Gaussian Process

Gaussian distribution is a common choice when dealing with symmetric data. However, other alternatives must be considered in applications with high tail-weight. One option is the randomization of the scale parameter for the Gaussian distribution, enabling a more flexible model for the tails albeit maintaining symmetry. Although any positive random variable can be used as a random scale parameter, Pareto distribution is a suitable choice in order to increase variance and tail-weight. Therefore, the aim of this work is to study the Pareto randomization of the scale parameter for symmetric distributions, in particular for the Gaussian distribution. Estimation problem is tackled and a simulation study is discussed. Finally, an application concerning the directions chosen by ants after a stimulus is provided. The results reveal that the proposed methodology works well both on simulated and real data.

The monograph provides a rigorous mathematics to the study of mind, inference, and intelligent behaviour — with the same fully explicit, derivation-first approach as the companion MCSE series.
The series opens with Bayesian inference and MCMC (Vol. I), then develops information geometry, probabilistic graphical models, Gaussian processes, neural networks, and deep learning, before moving into cognitive-science-specific territory: the Free Energy Principle and active inference (Friston), Integrated Information Theory (Tononi), neuro-symbolic integration, continual learning, formal semantics, and mathematical models of emotion. The final volume addresses AI alignment, covering inverse reinforcement learning, cooperative AI (Russell's corrigibility/deference framework), RLHF with Bradley-Terry reward modelling, and formal verification of neural policies. The unifying thread is the treatment of cognition — perception, inference, learning, language, consciousness — as mathematically tractable structure, while honestly acknowledging the limits: the symbol grounding problem, Goodhart's Law in reward modelling, and the gap between formal specifications and the irreducible informality of human values.

In this paper we address the problem of learning the structure of a Bayesian network in domains with continuous variables. This task requires a procedure for comparing different candidate structures. In the Bayesian framework, this is done by evaluating the marginal likelihood of the data given a candidate structure. This term can be computed in closed-form for standard paramet ric families (e.g., Gaussians), and can be approx imated, at some computational cost, for some semi-parametric families (e.g., mixtures of Gaus sians). We present a new family of continuous variable probabilistic networks that are based on Gaus sian Process priors. These priors are semi parametric in nature and can learn almost ar bitrary noisy functional relations. Using these priors, we can directly compute marginal like lihoods for structure learning. The resulting method can discover a wide range of functional dependencies in multivariate data. We develop the Bayesian score of Gaussian Process Net works and describe how to learn them from data. We present empirical results on artificial data as well as on real-life domains with non-linear de pendencies.

Forecasting often involves multiple time-series that are hierarchically organized (e.g., sales by geography). In that case, there is a constraint that the bottom level forecasts add-up to the aggregated ones. Common approaches use traditional forecasting methods to predict all levels in the hierarchy and then reconcile the forecasts to satisfy that constraint. We propose a new algorithm that automatically forecasts multiple hierarchically organized time-series. We introduce a combination of additive Gaussian processes (GPs) with a hierarchical piece-wise linear function to estimate, respectively, the stationary and non-stationary components of the time-series. We define a flexible structure of additive GPs generated by each aggregated group in the hierarchy of the data. This formulation aims to capture the nested information in the hierarchy while avoiding overfitting. We extended the piece-wise linear function to be hierarchical by defining hyperparameters shared across related time-series. From our experiments, our algorithm can estimate hundreds of time-series at once. To work at this scale, the estimation of the posterior distributions of the parameters is performed using mean-field approximation. We validate the proposed method in two different real-world datasets showing its competitiveness when compared to the state-of-the-art approaches. In summary, our method simplifies the process of hierarchical forecasting as no reconciliation is required. It is easily adapted to non-Gaussian likelihoods and multiple or non-integer seasonalities. The fact that it is a Bayesian approach makes modeling uncertainty of the forecasts trivial.

The problem of estimating the mean rain-flow fatigue damage in randomly vibrating structures, is considered. The excitations are assumed to be through a vector of mutually correlated, stationary Gaussian loadings. The load effect leading to fatigue damage is considered to be a nonlinear function of the vector of excitation loads and is thus non-Gaussian. Its probabilistic characteristics are however unknown. The fatigue damage is assumed to follow a linear damage accumulation rule. Though exact expressions for the mean fatigue damage are difficult to determine, approximations and bounds for the mean rain-flow fatigue damage can be developed. Computing these quantities require estimating the mean level crossing statistics for the associated non-Gaussian response. For the special case when the load effect can be expressed as quadratic combinations of Gaussian processes, analytical expressions are developed for computing the level crossing statistics. These, in turn, are used to determine approximations and the bounds for the mean fatigue damage. The applicability of the proposed method is demonstrated through a numerical example. With respect to this example, a comparative study on the quality of the bounds and the approximations is carried out viz-a-viz the predictions from existing techniques available in the literature.

