arXiv subjects

Mohamed Maama

Publications and source records attributed to Mohamed Maama.

13 recordsLinked to original sources

arXiv2026

Score-Based Martingale Posteriors for Deep Neural Networks

In this paper we investigate the efficacy of the score-based martingale posteriors (SMP) (Cui & Walker, 2025; Fong et al., 2023) in the context of modern and large-scale machine learning problems and its potential for meaningful uncertainty quantification. SMPs work with a stochastic gradient ascent-type recursion on the parameter space of stochastic models and construct a martingale on the parameter space. Under simple mathematical assumptions, the recursion can be built so that the parameters form a martingale sequence which possesses a limiting, in time, random variable, the latter of which can be simulated very quickly, in contrast to Monte Carlo-based methods such as Markov chain Monte Carlo. In this expository paper we explore the SMP for inferring the parameters of deep neural networks (DNNs) and, where feasible, compare our results to the state-of-the-art Monte Carlo methods aimed at inferring conventional Bayesian posteriors.

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arXiv2025

Antithetic Multilevel Methods for Elliptic and Hypo-Elliptic Diffusions with Applications

We present a new antithetic multilevel Monte Carlo (MLMC) method for the estimation of expectations with respect to laws of diffusion processes that can be elliptic or hypo-elliptic. In particular, we consider the case where one has to resort to time discretization of the diffusion and numerical simulation of such schemes. Inspired by recent works, we introduce a new MLMC estimator of expectations, which does not require any Lévy area simulation and has a strong error of order 2 and a weak error of order 2. We then show how this approach can be used in the context of the filtering problem associated to partially observed diffusions with discrete time observations. We illustrate that in numerical simulations our new approaches provide efficiency gains for several problems, particularly when the diffusion process is hypo-elliptic, relative to some existing methods.

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arXiv2025

Unbiased Parameter Estimation for Bayesian Inverse Problems

In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a solution of a differential equation and, even if exact solutions are available, an analytical intractability of the marginal likelihood and its associated gradient, which is used for parameter estimation. The focus of this article is to deliver unbiased estimates of the unknown parameters, that is, stochastic estimators that, in expectation, are equal to the maximize of the marginal likelihood, and possess no numerical approximation error. Based upon the ideas of [4] we develop a new approach for unbiased parameter estimation for Bayesian inverse problems. We prove unbiasedness and establish numerically that the associated estimation procedure is faster than the current state-of-the-art methodology for this problem. We demonstrate the performance of our methodology on a range of problems which include a PDE and ODE.

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arXiv2025

Stochastic Dynamics and Probability Analysis for a Generalized Epidemic Model with Environmental Noise

In this paper we consider a stochastic SEIQR (susceptible-exposed-infected-quarantined-recovered) epidemic model with a generalized incidence function. Using the Lyapunov method, we establish the existence and uniqueness of a global positive solution to the model, ensuring that it remains well-defined over time. Through the application of Young's inequality and Chebyshev's inequality, we demonstrate the concepts of stochastic ultimate boundedness and stochastic permanence, providing insights into the long-term behavior of the epidemic dynamics under random perturbations. Furthermore, we derive conditions for stochastic extinction, which describe scenarios where the epidemic may eventually die out, and V-geometric ergodicity, which indicates the rate at which the system's state converges to its equilibrium. Finally, we perform numerical simulations to verify our theoretical results and assess the model's behavior under different parameters.

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arXiv2025

Applied Probability Insights into Nonlinear Epidemic Dynamics with Independent Jumps

This paper focuses on the analysis of a stochastic SAIRS-type epidemic model that explicitly incorporates the roles of asymptomatic and symptomatic infectious individuals in disease transmission dynamics. Asymptomatic carriers, often undetected due to the lack of symptoms, play a crucial role in the spread of many communicable diseases, including COVID-19. Our model also accounts for vaccination and considers the stochastic effects of environmental and population-level randomness using Lévy processes. We begin by demonstrating the existence and uniqueness of a global positive solution to the proposed stochastic system, ensuring the model's mathematical validity. Subsequently, we derive sufficient conditions under which the disease either becomes extinct or persists over time, depending on the parameters and initial conditions. The analysis highlights the influence of random perturbations, asymptomatic transmission, and vaccination strategies on disease dynamics. Finally, we conduct comprehensive numerical simulations to validate the theoretical findings and illustrate the behavior of the model under various scenarios of randomness and parameter settings. These results provide valuable insights into the stochastic dynamics of epidemic outbreaks and inform strategies for disease management and control.

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arXiv2024

Multilevel Monte Carlo for a class of Partially Observed Processes in Neuroscience

In this paper we consider Bayesian parameter inference associated to a class of partially observed stochastic differential equations (SDE) driven by jump processes. Such type of models can be routinely found in applications, of which we focus upon the case of neuroscience. The data are assumed to be observed regularly in time and driven by the SDE model with unknown parameters. In practice the SDE may not have an analytically tractable solution and this leads naturally to a time-discretization. We adapt the multilevel Markov chain Monte Carlo method of [11], which works with a hierarchy of time discretizations and show empirically and theoretically that this is preferable to using one single time discretization. The improvement is in terms of the computational cost needed to obtain a pre-specified numerical error. Our approach is illustrated on models that are found in neuroscience.

