Entropy, Volume 19, Issue 5 (May 2017) – 54 articles

An open question in nonlinear dynamics is the relation between the Kolmogorov entropy and the largest Lyapunov exponent of a given orbit. Both have been shown to have diagnostic capability for phase transitions in thermodynamic systems. For systems with long-range interactions, the choice of boundary plays a critical role and appropriate boundary conditions must be invoked. In this work, we compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact expressions for time evolution of the tangent-space vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov entropy for each system for a given system size. Our approach forms an effective and approximation-free instrument for studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in small as well as large versions of the spatially periodic systems. When a phase transition exists, we find that the largest Lyapunov exponent serves as a precursor of the transition that becomes more pronounced as the system size increases. Full article

Using insights from cybernetics and an information-based understanding of biological systems, a precise, scientifically inspired, definition of free-will is offered and the essential requirements for an agent to possess it in principle are set out. These are: (a) there must be a self to self-determine; (b) there must be a non-zero probability of more than one option being enacted; (c) there must be an internal means of choosing among options (which is not merely random, since randomness is not a choice). For (a) to be fulfilled, the agent of self-determination must be organisationally closed (a “Kantian whole”). For (c) to be fulfilled: (d) options must be generated from an internal model of the self which can calculate future states contingent on possible responses; (e) choosing among these options requires their evaluation using an internally generated goal defined on an objective function representing the overall “master function” of the agent and (f) for “deep free-will”, at least two nested levels of choice and goal (d–e) must be enacted by the agent. The agent must also be able to enact its choice in physical reality. The only systems known to meet all these criteria are living organisms, not just humans, but a wide range of organisms. The main impediment to free-will in present-day artificial robots, is their lack of being a Kantian whole. Consciousness does not seem to be a requirement and the minimum complexity for a free-will system may be quite low and include relatively simple life-forms that are at least able to learn. Full article

This paper presents an application of Approximate Entropy (ApEn) and Sample Entropy (SampEn) in the analysis of heart rhythm, blood pressure and stroke volume for the diagnosis of vasovagal syndrome. The analyzed biosignals were recorded during positive passive tilt tests—HUTT(+). Signal changes and their entropy were compared in three main phases of the test: supine position, tilt, and pre-syncope, with special focus on the latter, which was analyzed in a sliding window of each signal. In some cases, ApEn and SampEn were equally useful for the assessment of signal complexity (p < 0.05 in corresponding calculations). The complexity of the signals was found to decrease in the pre-syncope phase (SampEn (RRI): 1.20–0.34, SampEn (sBP): 1.29–0.57, SampEn (dBP): 1.19–0.48, SampEn (SV): 1.62–0.91). The pattern of the SampEn (SV) decrease differs from the pattern of the SampEn (sBP), SampEn (dBP) and SampEn (RRI) decrease. For all signals, the lowest entropy values in the pre-syncope phase were observed at the moment when loss of consciousness occurred. Full article

This paper presents a coding theorem for linear coding over finite rings, in the setting of the Slepian–Wolf source coding problem. This theorem covers corresponding achievability theorems of Elias (IRE Conv. Rec. 1955, 3, 37–46) and Csiszár (IEEE Trans. Inf. Theory 1982, 28, 585–592) for linear coding over finite fields as special cases. In addition, it is shown that, for any set of finite correlated discrete memoryless sources, there always exists a sequence of linear encoders over some finite non-field rings which achieves the data compression limit, the Slepian–Wolf region. Hence, the optimality problem regarding linear coding over finite non-field rings for data compression is closed with positive confirmation with respect to existence. For application, we address the problem of source coding for computing, where the decoder is interested in recovering a discrete function of the data generated and independently encoded by several correlated i.i.d. random sources. We propose linear coding over finite rings as an alternative solution to this problem. Results in Körner–Marton (IEEE Trans. Inf. Theory 1979, 25, 219–221) and Ahlswede–Han (IEEE Trans. Inf. Theory 1983, 29, 396–411, Theorem 10) are generalized to cases for encoding (pseudo) nomographic functions (over rings). Since a discrete function with a finite domain always admits a nomographic presentation, we conclude that both generalizations universally apply for encoding all discrete functions of finite domains. Based on these, we demonstrate that linear coding over finite rings strictly outperforms its field counterpart in terms of achieving better coding rates and reducing the required alphabet sizes of the encoders for encoding infinitely many discrete functions. Full article