B D@ Fq rG p vu @ Fq rd 9E w@ Q8 yx q H@ Fq d q d eE T P 7 4T d 9 p Y d 9q b Q x 0 p P p T # © wd e f g vh 4i jh k th ml nh mo k p rq ns et uk vp ck 3w x h y Cz {s q |t Hp ¥q |i ~} (s y h l |l nh y k ei ~k t H wk ey k vp rq | c s v r} s v k o rh m mp rs e o k vl n h y " &k e vi ji jq nk vt %z rs H mh i ji £ q p rg 3k r vq p j k v j a} (h k vt F y t 4k ez vz vk v rh t Hp Ik o rh q |i k z wk jp &s e p rg vh i jh k rh s e y s i jh m ro rh y ( {h mp 9 h h t #p e s s l |l |s q |t 1 vz { m rs ei ji jq |t v i s v Qp rg vh i Ap rq |l |l k vp rh m l |h o h l k y y l |h z {s eq nt Hp } (h mp rg vs y (q |i vi jh y (p rs k ez z rs x q |} #k vp rh p rg vh q nt Hp rh t i jq p e 1 0 i rk H Hp rg vh i jh k l nh mo rh l { m rs ei ji jh i p rg vh l |h o rh l { Q f g vh £q |t Hp rh t i jq |p 9 #q |i p rg vh t # i jh y #p rs h i Ap rq |} #k vp rh } (s e rh s } (z l |q n k vp rh y ( Ik o rh ¡g vk v k e ¡p rh m rq |i Ap rq n mi l |q |¢ rh Ik o rh z {h m rq |s y ~k et vy ¥ m rh i Ap g vh mq n eg Hp £ £ v} (h ¡ rq n k el h mx k e} (z l nh mi k rh q |o rh t ¤ ¥¥ ¦ F § ¨ v© X ª 3« rh i Ap 6y q |i Ap j rq | v p rq |s et F t s t ¬ A &k v vi ji jq nk et i jh k I® q | h ¯i ( s v r} vl nk Ik o rh ¥z {h m rq |s y Ck et vy k e} (z vl |q p r wy h v p rs e¢ rh i I Ik o rh i rÚ eÙ rÛ Ü 9Ý eÞ 4Ú 9ß rà rà á Ü ãâ eÙ rä cå ae 4à HÛ Ü 9â ç eè ~â ãÞ HÙ é ae eß râ &Û ae Hä aê ë å Ý rÙ 1ì Û ë ë Ù jae vç Ù Ü 9í ~î Há ß rae Hä Û â 9å á ae "â 9Þ rÜ eá ß Hí Þ aâ ãÞ HÙ #ï á â ãÞ HÙ jae eç rß rÜ eí ð â 9á eÝ 9Þ HÛ Ú 9â ãå Ý £ñ ©Ù jae vâ eÜ 9Ù ò ¢ QI R¤ F ¦ ¨ § ¢ QI R¤

In this paper a new model for random loads -the Laplace driven moving average -is presented. The model is second order, non-Gaussian, and strictly stationary. It shares with its Gaussian counterpart the ability to model any spectrum but has additional flexibility to model the skewness and kurtosis of the marginal distribution. Unlike most other non-Gaussian models proposed in the literature, such as the transformed Gaussian or Volterra series models, the new model is no longer derivable from Gaussian processes. In the paper a summary of the properties of the new model is given and its upcrossing intensities are evaluated. Then it is used to estimate fatigue damage both from simulations and in terms of an upper bound that is of particular use for narrowband spectra.

The envelope process is an analytical tool often used to study extremes and wave groups. In an approach to approximate the first passage probability for the underlying response the average number of envelope crossings is used to obtain an upper bound. improved this approximation by accounting for the proportion of empty excursions of the envelope. proposed an accurate approximation of this proportion by using the Slepian model method. This approximation was further

Fatigue damage due to a special class of non-Gaussian broadband loadings is considered. Specifically, the loadings are considered to be non-monotonic transformation of stationary, Gaussian random processes. A linear damage accumulation rule is assumed and the rain-flow cycle counting method is considered. Analytical expressions are developed for estimating the expected rain-flow fatigue damage. Easy to compute approximations, in terms of the upper bounds for the expected rain-flow fatigue damage, have been developed. The proposed method has been demonstrated through a set of illustrative numerical examples. The method developed is of use in estimating the time variant reliability of ageing structures under random dynamic loadings.

