Maximum Marginal Likelihood Estimation

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11 papers1 paper in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Oct 8, 2026

Feature Space Adaptation for Effortless Gaussian Process Flows

Outside the linear-Gaussian regime, conditional sampling from Gaussian processes (GPs) is challenging. Recent methods such as FlowGP (Moss et al., (2026)) can condition on arbitrary non-linear and non-Gaussian statements, but at considerable cost: an expensive iterative and high-dimensional diffusion that requires hand-specified kernel hyperparameters. In this paper, we alleviate two significant drawbacks of FlowGP by (1) introducing kernel approximations that enable scaling to high-resolution domains and (2) proposing a way to obtain the marginal likelihood by measuring the work needed to steer the diffusion towards conditioning statements. We enable, for the first time, hyperparameter optimisation within FlowGP and demonstrate our approach on probabilistic downscaling from areal summary statistics, PDE solution inference on irregular domains, and recovery of sea level anomaly fields from non-Gaussian satellite observations.

Thomas Cowperthwaite, Louis Sharrock, Lachlan Astfalck +1

University of Cambridge · University College London · University of New South Wales +1

Maximum Marginal Likelihood EstimationGaussian ProcessesBayesian InferenceDiffusion SamplingKernel Methods
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Sep 14, 2026

Quenched Ensemble Sampling

Some of the sharpest challenges in sampling from the energy functions of physical systems arise at phase transitions, where the density of states changes abruptly and many sampling algorithms stall. Nested sampling is a particle method that traverses the density of states under a hard energy constraint and is known to be robust to such transitions, but its application in high dimension is limited by the difficulty of sampling under that constraint. In this work we introduce Quenched Ensemble Sampling, which generalises the hard constraint to a family of repulsive potentials at the energy boundary. This preserves the quenched path of monotonically decreasing energy while making the constrained target amenable to scalable gradient-based kernels. We demonstrate on synthetic models of phase transitions that our method estimates the marginal likelihood and draws posterior samples across a first-order transition where popular alternatives such as tempering fail. We apply the procedure to marginal likelihood estimation in Bayesian neural networks, enabling model comparison between network architectures. Finally, in a high-dimensional continuous lattice field theory, we show that this method traverses a first-order transition and estimates the partition function.

David Yallup

Kavli Institute for Cosmology Cambridge Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK

Bayesian InferencePosterior SamplingMaximum Marginal Likelihood EstimationConstrained Sampling
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Jul 21, 2026

ROMS-IMLE: A Minimalist Approach to Competitive Single-Step Generative Modelling

Generative models have undergone many generations of evolution, from VAEs/GANs to diffusion/flow matching. Along the way, the underlying techniques have become more complicated and various beliefs about what drives strong empirical performance have taken hold. Due to the success of diffusion models and flow matching, one of the more common beliefs is the importance of transforming the noise distribution to the data distribution gradually through many small transformations. We ask whether this is truly necessary, and take a minimalist approach to designing a competitive generative model. We start with the bare-bones essentials, namely just a training objective and a model. We purposefully make both simple. For the training objective, we choose Implicit Maximum Likelihood Estimation (IMLE), and eschew more complicated alternatives such as variational inference, adversarial training and numerical integration. For the model, we eschew transformers and instead choose a moderately sized convolutional network. Then we judiciously added elements that are truly essential, which surprisingly do not include iterative denoising. The result is a single-step parameter-efficient generative model that produces high quality samples at fast speed: it achieves an FID of 2.56 on ImageNet 256 and simultaneously attains good precision and recall.

Chirag Vashist, Ke Li

APEX Lab · Simon Fraser University · Simon Fraser University, Amii and CIFAR

Image GenerationOne-Step Generative ModelingMaximum Marginal Likelihood Estimation
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May 27, 2026

Gradient-Flow Optimization as Dynamic Random-Effects Inference: Testing and Early Stopping with Applications to Deep Learning

Gradient-flow optimization is usually viewed as an algorithmic procedure for minimizing empirical loss, with training duration selected by validation or heuristic early stopping rules. We develop a statistical inference framework for gradient-flow training. We show that whenever fitted values evolve through a time-invariant positive semidefinite training operator, the output at each time is equivalent to the best linear unbiased predictor under a corresponding random-effects model. Training time then becomes a variance-component parameter governing variance reallocation from residual noise to structured signal. This turns two training decisions into inferential problems: whether training is needed becomes a variance-component test for signal beyond initialization, and how long to train becomes restricted maximum likelihood (REML) estimation of the training-time variance component. We show that the REML-guided early stopping rule selects the time at which optimized spectral losses become decorrelated from the training-operator eigenvalues. The asymptotic prediction optimality of the REML-guided early stopping time is established for fixed-design in-sample risk and random-design out-of-sample risk. Deep learning models in fixed-kernel gradient regimes provide canonical instantiations for our results. Numerical experiments and a UK Biobank proteomics application show competitive accuracy of the REML-guided early stopping time with reduced reliance on validation splits and repeated checkpoint evaluation.