In classical Maxwell electrodynamics, charged particles following deterministic trajectories are described by currents that induce fields, mediating interactions with other particles. Statistical methods are used when needed to treat complex particle and/or field configurations. In Stueckelberg–Horwitz–Piron (SHP) electrodynamics, the classical trajectories are traced out dynamically, through the evolution of a 4D spacetime event $x^{\mu }\left(\tau \right)$ as $\tau$ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time $x^0=ct$ (measured by laboratory clocks) and chronology $\tau$ (the temporal ordering of event occurrence) in order to describe antiparticles and resolve problems of irreversibility such as grandfather paradoxes. Consequently, in SHP theory, the elementary object is not a particle (a 4D curve in spacetime) but rather an event (a single point along the dynamically evolving curve). Following standard deterministic methods in classical relativistic field theory, one is led to Maxwell-like field equations that are $\tau$ -dependent and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width $\lambda$ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As $\lambda$ becomes very large, the photon mass goes to zero and the field equations become $\tau$ -independent Maxwell’s equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which $\lambda$ is larger than any other relevant time scale. Thus, statistical mechanics is a fundamental ingredient in SHP electrodynamics, and its insights are required to give meaning to the concept of a particle. Full article

Entropy methods enable a convenient general approach to providing a probability distribution with partial information. The minimum cross-entropy principle selects the distribution that minimizes the Kullback–Leibler divergence subject to the given constraints. This general principle encompasses a wide variety of distributions, and generalizes other methods that have been proposed independently. There remains, however, some confusion about the breadth of entropy methods in the literature. In particular, the asymmetry of the Kullback–Leibler divergence provides two important special cases when the target distribution is uniform: the maximum entropy method and the maximum log-probability method. This paper compares the performance of both methods under a variety of conditions. We also examine a generalized maximum log-probability method as a further demonstration of the generality of the entropy approach. Full article

According to non-stationary characteristic of the acoustic emission signal of rolling element bearings, a novel fault diagnosis method based on empirical wavelet transform (EWT) and ambiguity correlation classification (ACC) is proposed. In the proposed method, the acoustic emission signal acquired from a one-channel sensor is firstly decomposed using the EWT method, and then the mutual information of decomposed components and the original signal is computed and used to extract the noiseless component in order to obtain the reconstructed signal. Afterwards, the ambiguity correlation classifier, which has the advantages of ambiguity functions in the processing of the non-stationary signal, and the combining of correlation coefficients, is applied. Finally, multiple datasets of reconstructed signals for different operative conditions are fed to the ambiguity correlation classifier for training and testing. The proposed method was verified by experiments, and experimental results have shown that the proposed method can effectively diagnose three different operative conditions of rolling element bearings with higher detection rates than support vector machine and back-propagation (BP) neural network algorithms. Full article

We present a first formal analysis of specific and complete local integration. Complete local integration was previously proposed as a criterion for detecting entities or wholes in distributed dynamical systems. Such entities in turn were conceived to form the basis of a theory of emergence of agents within dynamical systems. Here, we give a more thorough account of the underlying formal measures. The main contribution is the disintegration theorem which reveals a special role of completely locally integrated patterns (what we call ι-entities) within the trajectories they occur in. Apart from proving this theorem we introduce the disintegration hierarchy and its refinement-free version as a way to structure the patterns in a trajectory. Furthermore, we construct the least upper bound and provide a candidate for the greatest lower bound of specific local integration. Finally, we calculate the $\iota$ -entities in small example systems as a first sanity check and find that $\iota$ -entities largely fulfil simple expectations. Full article