This paper presents a rigorous mathematical analysis of optimization algorithms central to deep learning, including Gradient Descent (GD), Stochastic Gradient Descent (SGD), Momentum, Adam, and AMSGrad. We compare and discuss the update rules for each algorithm, delving into their underlying mathematical techniques such as Taylor expansions for approximating loss functions and gradients, and the theory of dynamical systems for understanding acceleration properties. We prove their convergence properties under standard assumptions, including convexity, smoothness (Lipschitz continuity of gradients), and strong convexity. Furthermore, we analyze their rates of convergence for various scenarios, such as O(1/t) for convex and smooth functions in GD, and O(1/ √ t) for stochastic methods in non-convex settings. We also consider the impact of bounded gradients in stochastic settings and the use of Lyapunov functions for proving convergence. Through this analysis, we aim to bridge the gap between theory and practice, offering insights into the design and application of optimization algorithms in deep learning.

Image classification is essential in artificial intelligence, with applications in medical diagnostics, autonomous navigation, and industrial automation. Traditional training methods like stochastic gradient descent (SGD) often suffer from slow convergence and local minima. This research presents a hybrid Particle Swarm Optimization (PSO)-Genetic Algorithm (GA)-Backpropagation framework to enhance neural network training. By integrating AdaGrad and PSO for weight optimization, GA for refinement, and backpropagation for fine-tuning, the model improves performance. Results show a 97.5% accuracy on MNIST, a 5% improvement over Adam, and 40% faster convergence than SGD. This approach enhances efficiency, accuracy, and generalization, making it valuable for high-dimensional AI tasks.

We develop a code length principle which is invariant to the choice of parameterization on the model distributions. An invariant approximation formula for easy computation of the marginal distribution is provided for gaussian likelihood models. We provide invariant estimators of the model parameters and formulate conditions under which these estimators are essentially posteriori unbiased for gaussian models. An upper bound on the coarseness of discretization on the model parameters is deduced. We introduce a discrimination measure between probability distributions and use it to construct probability distributions on model classes. The total code length is shown to equal the NML code length of Rissanen to within an additive constant when choosing Jeffreys prior distribution on the model parameters together with a particular choice of prior distribution on the model classes. Our model selection principle is applied to a gaussian estimation problem for data in a wavelet representation and its performance is tested and compared to alternative wavelet-based estimation methods in numerical experiments.

Gaussian mixture filters model the evolution of the probability density function, inter alia, when the noise on the linear measurements is a gaussian mixture. Such a model is relevant when the measurements are subject to disturbances whose statistcs may vary abruptly, like the GNSS signals in avionics, which are subject to reflections, scintillation, multipath... The naturally resulting probability density of the hidden state given the observations up to the current time is also a Gaussian mixture, and thus many computations of interest can be done by using well known numerical routines. In particular, it enables to use an algorithm which aims at estimating a position such as its associated protection levels are the smallest (such as the radius of the confidence ball is the smallest). Also, since the number of components of the gaussian mixture grows exponentially with time, we use an algorithm to approximate the probability density at each step by a fixed amount of gaussians. This algorithm iteratively merges the components which result in the smallest changes according to a Kullback-Leibler based metric. Finally, we assess this algorithm that approximates the optimal filter by comparing the empirical integrity (obtained through simulations) to the targeted integrity. We also compare this algorithm with a least squares state of the art estimator by measuring the ratio of their protection levels.

Physical phenomena are observed in many fields (science and engineering) and are often studied by time-consuming computer codes. These codes are analyzed with statistical models, often called emulators. In many situations, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all input variables. Our aim is to build a model capable of incorporating both data interpolation and inequality constraints into a Gaussian process emulator. By using a functional decomposition, we propose a finite-dimensional approximation of Gaussian processes such that all conditional simulations satisfy the inequality constraints in the entire domain. The inequality mean and mode (i.e. mean and maximum a posteriori ) of the conditional Gaussian process are calculated and prediction intervals are quantified. To show the performance of the proposed model, some conditional simulations with inequality constraints such as boundedness, monotonicity or convexity conditions in one and two dimensions are given. A simulation study to investigate the efficiency of the method in terms of prediction and uncertainty quantification is included.

Statistical researchers have shown increasing interest in generating truncated multivariate normal distributions. In this paper, we only assume that the acceptance region is convex and we focus on rejection sampling. We propose a new algorithm that outperforms crude rejection method for the simulation of truncated multivariate Gaussian random variables. The proposed algorithm is based on a generalization of Von Neumann's rejection technique which requires the determination of the mode of the truncated multivariate density function. We provide a theoretical upper bound for the ratio of the target probability density function over the proposal probability density function. The simulation results show that the method is especially efficient when the probability of the multivariate normal distribution of being inside the acceptance region is low.