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arXiv2024

Parameter Estimation for Partially Observed McKean-Vlasov Diffusions

In this article we consider likelihood-based estimation of static parameters for a class of partially observed McKean-Vlasov (POMV) diffusion process with discrete-time observations over a fixed time interval. In particular, using the framework of [5] we develop a new randomized multilevel Monte Carlo method for estimating the parameters, based upon Markovian stochastic approximation methodology. New Markov chain Monte Carlo algorithms for the POMV model are introduced facilitating the application of [5]. We prove, under assumptions, that the expectation of our estimator is biased, but with expected small and controllable bias. Our approach is implemented on several examples.

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arXiv2024

Modeling of Measurement Error in Financial Returns Data

In this paper we consider the modeling of measurement error for fund returns data. In particular, given access to a time-series of discretely observed log-returns and the associated maximum over the observation period, we develop a stochastic model which models the true log-returns and maximum via a Lévy process and the data as a measurement error there-of. The main technical difficulty of trying to infer this model, for instance Bayesian parameter estimation, is that the joint transition density of the return and maximum is seldom known, nor can it be simulated exactly. Based upon the novel stick breaking representation of [12] we provide an approximation of the model. We develop a Markov chain Monte Carlo (MCMC) algorithm to sample from the Bayesian posterior of the approximated posterior and then extend this to a multilevel MCMC method which can reduce the computational cost to approximate posterior expectations, relative to ordinary MCMC. We implement our methodology on several applications including for real data.

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arXiv2023

Multilevel Particle Filters for a Class of Partially Observed Piecewise Deterministic Markov Processes

In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes (PDMPs). In particular, we assume that an ordinary differential equation (ODE) drives the deterministic element and can only be solved numerically via a time discretization. We develop, based upon the approach in [20], a new particle and multilevel particle filter (MLPF) in order to approximate the filter associated to the discretized ODE. We provide a bound on the mean square error associated to the MLPF which provides guidance on setting the simulation parameter of that algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples.

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arXiv2023

Consensus dynamics and coherence in hierarchical small-world networks

The hierarchical small-world network is a real-world network. It models well the benefit transmission web of the pyramid selling in China and many other countries. In this paper, by applying the spectral graph theory, we study three important aspects of the consensus problem in the hierarchical small-world network: convergence speed, communication time-delay robustness, and network coherence. Firstly, we explicitly determine the Laplacian eigenvalues of the hierarchical small-world network by making use of its treelike structure. Secondly, we find that the consensus algorithm on the hierarchical small-world network converges faster than that on some well-studied sparse networks, but is less robust to time delay. The closed-form of the first-order and the second-order network coherence are also derived. Our result shows that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Therefore, we provide a positive answer to two open questions of Yi \emph{et al}. Finally, we argue that some network structure characteristics, such as large maximum degree, small average path length, and large vertex and edge connectivity, are responsible for the strong robustness with respect to external perturbations.

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arXiv2023

An Improved Unbiased Particle Filter

In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. We assume that, for numerical reasons, one has to time-discretize the diffusion process which typically leads to filtering that is subject to discretization bias. The approach in [16] establishes that when only having access to the time-discretized diffusion it is possible to remove the discretization bias with an estimator of finite variance. We improve on the method in [16] by introducing a modified estimator based on the recent work of [17]. We show that this new estimator is unbiased and has finite variance. Moreover, we conjecture and verify in numerical simulations that substantial gains are obtained. That is, for a given mean square error (MSE) and a particular class of multi-dimensional diffusion, the cost to achieve the said MSE falls.

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arXiv2023

Antithetic Multilevel Particle Filters

In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of \cite{mlpf}. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of \cite{ml_anti}. We show that for a class of diffusion problems, for $ε>0$ given, that the cost to produce a mean square error (MSE) in estimation of the filter, of $\mathcal{O}(ε^2)$ is $\mathcal{O}(ε^{-2}\log(ε)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of \cite{mlpf} has a cost of $\mathcal{O}(ε^{-2.5})$ to achieve the same MSE. We support our theory with numerical results in several examples.

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arXiv2022

Bayesian Parameter Inference for Partially Observed SDEs driven by Fractional Brownian Motion

In this paper we consider Bayesian parameter inference for partially observed fractional Brownian motion (fBM) models. The approach we follow is to time-discretize the hidden process and then to design Markov chain Monte Carlo (MCMC) algorithms to sample from the posterior density on the parameters given data. We rely on a novel representation of the time discretization, which seeks to sample from an approximation of the posterior and then corrects via importance sampling; the approximation reduces the time (in terms of total observation time T) by O(T). This method is extended by using a multilevel MCMC method which can reduce the computational cost to achieve a given mean square error (MSE) versus using a single time discretization. Our methods are illustrated on simulated and real data.

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