Li₂O belongs to the family of antifluorites that show superionic behavior at high temperatures. While some of the superionic characteristics of Li₂O are well-known, the mechanistic details of ionic conduction processes are somewhat nebulous. In this work, we first establish an onset of superionic conduction that is emblematic of a gradual disordering process among the Li ions at a characteristic temperature T_α (~1000 K) using reported neutron diffraction data and atomistic simulations. In the superionic state, the Li ions are observed to portray dynamic disorder by hopping between the tetrahedral lattice sites. We then show that string-like ionic diffusion pathways are established among the Li ions in the superionic state. The diffusivity of these dynamical string-like structures, which have a finite lifetime, shows a remarkable correlation to the bulk diffusivity of the system. Full article

To investigate the intra- and inter-limb muscle coordination mechanism of human hands-and-knees crawling by means of muscle synergy analysis, surface electromyographic (sEMG) signals of 20 human adults were collected bilaterally from 32 limb related muscles during crawling with hands and knees at different speeds. The nonnegative matrix factorization (NMF) algorithm was exerted on each limb to extract muscle synergies. The results showed that intra-limb coordination was relatively stable during human hands-and-knees crawling. Two synergies, one relating to the stance phase and the other relating to the swing phase, could be extracted from each limb during a crawling cycle. Synergy structures during different speeds kept good consistency, but the recruitment levels, durations, and phases of muscle synergies were adjusted to adapt the change of crawling speed. Furthermore, the ipsilateral phase lag (IPL) value which was used to depict the inter-limb coordination changed with crawling speed for most subjects, and subjects using the no-limb-pairing mode at low speed tended to adopt the trot-like mode or pace-like mode at high speed. The research results could be well explained by the two-level central pattern generator (CPG) model consisting of a half-center rhythm generator (RG) and a pattern formation (PF) circuit. This study sheds light on the underlying control mechanism of human crawling. Full article

Face verification for unrestricted faces in the wild is a challenging task. This paper proposes a method based on two deep convolutional neural networks (CNN) for face verification. In this work, we explore using identification signals to supervise one CNN and the combination of semi-verification and identification to train the other one. In order to estimate semi-verification loss at a low computation cost, a circle, which is composed of all faces, is used for selecting face pairs from pairwise samples. In the process of face normalization, we propose using different landmarks of faces to solve the problems caused by poses. In addition, the final face representation is formed by the concatenating feature of each deep CNN after principal component analysis (PCA) reduction. Furthermore, each feature is a combination of multi-scale representations through making use of auxiliary classifiers. For the final verification, we only adopt the face representation of one region and one resolution of a face jointing Joint Bayesian classifier. Experiments show that our method can extract effective face representation with a small training dataset and our algorithm achieves 99.71% verification accuracy on Labeled Faces in the Wild (LFW) dataset. Full article

In this paper we investigate the relationship between the information entropy of the distribution of intraday returns and intraday and daily measures of market risk. Using data on the EUR/JPY exchange rate, we find a negative relationship between entropy and intraday Value-at-Risk, and also between entropy and intraday Expected Shortfall. This relationship is then used to forecast daily Value-at-Risk, using the entropy of the distribution of intraday returns as a predictor. Full article

Sleep is a central activity in human adults and characterizes most of the newborn infant life. During sleep, autonomic control acts to modulate heart rate variability (HRV) and respiration. Mechanisms underlying cardiorespiratory interactions in different sleep states have been studied but are not yet fully understood. Signal processing approaches have focused on cardiorespiratory analysis to elucidate this co-regulation. This manuscript proposes to analyze heart rate (HR), respiratory variability and their interrelationship in newborn infants to characterize cardiorespiratory interactions in different sleep states (active vs. quiet). We are searching for indices that could detect regulation alteration or malfunction, potentially leading to infant distress. We have analyzed inter-beat (RR) interval series and respiration in a population of 151 newborns, and followed up with 33 at 1 month of age. RR interval series were obtained by recognizing peaks of the QRS complex in the electrocardiogram (ECG), corresponding to the ventricles depolarization. Univariate time domain, frequency domain and entropy measures were applied. In addition, Transfer Entropy was considered as a bivariate approach able to quantify the bidirectional information flow from one signal (respiration) to another (RR series). Results confirm the validity of the proposed approach. Overall, HRV is higher in active sleep, while high frequency (HF) power characterizes more quiet sleep. Entropy analysis provides higher indices for SampEn and Quadratic Sample entropy (QSE) in quiet sleep. Transfer Entropy values were higher in quiet sleep and point to a major influence of respiration on the RR series. At 1 month of age, time domain parameters show an increase in HR and a decrease in variability. No entropy differences were found across ages. The parameters employed in this study help to quantify the potential for infants to adapt their cardiorespiratory responses as they mature. Thus, they could be useful as early markers of risk for infant cardiorespiratory vulnerabilities. Full article