Due to their flexibility Gaussian processes are a well-known Bayesian framework for nonparametric function estimation. Integrating inequality constraints, such as monotonicity, convexity, and boundedness, into Gaussian process models significantly improves prediction accuracy and yields more realistic credible intervals in various real-world data applications. The Gaussian process approximation, originally proposed in is considered. It satisfies interpolation conditions and handles a wide range of inequality constraints everywhere. Our contribution in this paper is threefold. First, we extend this approach to handle noisy observations and multiple, more general convex and non-convex constraints. Second, we propose new basis functions in order to extend the smoothness of sample paths to differentiability of class C p , for any p ≥ 1. Third, we examine its behavior in specific scenarios such as monotonicity with flat regions and boundedness near lower and/or upper bounds. In that case, we show that, unlike the Maximum a posteriori (MAP) estimate, the mean a posteriori (mAP) estimate fails to capture flat regions. To address this issue, we propose incorporating multiple constraints, such as monotonicity with bounded slope constraints. According to the theoretical convergence and based on a variety of numerical experiments, the MAP estimate behaves well and outperforms the mAP estimate in terms of prediction accuracy. The performance of the proposed approach is confirmed through real-world data studies.

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) by adding convex constraints to the problem. Through a sequence of approximating Hilbertian subspaces and a discretized model, we prove that the Maximum a posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper , where it is proved that the optimal smoothing solution is the mean of the posterior distribution.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

This paper collects the contributions which were presented during the session devoted to Gaussian processes at the Journées MAS 2016. First, an introduction to Gaussian processes is provided, and some current research questions are discussed. Then, an application of Gaussian process modeling under linear inequality constraints to financial data is presented. Also, an original procedure for handling large data sets is described. Finally, the case of Gaussian process based iterative optimization is discussed. MAS 2016 dédiée aux processus gaussiens. Tout d'abord, une introduction aux processus gaussiens est donnée, et certaines questions de recherche actuelles sont discutées. Ensuite, une application d'un modèle de processus gaussien sous contraintes de type inégalité, à des données financières, est présentée. Puis, une procédure originale, pour gérer des volumes de données importants, est présentée. Enfin, le cas de l'optimisation séquentielle par processus gaussiens est discuté.

Physical phenomena are observed in many fields (science and engineering) and are often studied by time-consuming computer codes. These codes are analyzed with statistical models, often called emulators. In many situations, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all input variables. Our aim is to build a model capable of incorporating both data interpolation and inequality constraints into a Gaussian process emulator. By using a functional decomposition, we propose a finite-dimensional approximation of Gaussian processes such that all conditional simulations satisfy the inequality constraints in the entire domain. The inequality mean and mode (i.e. mean and maximum a posteriori ) of the conditional Gaussian process are calculated and prediction intervals are quantified. To show the performance of the proposed model, some conditional simulations with inequality constraints such as boundedness, monotonicity or convexity conditions in one and two dimensions are given. A simulation study to investigate the efficiency of the method in terms of prediction and uncertainty quantification is included.

In this paper, we extend the correspondence between Bayes' estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of convex constraints such as boundedness, monotonicity or convexity. In the unconstrained interpolation case, the mean of the posterior distribution of a Gaussian Process (GP) given data interpolation is known to be the optimal interpolation function minimizing the norm in the RKHS associated to the GP. In the constrained case, we prove that the Maximum A Posteriori (MAP) or Mode of the posterior distribution is the optimal constrained interpolation function in the RKHS. So, the general correspondence is achieved with the MAP estimator and not the mean of the posterior distribution. A numerical example is given to illustrate this last result.

We develop the canonical Volterra representation for a self-similar Gaussian process by using the Lamperti transformation of the corresponding stationary Gaussian process, where this latter one admits a canonical integral representation under the assumption of pure non-determinism. We apply the representation obtained to the equivalence in law for self-similar Gaussian processes.

An increasing amount of commercial measurement instruments implementing a wide range of measurement technologies is rapidly becoming available for dimensional and geometric verification. Multiple solutions are often acquired within the shop-floor with the aim of providing alternatives to cover a wider array of measurement needs, thus overcoming the limitations of individual instruments and technologies. In such scenarios, multisensor data fusion aims at going one step further by seeking original and different ways to analyze and combine multiple measurement datasets taken from the same measurand, in order to produce synergistic effects and ultimately obtain overall better measurement results. In this work an original approach to multisensor data fusion is presented, based on the development of Gaussian process models (the technique also known as kriging), starting from point sets acquired from multiple instruments. The approach is illustrated and validated through the application to a simulated test case and two real-life industrial metrology scenarios involving structured light scanners and coordinate measurement machines. The results show that not only the proposed approach allows for obtaining final measurement results whose metrological quality transcends that of the original single-sensor datasets, but also it allows to better characterize metrological performance and potential sources of measurement error originated from within each individual sensor.

