linear depolarization ratio (LDR), spectral width, and vertical gradients of reflectivity factor and radial velocity corrected for vertical air motion. Results highlight that the optimal supercooled liquid water detection skill levels are realized for the radar variable combination of spectral width and reflectivity vertical gradient, suggesting that radar-based polarimetry, in the absence of full LDR spectra, is not as critical as Doppler capabilities. The cloud phase masking technique is proven particularly reliable when applied to cloud tops with an Equitable Threat Score (ETS) of 65%; the detection of embedded supercooled layers remains much more uncertain (ETS = 27%).