This article proposes a methodology for the classification of fractal signals as stationary or nonstationary. The methodology is based on the theoretical behavior of two-parameter wavelet entropy of fractal signals. The wavelet $(q,q^{\prime })$ -entropy is a wavelet-based extension of the $(q,q^{\prime })$ -entropy of Borges and is based on the entropy planes for various q and $q^{\prime }$ ; it is theoretically shown that it constitutes an efficient and effective technique for fractal signal classification. Moreover, the second parameter $q^{\prime }$ provides further analysis flexibility and robustness in the sense that different $(q,q^{\prime })$ pairs can analyze the same phenomena and increase the range of dispersion of entropies. A comparison study against the standard signal summation conversion technique shows that the proposed methodology is not only comparable in accuracy but also more computationally efficient. The application of the proposed methodology to physiological and financial time series is also presented along with the classification of these as stationary or nonstationary. Full article

The change of zero order entropy is studied over different strategies of grammar production rule selection. The two major rules are distinguished: transformations leaving the message size intact and substitution functions changing the message size. Relations for zero order entropy changes were derived for both cases and conditions under which the entropy decreases were described. In this article, several different greedy strategies reducing zero order entropy, as well as message sizes are summarized, and the new strategy MinEnt is proposed. The resulting evolution of the zero order entropy is compared with a strategy of selecting the most frequent digram used in the Re-Pair algorithm. Full article

This study aimed to investigate the fatiguing characteristics of muscle-tendon units (MTUs) within skeletal muscles during static isometric contraction tasks. The deltoid was selected as the target muscle and three head-related static isometric contraction tasks were designed to activate three heads of the deltoid in different modes. Nine male subjects participated in this study. Surface electromyography (SEMG) signals were collected synchronously from the three heads of the deltoid. The performances of five SEMG parameters, including root mean square (RMS), mean power frequency (MPF), the first coefficient of autoregressive model (ARC1), sample entropy (SE) and Higuchi’s fractal dimension (HFD), in quantification of fatigue, were evaluated in terms of sensitivity to variability ratio (SVR) and consistency firstly. Then, the HFD parameter was selected as the fatigue index for further muscle fatigue analysis. The experimental results demonstrated that the three deltoid heads presented different activation modes during three head-related fatiguing contractions. The fatiguing characteristics of the three heads were found to be task-dependent, and the heads kept in a relatively high activation level were more prone to fatigue. In addition, the differences in fatiguing rate between heads increased with the increase in load. The findings of this study can be helpful in better understanding the underlying neuromuscular control strategies of the central nervous system (CNS). Based on the results of this study, the CNS was thought to control the contraction of the deltoid by taking the three heads as functional units, but a certain synergy among heads might also exist to accomplish a contraction task. Full article

We apply the geometric quantization procedure via symplectic groupoids to the setting of epistemically-restricted toy theories formalized by Spekkens (Spekkens, 2016). In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra of the toy theories to the Frobenius algebra of stabilizer quantum mechanics. Full article

Reaction coordinates are vital tools for qualitative and quantitative analysis of molecular processes. They provide a simple picture of reaction progress and essential input for calculations of free energies and rates. Iso-committor surfaces are considered the optimal reaction coordinate. We present an algorithm to compute efficiently a sequence of isocommittor surfaces. These surfaces are considered an optimal reaction coordinate. The algorithm analyzes Milestoning results to determine the committor function. It requires only the transition probabilities between the milestones, and not transition times. We discuss the following numerical examples: (i) a transition in the Mueller potential; (ii) a conformational change of a solvated peptide; and (iii) cholesterol aggregation in membranes. Full article