This study conducts a systematic comparative analysis of nine principal international seismic codes, focusing on their provisions for non-structural elements and their adequacy for protecting museum artifacts housed within buildings. While these standards share a consistent force-based framework for general non-structural components, they rarely address the distinct vulnerabilities of museum objects, such as low fragility thresholds, irreplaceability, and conservation constraints that often prohibit invasive anchoring. The results indicate that, although fundamental parameters (e.g., component weight and spectral acceleration) are universally included, advanced modifying factors, such as response reduction, resonance, and strength-related terms, are incorporated in only a limited number of codes, with overall parameter coverage ranging from 4 to 11. Even in the most parameter-rich formulations, the added complexity primarily refines inertial force scaling rather than governing artifact-specific stability mechanisms, including sliding, rocking, and overturning. The findings demonstrate that current seismic codes provide limited reliability for protecting museum objects when applied without adaptation. This research highlights the need for mechanism-aware extensions to existing code frameworks. It provides a foundation for the development of museum-specific seismic provisions to safeguard irreplaceable cultural heritage from earthquake hazards.

We propose a Bayesian framework for regression
modeling that uses Pearson Type VII Processes (P7Ps) as an
adaptable generalization of Gaussian Processes (GPs). The
P7P is a scale mixture of Normal and Gamma distributions. It
has flexible heavy-tailed properties that enhance its robustness
against outliers and render it suitable for modelling heavytailed
data. The P7P framework outperforms GP functions
by providing explicit control over tail behavior and variance
modulation via the shape and scale parameters. These factors
dictate the weight of the tails and the overall dispersion
of the process, facilitating a seamless transition between
Gaussian and heavy-tailed regimes, thus improving modelling
flexibility and robustness. We derive the predictive distribution
for new data points and use the Laplace approximation for
efficient posterior inference under the Pearson Type VII (P7)
prior, which ensures scalability while maintaining analytical
tractability. The Pearson Type VII Process for regression
(P7PR) outperforms Gaussian Process for regression (GPR)
in terms of resilience, adaptability and predictive accuracy,
especially in challenging scenarios with heavy-tailed noise
and outliers, as evidenced using comparative analysis based
on Mean Square Error (MSE), Mean Absolute Error (MAE)
and predictive log-likelihood (PLL), using both simulated
and real-world datasets. These results validate the benefits of
heavy-tailed priors in modelling non-Gaussian distributions
and illustrate the resilience of P7PR as a Bayesian substitute
for conventional GPR. Although the proposed P7PR framework
exhibits robustness and strong predictive accuracy, its
computational cost increases remarkably as the size of the
datasets, the complexity of the model, and the posterior space
expand. Consequently, in order to preserve scalability without
compromising performance, practical implementations may
necessitate moderate model sizes or approximate inference
methods.

In space-filling designs, Latin Hypercube Design (LHD) is a common choice of experimental design strategy for computer experiments. Prediction variance describes the error involved with making a prediction using a response surface model. This study provides a theoretical bound on the prediction variance of LHD, which is useful for analyzing the efficiency of space-filling design in computer experiments. The LHDs examined in this study have one-dimensional uniformity such that, for each input variable, its range is divided into the same number of equally-spaced intervals as the number of observations. This is done for factors k = 2-7 within the second-and third-order response surface models. The G-optimality, I-optimality criteria, and Fraction of Design Space (FDS) plots were the methods employed to assess the predictive capabilities and accuracy of these LHDs. The findings revealed that LHDs performed better under third-order models when evaluated using the G-optimality criterion, while using the I-optimality criterion, LHDs performed better in second-order models. The FDS plots further indicate that as the number of factors increases, the prediction errors across the models become approximately similar. The results highlight LHDs' versatility in handling high-dimensional problems which underscores practical interest in LHDs for engineering simulations, reinforcing their role in minimizing prediction variance.

This study conducts a systematic comparative analysis of nine principal international seismic codes, focusing on their provisions for non-structural elements and their adequacy for protecting museum artifacts housed within buildings. While these standards share a consistent force-based framework for general non-structural components, they rarely address the distinct vulnerabilities of museum objects, such as low fragility thresholds, irreplaceability, and conservation constraints that often prohibit invasive anchoring. The results indicate that, although fundamental parameters (e.g., component weight and spectral acceleration) are universally included, advanced modifying factors, such as response reduction, resonance, and strength-related terms, are incorporated in only a limited number of codes, with overall parameter coverage ranging from 4 to 11. Even in the most parameter-rich formulations, the added complexity primarily refines inertial force scaling rather than governing artifact-specific stability mechanisms, including sliding, rocking, and overturning. The findings demonstrate that current seismic codes provide limited reliability for protecting museum objects when applied without adaptation. This research highlights the need for mechanism-aware extensions to existing code frameworks. It provides a foundation for the development of museum-specific seismic provisions to safeguard irreplaceable cultural heritage from earthquake hazards.