Minhao Yao, Ruoyu Wang, Xihong Lin +2

Centre for Biomedical Data Science, Duke-NUS Medical School, National University of Singapore · Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, MA, USA · Institute of Natural Sciences, MOE-LSC, School of Mathematical Sciences, CMA-Shanghai, SJTU-Yale2026 Joint Center of Biostatistics and Data Science, Shanghai Jiao Tong University +1

Early StoppingNeural Network Training DynamicsMaximum Marginal Likelihood EstimationGradient Descent Dynamics
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May 14, 2026

One Pass Is Not Enough: Recursive Latent Refinement for Generative Models

Despite remarkable progress, image generation is far from solved. The dominant metric, FID, conflates sample fidelity with mode coverage and is close to being saturated. Yet a model can still exhibit mode collapse while achieving a low FID, since a handful of sharp, near-duplicate images can outscore a model that faithfully covers the full data distribution. We argue that precision and recall are essential complements to FID, and that because FID is already saturated, the more meaningful goal is to improve diversity and coverage. Achieving high recall requires a model that explicitly prioritizes mode coverage, unlike most generative models, which optimize sample fidelity. We introduce RTM, which replaces the single-pass latent mapping in style-based generators with an iterative refinement process, and show that this consistently improves both quality and diversity. Integrated with Implicit Maximum Likelihood Estimation (IMLE), which optimizes mode coverage by design, RTM achieves the highest precision and recall among current state-of-the-art approaches while maintaining competitive FID, with improvements across CIFAR-10, CelebA-HQ at 256x256, and nine few-shot benchmarks. RTM also improves StyleGAN2 and StyleGAN2-ADA on CIFAR-10 and AFHQ-v1 at 512x512, demonstrating that the benefit is not specific to IMLE. Unlike flow-matching baselines that achieve competitive FID at the expense of coverage, recursive refinement improves both quality and diversity simultaneously.

Mehdi Esmaeilzadeh, Alexia Jolicoeur-Martineau, Chirag Vashist +1

1Simon Fraser University · 2Independent

Generative ModelingImage Generation EvaluationIterative RefinementMaximum Marginal Likelihood Estimation
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Apr 8, 2026

Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data

This research considers a scalable inference for spatial data modeled through Gaussian intrinsic conditional autoregressive (ICAR) structures. The classical estimation method, restricted maximum likelihood (REML), requires repeated inversion and factorization of large, sparse precision matrices, which makes this computation costly. To sort this problem out, we propose a variational restricted maximum likelihood (VREML) framework that approximates the intractable marginal likelihood using a Gaussian variational distribution. By constructing an evidence lower bound (ELBO) on the restricted likelihood, we derive a computationally efficient coordinate-ascent algorithm for jointly estimating the spatial random effects and variance components. In this article, we theoretically establish the monotone convergence of ELBO and mathematically exhibit that the variational family is exact under Gaussian ICAR settings, which is an indication of nullifying approximation error at the posterior level. We empirically establish the supremacy of our VREML over MLE and INLA.

Debjoy Thakur

Washington University in St. Louis, USA · Ahmedabad University, India

Maximum Marginal Likelihood EstimationVariational Inference
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Apr 4, 2026

New insights into Elo algorithm for practitioners and statisticians

This work reconciles two perspectives on the Elo ranking that coexist in the literature: the practitioner's view as a heuristic feedback rule, and the statistician's view as online maximum likelihood estimation via stochastic gradient ascent. Both perspectives coincide exactly in the binary case (iff the expected score is the logistic function). However, estimation noise forces a principled decoupling between the model used for ranking and the model used for prediction: the effective scale and home-field advantage parameter must be adjusted to account for the noise. We provide both closed-form corrections and a data-driven identification procedure. For multilevel outcomes, an exact relationship exists when outcome scores are uniformly spaced, but approximations are preferred in general: they account for estimation noise and better fit the data. The decoupled approach substantially outperforms the conventional one that reuses the ranking model for prediction, and serves as a diagnostic of convergence status. Applied to six years of FIFA men's ranking, we find that the ranking had not converged for the vast majority of national teams. The paper is written in a semi-tutorial style accessible to practitioners, with all key results accompanied by closed-form expressions and numerical examples.