Stochastic distribution control (SDC) systems are a group of systems where the outputs considered is the measured probability density function (PDF) of the system output whilst subjected to a normal crisp input. The purpose of the active fault tolerant control of such systems is to use the fault estimation information and other measured information to make the output PDF still track the given distribution when the objective PDF is known. However, if the target PDF is unavailable, the PDF tracking operation will be impossible. Minimum entropy control of the system output can be considered as an alternative strategy. The mean represents the center location of the stochastic variable, and it is reasonable that the minimum entropy fault tolerant controller can be designed subjected to mean constraint. In this paper, using the rational square-root B-spline model for the shape control of the system output probability density function (PDF), a nonlinear adaptive observer based fault diagnosis algorithm is proposed to diagnose the fault. Through the controller reconfiguration, the system entropy subjected to mean restriction can still be minimized when fault occurs. An illustrative example is utilized to demonstrate the use of the minimum entropy fault tolerant control algorithms. Full article

This paper describes and proves two important theorems that compose the Law of Large Numbers for the non-Euclidean ${\mathcal{L}}^p$ -means, known to be true for the Euclidean ${\mathcal{L}}^2$ -means: Let the ${\mathcal{L}}^p$ -mean estimator, which constitutes the specific functional that estimates the ${\mathcal{L}}^p$ -mean of N independent and identically distributed random variables; then, (i) the expectation value of the ${\mathcal{L}}^p$ -mean estimator equals the mean of the distributions of the random variables; and (ii) the limit $N\to \infty$ of the ${\mathcal{L}}^p$ -mean estimator also equals the mean of the distributions. Full article

Cauchy principal value is a standard method applied in mathematical applications by which an improper, and possibly divergent, integral is measured in a balanced way around singularities or at infinity. On the other hand, entropy prediction of systems behavior from a thermodynamic perspective commonly involves contour integrals. With the aim of facilitating the calculus of such integrals in this entropic scenario, we revisit the generalization of Cauchy principal value to complex contour integral, formalize its definition and—by using residue theory techniques—provide an useful way to evaluate them. Full article

In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nosé–Hoover equations, the Lü system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of molecular dynamics, chaotic theory and fluid mechanics, respectively. Our main results show that all these considered systems are, in fact, non-integrable in nearly all parameters. Full article

Representing patterns as labeled graphs is becoming increasingly common in the broad field of computational intelligence. Accordingly, a wide repertoire of pattern recognition tools, such as classifiers and knowledge discovery procedures, are nowadays available and tested for various datasets of labeled graphs. However, the design of effective learning procedures operating in the space of labeled graphs is still a challenging problem, especially from the computational complexity viewpoint. In this paper, we present a major improvement of a general-purpose classifier for graphs, which is conceived on an interplay between dissimilarity representation, clustering, information-theoretic techniques, and evolutionary optimization algorithms. The improvement focuses on a specific key subroutine devised to compress the input data. We prove different theorems which are fundamental to the setting of the parameters controlling such a compression operation. We demonstrate the effectiveness of the resulting classifier by benchmarking the developed variants on well-known datasets of labeled graphs, considering as distinct performance indicators the classification accuracy, computing time, and parsimony in terms of structural complexity of the synthesized classification models. The results show state-of-the-art standards in terms of test set accuracy and a considerable speed-up for what concerns the computing time. Full article

Collective organization in matter plays a significant role in its expressed physical properties. Typically, it is detected via an order parameter, appropriately defined for each given system’s observed emergent patterns. Recent developments in information theory, however, suggest quantifying collective organization in a system- and phenomenon-agnostic way: decomposing the system’s thermodynamic entropy density into a localized entropy, that is solely contained in the dynamics at a single location, and a bound entropy, that is stored in space as domains, clusters, excitations, or other emergent structures. As a concrete demonstration, we compute this decomposition and related quantities explicitly for the nearest-neighbor Ising model on the 1D chain, on the Bethe lattice with coordination number $k=3$ , and on the 2D square lattice, illustrating its generality and the functional insights it gives near and away from phase transitions. In particular, we consider the roles that different spin motifs play (in cluster bulk, cluster edges, and the like) and how these affect the dependencies between spins. Full article