Extracting meaningful information from high-dimensional data poses a formidable modeling challenge, particularly when the data is obscured by noise or represented through different modalities. This research proposes a novel non-parametric modeling approach, leveraging the Gaussian process (GP), to characterize high-dimensional data by mapping it to a latent low-dimensional manifold. This model, named the latent discriminative generative decoder (LDGD), employs both the data and associated labels in the manifold discovery process. We derive a Bayesian solution to infer the latent variables, allowing LDGD to effectively capture inherent stochasticity in the data. We demonstrate applications of LDGD on both synthetic and benchmark datasets. Not only does LDGD infer the manifold accurately, but its accuracy in predicting data points' labels surpasses state-ofthe-art approaches. In the development of LDGD, we have incorporated inducing points to reduce the computational complexity of Gaussian processes for large datasets, enabling batch training for enhanced efficient processing and scalability. Additionally, we show that LDGD can robustly infer manifold and precisely predict labels for scenarios in that data size is limited, demonstrating its capability to efficiently characterize high-dimensional data with limited samples. These collective attributes highlight the importance of developing non-parametric modeling approaches to analyze high-dimensional data.

The Linear Model of Co-regionalization (LMC) is a very general multitask gaussian process model for regression or classification. While its expressiveness and conceptual simplicity are appealing, naive implementations have cubic complexity in the product (number of datapoints × number of tasks), making approximations mandatory for most applications. However, recent work has shown that in some settings the latent processes of the model can be decoupled, leading to a complexity that is only linear in the number of said processes. We here extend these results, showing from the most general assumptions that the only condition necessary to an efficient exact computation of the LMC is a mild hypothesis on the noise model. We introduce

Finding the probability that a stochastic system stays in a certain region of its state space over a specified time—a long-standing problem both in computational physics and in applied and theoretical mathematics—is approached through the extended and multivariate Rice formula. In principle, it applies to any smooth process multivariate both in argument and in value given that efficient numerical implementations of the high-dimensional integration are available. The computational method offers an exact integral representation yielding remarkably accurate results and provides an alternative method of computing persistency probability and exponent for a physical system. It can be viewed as an implementation of path integration for a smooth Gaussian process with an arbitrary covariance. Its high accuracy is due to efficient computation of expectations with respect to high-dimensional nearly singular Gaussian distributions. For Gaussian processes, the computations are effective and more...

The classical Ornstein-Uhlenbeck diffusion neuronal model is generalized by inclusion of a time-dependent input whose strength exponentially decreases in time. The behavior of the membrane potential is consequently seen to be modeled by a process whose mean and covariance classify it as Gaussian -Markov. The effect of the input on the neuron's firing characteristics is investigated by comparing the firing probability densities and distributions for such a process with the corresponding ones of the Ornstein -Uhlenbeck model. All numerical results are obtained by implementation of a recently developed computational method.

Holder-extendable options are characterized by two maturity dates, which means the option can be exercised at either the expiration date or the extended maturity date. This paper develops a pricing framework for holder-extendable options which deals with the extended version of a stochastic volatility model with an Ornstein-Uhlenbeck (OU) process. The extended model allows correlation between volatility and asset returns. The method uses Fourier inversion techniques that does not require an initial guess of the characteristic functions. A closed-form pricing formula for holder-extendable options is derived for logarithmic asset price dynamics.

Accurate prediction of the output power of distributed photovoltaic (PV) systems is crucial for achieving efficient renewable energy integration and ensuring stable grid operation. Given that the power output of distributed PV systems is significantly influenced by meteorological factors and exhibits strong randomness and volatility, this study takes a distributed PV system in a region of Guangdong Province as the research object and constructs a PV power generation calculation model based on real meteorological data and system parameters. For the prediction approach, traditional time series methods are first employed as a benchmark for comparison. Subsequently, a GWO-LSTM model is proposed, in which the Grey Wolf Optimizer (GWO) is used to optimize the hyperparameters of the Long Short-Term Memory (LSTM) model. The experimental evaluation employs mean squared error (MSE), root mean squared error (RMSE), and mean absolute error (MAE) as performance metrics. The results indicate that the MSE of the GWO-LSTM model is reduced by approximately 85% compared with traditional methods, while the RMSE and MAE are reduced to around 38% and 33% of those of the traditional methods, respectively. This model demonstrates significantly higher prediction accuracy than conventional time series approaches, verifying the effectiveness and superiority of using GWO to optimize LSTM hyperparameters in distributed PV power forecasting.