Leszek Szczecinski

Maximum Marginal Likelihood EstimationSports Outcome Prediction
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Mar 14, 2026

Implicit Maximum Likelihood Estimation for Real-time Generative Model Predictive Control

Diffusion-based models have recently shown strong performance in trajectory planning, as they are capable of capturing diverse, multimodal distributions of complex behaviors. A key limitation of these models is their slow inference speed, which results from the iterative denoising process. This makes them less suitable for real-time applications such as closed-loop model predictive control (MPC), where plans must be generated quickly and adapted continuously to a changing environment. In this paper, we investigate Implicit Maximum Likelihood Estimation (IMLE) as an alternative generative modeling approach for planning. IMLE offers strong mode coverage while enabling inference that is two orders of magnitude faster, making it particularly well suited for real-time MPC tasks. Our results demonstrate that IMLE achieves competitive performance on standard offline reinforcement learning benchmarks compared to the standard diffusion-based planner, while substantially improving planning speed in both open-loop and closed-loop settings. We further validate IMLE in a closed-loop human navigation scenario, operating in real-time, demonstrating how it enables rapid and adaptive plan generation in dynamic environments. Real-world videos and code are available at https://gmpc-imle.github.io/.

Grayson Lee, Minh Bui, Shuzi Zhou +3

School of Computing Science, Simon Fraser University, Burnaby, BC, Canada.

Maximum Marginal Likelihood EstimationModel Predictive ControlDiffusion-Based Planning
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Mar 10, 2026

From Flow to One Step: Real-Time Multi-Modal Trajectory Policies via Implicit Maximum Likelihood Estimation-based Distribution Distillation

Generative policies based on diffusion and flow matching achieve strong performance in robotic manipulation by modeling multi-modal human demonstrations. However, their reliance on iterative Ordinary Differential Equation (ODE) integration introduces substantial latency, limiting high-frequency closed-loop control. Recent single-step acceleration methods alleviate this overhead but often exhibit distributional collapse, producing averaged trajectories that fail to execute coherent manipulation strategies. We propose a framework that distills a Conditional Flow Matching (CFM) expert into a fast single-step student via Implicit Maximum Likelihood Estimation (IMLE). A bi-directional Chamfer distance provides a set-level objective that promotes both mode coverage and fidelity, enabling preservation of the teacher multi-modal action distribution in a single forward pass. A unified perception encoder further integrates multi-view RGB, depth, point clouds, and proprioception into a geometry-aware representation. The resulting high-frequency control supports real-time receding-horizon re-planning and improved robustness under dynamic disturbances.

Ju Dong, Liding Zhang, Lei Zhang +7

TAMS (Technical Aspects of Multimodal Systems), Department of Informatics, University of Hamburg, Hamburg, Germany. · Technical University of Munich, Germany. · Agile Robots SE, Munich, Germany.

Maximum Marginal Likelihood EstimationRobot Imitation LearningPolicy DistillationEfficient Multimodal InferenceFlow-Based Generative Modeling
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Mar 9, 2026

Momentum SVGD-EM for Accelerated Maximum Marginal Likelihood Estimation

Maximum marginal likelihood estimation (MMLE) can be formulated as the optimization of a free energy functional. From this viewpoint, the Expectation-Maximisation (EM) algorithm admits a natural interpretation as a coordinate descent method over the joint space of model parameters and probability measures. Recently, a significant body of work has adopted this perspective, leading to interacting particle algorithms for MMLE. In this paper, we propose an accelerated version of one such procedure, based on Stein variational gradient descent (SVGD), by introducing Nesterov acceleration in both the parameter updates and in the space of probability measures. The resulting method, termed Momentum SVGD-EM, consistently accelerates convergence in terms of required iterations across various tasks of increasing difficulty, demonstrating effectiveness in both low- and high-dimensional settings.

Adam Rozzio, Rafael Athanasiades, O. Deniz Akyildiz

Department of Mathematics, Ecole Normale Supérieure Paris-Saclay, France · Department of Mathematics, Imperial College London, UK

Maximum Marginal Likelihood EstimationExpectation-MaximizationStein Variational Gradient DescentAccelerated Gradient Methods
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Oct 14, 2025

Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we empirically validate the effectiveness of our method on synthetic and image datasets and demonstrate that using a particle based approach offers significant improvement in computational efficiency.

Joanna Marks, Tim Y. J. Wang, O. Deniz Akyildiz

Department of Mathematics, Imperial College London

Parameter EstimationLatent Variable ModelsEnergy-Based ModelsMaximum Marginal Likelihood Estimation
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