In transport planning, one should allow passengers to travel through the complicated transportation scheme with efficient use of different modes of transport. In this paper, we propose the use of a cockroach swarm optimization algorithm for determining paths with the shortest travel time. In our approach, this algorithm has been modified to work with the time-expanded model. Therefore, we present how the algorithm has to be adapted to this model, including correctly creating solutions and defining steps and movement in the search space. By introducing the proposed modifications, we are able to solve journey planning. The results have shown that the performance of our approach, in terms of converging to the best solutions, is satisfactory. Moreover, we have compared our results with Dijkstra’s algorithm and a particle swarm optimization algorithm. Full article

A dynamical estimate is given for the Boltzmann entropy of the Universe, under the simplifying assumptions provided by Newtonian cosmology. We first model the cosmological fluid as the probability fluid of a quantum-mechanical system. Next, following current ideas about the emergence of spacetime, we regard gravitational equipotentials as isoentropic surfaces. Therefore, gravitational entropy is proportional to the vacuum expectation value of the gravitational potential in a certain quantum state describing the matter contents of the Universe. The entropy of the matter sector can also be computed. While providing values of the entropy that turn out to be somewhat higher than existing estimates, our results are in perfect compliance with the upper bound set by the holographic principle. Full article

This paper presents a low probability of interception (LPI) radar waveform design method with a fixed average power constraint based on information theory. The Kullback–Leibler divergence (KLD) between the intercept signal and background noise is presented as a practical metric to evaluate the performance of the adversary intercept receiver in this paper. Through combining it with the radar performance metric, that is, the mutual information (MI), a multi-objective optimization model of LPI waveform design is developed. It is a trade-off between the performance of radar and enemy intercept receiver. After being transformed into a single-objective optimization problem, it can be solved by using an interior point method and a sequential quadratic programming (SQP) method. Simulation results verify the correctness and effectiveness of the proposed LPI radar waveform design method. Full article

In this work, we present a numerical study of the influence of matrix degrading enzyme (MDE) dynamics and haptotaxis on the development of vessel networks in tumor-induced angiogenesis. Avascular tumors produce growth factors that induce nearby blood vessels to emit sprouts formed by endothelial cells. These capillary sprouts advance toward the tumor by chemotaxis (gradients of growth factor) and haptotaxis (adhesion to the tissue matrix outside blood vessels). The motion of the capillaries in this constrained space is modelled by stochastic processes (Langevin equations, branching and merging of sprouts) coupled to continuum equations for concentrations of involved substances. There is a complementary deterministic description in terms of the density of actively moving tips of vessel sprouts. The latter forms a stable soliton-like wave whose motion is influenced by the different taxis mechanisms. We show the delaying effect of haptotaxis on the advance of the angiogenic vessel network by direct numerical simulations of the stochastic process and by a study of the soliton motion. Full article

Asymptotic behavior of qualitative variation statistics, including entropy measures, can be modeled well by normal distributions. In this study, we test the normality of various qualitative variation measures in general. We find that almost all indices tend to normality as the sample size increases, and they are highly correlated. However, for all of these qualitative variation statistics, maximum uncertainty is a serious factor that prevents normality. Among these, we study the properties of two qualitative variation statistics; VarNC and StDev statistics in the case of maximum uncertainty, since these two statistics show lower sampling variability and utilize all sample information. We derive probability distribution functions of these statistics and prove that they are consistent. We also discuss the relationship between VarNC and the normalized form of Tsallis (α = 2) entropy in the case of maximum uncertainty. Full article

In this paper, we provide an entropy inference method that is based on an objective Bayesian approach for upper record values having a two-parameter logistic distribution. We derive the entropy that is based on the i-th upper record value and the joint entropy that is based on the upper record values. Moreover, we examine their properties. For objective Bayesian analysis, we obtain objective priors, namely, the Jeffreys and reference priors, for the unknown parameters of the logistic distribution. The priors are based on upper record values. Then, we develop an entropy inference method that is based on these objective priors. In real data analysis, we assess the quality of the proposed models under the objective priors and compare them with the model under the informative prior. Full article