We address Problem 7.5 from our companion papers: extend the framework linking fractional Brownian motion, Chebyshev approximation, and the Riemann Hypothesis to the two-dimensional setting. We present two independent strategies. Strategy I (Algebraic): We define a two-dimensional summatory function Fractional Brownian Sheet and Two-Dimensional Mobius Functions

We present a computationally efficient segmentation-restoration method, based on a probabilistic formulation, for the joint estimation of the label map (segmentation) and the parameters of the feature generator models (restoration). Our algorithm computes an estimation of the posterior marginal probability distributions of the label field based on a Gauss Markov Random Measure Field model. Our proposal introduces an explicit entropy control for the estimated posterior marginals, therefore it improves the parameter estimation step. If the model parameters are given, our algorithm computes the posterior marginals as the global minimizers of a quadratic, linearly constrained energy function; therefore, one can compute very efficiently the optimal (Maximizer of the Posterior Marginals or MPM) estimator for multi-class segmentation problems. Moreover, a good estimation of the posterior marginals allows one to compute estimators different from the MPM for restoration problems, denoising and optical flow computation. Experiments demonstrate better performance over other state of the art segmentation approaches. This work was partially supported by CONACYT, Mexico (grants 40722 and 46270).

This paper introduces a Bayesian framework for estimating individualized effects (ITE) in highobservational data. The proposed model integrates flexible regression with based outcome modeling, enabling the estimation of heterogeneous effects while for parameter uncertainty and complex covariate. We evaluate the model extensive simulations and real data analysis using the IHDP. Results show that the approach consistently outperforms traditional methods including BART, Baye ian LASSO, BCF, and CEVAE in terms of predictive , bias reduction, and interval coverage. This highlights the model's effectiveness in robust, causal inferences in settings characterized by high and nonlinear .

We develop a unified master-theorem framework for improper integrals generated by analytic compositions of the form f (α + βe iθx) and, in a complementary Jackson setting, by affine analytic compositions f (α + βx) on the q-lattice. The rational-kernel core is based on a residue-finite-part mechanism consisting of a universal whole-line residue theorem with an explicit correction term at infinity, corresponding half-line formulas for even kernels and the symmetric/antisymmetric combinations S f (x) := f (α + βe iθx) + f (α + βe-iθx), A f (x) := f (α + βe iθx)-f (α + βe-iθx) i , and a Hadamard finite-part treatment of real poles. This framework yields universal P/Q r theorems, finite-rank sampling formulas, tensor-product extensions, and exact rationalkernel applications, including multiscale Cauchy quadratures, logarithmic identities, and double-pole sampling. We then develop four rigorous extensions. First, we prove fractional Cauchy-Bessel master theorems for (x 2 + a 2)-s and multiscale product kernels, showing that the integer-order residue formulas arise as half-integer specializations of a broader Bessel-K transform. Second, we establish a coefficient-multiplier transfer principle leading to exact master-integral formulas for Poisson, heat, wave, Schrödinger, and modified Helmholtz evolutions. Third, we prove a Jackson q-Beta master theorem together with its classical limit and basic-hypergeometric applications. Fourth, we derive modular and q-series bridges for Lambert terms, Eisenstein series, the Dedekind eta function, and Jacobi theta functions through explicit Cauchy-kernel representations. Highlights • One residue-finite-part core unifies and sharpens the six 2022 master theorems. • Fractional Cauchy and product kernels lead to exact Bessel-K master expansions. • Analytic coefficient multipliers generate exact Poisson, heat, wave, Schrödinger, and modified Helmholtz master formulas. • A rigorous Jackson q-Beta master theorem provides a submission-safe q-analog branch.

In statistical signal processing, parametric modeling of non-Gaussian processes experiencing noise interference is a very important research area. The autoregressive moving average (ARMA) model is the most general and important tool of modeling system. This paper develops an algorithm for the selection of the proper ARMA model orders. The proposed technique is based on forming a third order cumulant matrix from the observed data sequence. The observed sequence is modeled as the output of an ARMA system that is excited by an unobservable input, and is corrupted by zero-mean Gaussian additive noise of unknown variance. Examples are given to demonstrate the performance of the proposed algorithm.

Expert assessments are routinely used to inform management and other decision making. However, often these assessments contain considerable biases and uncertainties for which reason they should be calibrated if possible. Moreover, coherently combining multiple expert assessments into one estimate poses a long-standing problem in statistics since modeling expert knowledge is often difficult. Here, we present a hierarchical Bayesian model for expert calibration in a task of estimating a continuous univariate parameter. The model allows experts’ biases to vary as a function of the true value of the parameter and according to the expert’s background. We follow the fully Bayesian approach (the so-called supra-Bayesian approach) and model experts’ bias functions explicitly using hierarchical Gaussian processes. We show how to use calibration data to infer the experts’ observation models with the use of bias functions and to calculate the bias corrected posterior distributions for an unkno...

Expert assessments are routinely used to inform management and other decision making. However, often these assessments contain considerable biases and uncertainties for which reason they should be calibrated if possible. Moreover, coherently combining multiple expert assessments into one estimate poses a long-standing problem in statistics since modeling expert knowledge is often difficult. Here, we present a hierarchical Bayesian model for expert calibration in a task of estimating a continuous univariate parameter. The model allows experts' biases to vary as a function of the true value of the parameter and according to the expert's background. We follow the fully Bayesian approach (the so-called supra-Bayesian approach) and model experts' bias functions explicitly using hierarchical Gaussian processes. We show how to use calibration data to infer the experts' observation models with the use of bias functions and to calculate the bias corrected posterior distributions for an unknown system parameter of interest. We demonstrate and test our model and methods with simulated data and a real case study on data-limited fisheries stock assessment. The case study results show that experts' biases vary with respect to the true system parameter value and that the calibration of the expert assessments improves the inference compared to using uncalibrated expert assessments or a vague uniform guess. Moreover, the bias functions in the real case study show important differences between the reliability of alternative experts. The model and methods presented here can be also straightforwardly applied to other applications than our case study.

We present a maximum-margin sparse Gaussian Process (MM-SGP) for active learning (AL) of classification models for multi-class problems. The proposed model makes novel extensions to a GP by integrating maximummargin constraints into its learning process, aiming to further improve its predictive power while keeping its inherent capability for uncertainty quantification. The MM constraints ensure small "effective size" of the model, which allows MM-SGP to provide good predictive performance by using limited "active" data samples, a critical property for AL. Furthermore, as a Gaussian process model, MM-SGP will output both the predicted class distribution and the predictive variance, both of which are essential for defining a sampling function effective to improve the decision boundaries of a large number of classes simultaneously. Finally, the sparse nature of MM-SGP ensures that it can be efficiently trained by solving a low-rank convex dual problem. Experiment results on both synthetic and real-world datasets show the effectiveness and efficiency of the proposed AL model.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

An off-policy Bayesian nonparameteric approximate reinforcement learning framework, termed as GPQ, that employs a Gaussian Processes (GP) model of the value (Q) function is presented in both the batch and online settings. Sufficient conditions on GP hyperparameter selection are established to guarantee convergence of off-policy GPQ in the batch setting, and theoretical and practical extensions are provided for the online case. Empirical results demonstrate GPQ has competitive learning speeds in addition to its convergence guarantees and its ability to automatically choose its own bases locations.

We present a combination of heuristic and rigorous arguments indicating that both the pure state structure and the overlap structure of realistic spin glasses should be relatively simple: in a large finite volume with coupling-independent boundary conditions, such as periodic, at most a pair of flip-related (or the appropriate number of symmetry-related in the non-Ising case) states appear, and the Parisi overlap distribution correspondingly exhibits at most a pair of δ-functions at ±q EA . This rules out the nonstandard SK picture introduced by us earlier, and when combined with our previous elimination of more standard versions of the mean field picture, argues against the possibility of even limited versions of mean field ordering in realistic spin glasses. If broken spin flip symmetry should occur, this leaves open two main possibilities for ordering in the spin glass phase: the droplet/scaling two-state picture, and the chaotic pairs many-state picture introduced by us earlier. We present scaling arguments which provide a possible physical basis for the latter picture, and discuss possible reasons behind numerical observations of more complicated overlap structures in finite volumes.

A matlleinatical cl~a.racterization of tlle ineinbrane potentid as an instanta.nm~~s rct,urn process in tllc presence of rcfract,orincss is in-~cst~iga~tcd for difftrsion inoclcls of single ncuron's act,ivity. The sta.tist,ical features of the random variable modeling the nuinber of neuronal firings is arlalyzed by including t,he additional assuirlption of the exist,erlce of ncuronal rcfra.ctorincss. Asymptot,ic cxa.ct forinulas for the multiple firing probabili~ies and for tile expecled nuinber of produced firings are finally given.

A mathematical characterization of the membrane potential as an instantaneous return process in the presence of refractoriness is investigated for diffusion models of single neuron’s activity. The statistical features of the random variable modeling the number of neuronal firings is analyzed by including the additional assumption of the existence of neuronal refractoriness. Asymptotic exact formulas for the multiple firing probabilities and for the expected number of produced firings are finally given.