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The maximum likelihood estimation is a general estimation procedure. It is often compared to estimation procedures like the ordinary least squares regression or generalized method of moments, to name a few. We discuss some basics about the maximum likelihood estimation, its advantages and disadvantages, and provide an example application to a gamma distribution function.
The maximum likelihood estimation is a general estimation procedure. It is often compared to estimation procedures like the ordinary least squares regression or generalized method of moments, to name a few. We discuss some basics about the maximum likelihood estimation, its advantages and disadvantages, and provide an example application to a gamma distribution function.
We establish parameter inference for the Poisson canonical polyadic (PCP) tensor model through a latent-variable formulation. Our approach exploits the observation that any random PCP tensor can be derived by marginalizing an unobservable random tensor of one dimension larger. The loglikelihood of this larger dimensional tensor, referred to as the “complete” loglikelihood, is comprised of multiple rank one PCP loglikelihoods. Using this methodology, we first derive maximum likelihood estimators for the PCP model and demonstrate that several existing algorithms for fitting non-negative matrix and tensor factorizations are Expectation-Maximization algorithms. Next, we derive the observed and expected Fisher information matrices for the PCP model. The Fisher information provides us crucial insights into the well-posedness of the tensor model, such as the role that tensor rank plays in identifiability and indeterminacy. For the special case of rank one PCP models, we demonstrate that these results are greatly simplified.
Abstract Bayesian optimal experimental design (BOED) seeks to maximize the expected information gain (EIG) of experiments. This requires a likelihood estimate, which in many settings is intractable. Simulation-based inference (SBI) provides powerful tools for this regime. However, existing work explicitly connecting SBI and BOED is restricted to a single contrastive EIG bound. We show that the EIG admits multiple formulations which can directly leverage modern SBI density estimators, encompassing neural posterior, likelihood, and ratio estimation. Building on this perspective, we define a novel EIG estimator using neural likelihood estimation. Further, we identify optimization as a key bottleneck of gradient based EIG maximization and show that a simple multi-start parallel gradient ascent procedure can substantially improve reliability and performance. With these innovations, our SBI-based BOED methods are able to match or outperform by up to 22% existing state-of-the-art approaches across standard BOED benchmarks.
Abstract We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator (LFE) for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using LFEs. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging.
Neutron imaging systems are important diagnostic tools for characterizing the physics of inertial confinement fusion reactions at the National Ignition Facility (NIF). In particular, neutron images give diagnostic information on the size, symmetry, and shape of the fusion hot spot and surrounding cold fuel. Images are formed via collection of neutron flux from the source using a system of aperture arrays and scintillator-based detectors. Currently, reconstruction of fusion source geometry from the collected neutron images is accomplished by solving a computationally intensive maximum likelihood estimation problem via expectation maximization. In contrast, it is often useful to have simple representations of the overall source geometry that can be computed quickly. In this work, we develop convolutional neural networks (CNNs) to reconstruct the outer contours of simple source geometries. We compare the performance of the CNN for penumbral and pinhole data and provide experimental demonstrations of our methods on both non-noisy and noisy data.
ABSTRACT Cosmological N-body simulations of galaxies operate at the level of ‘star particles’ with a mass resolution on the scale of thousands of solar masses. Turning these simulations into stellar mock catalogues requires ‘upsampling’ the star particles into individual stars following the same phase-space density. In this paper, we introduce two new upsampling methods. First, we describe GalaxyFlow, a sophisticated upsampling method that utilizes normalizing flows to both estimate the stellar phase-space density and sample from it. Secondly, we improve on existing upsamplers based on adaptive kernel density estimation (KDE), using maximum likelihood estimation to fine-tune the bandwidth for such algorithms in a way that improves both the density estimation accuracy and upsampling results. We demonstrate our upsampling techniques on a neighbourhood of the Solar location in two simulated galaxies: Auriga 6 and h277. Both yield smooth stellar distributions that closely resemble the stellar densities seen in the Gaia DR3 catalogue. Furthermore, we introduce a novel multimodel classifier test to compare the accuracy of different upsampling methods quantitatively. This test confirms that GalaxyFlow more accurately estimates the density of the underlying star particles than methods based on KDE, at the cost of being more computationally intensive.
Scientific machine learning (SciML) often operates in ill-conditioned, weakly identifiable regimes due to limited data or indirect observations. In such settings, optimization and inference are highly sensitive to the starting point, making initialization--often under-reported--a consequential degree of freedom. Random initialization is not a neutral default as it induces an implicit prior over candidate solutions and can systematically bias the result, producing large run-to-run variability. Here, we formalize this view by treating initialization as a hidden confounder in SciML and develop a unifying theory for structure-aware initialization via numerical continuation, constructing warm starts from related problem instances. Across representative tasks, including physics-informed neural networks, maximum likelihood estimation, and variational inference, warm starts have been shown to consistently reduce optimization effort and improve reliability.
IceCube has observed a diffuse astrophysical neutrino flux over the energy region from a few TeV to a few PeV. At PeV energies, the spectral shape is not yet well measured due to the low statistics of the data. This analysis probes the gap between 1 and 10 PeV by using high-energy downgoing muon neutrinos. Here, to reject the large atmospheric muon background, two complementary techniques are combined. The first technique selects events with high stochasticity to reject atmospheric muon bundles whose stochastic energy losses are smoothed due to high muon multiplicity. The second technique vetoes atmospheric muons with the IceTop surface array. Using 9 yrs of data, we found two neutrino candidate events in the signal region, consistent with expectation from background, each with relatively high signal probabilities. A joint maximum likelihood estimation is performed using this sample and an independent 9.5-yr sample of tracks to measure the neutrino spectrum. A likelihood ratio test is done to compare the single power-law (SPL) vs SPL+cutoff hypothesis; the SPL+cutoff model is not significantly better than the SPL. High-energy astrophysical objects from four source catalogs are also checked around the direction of the two events. No significant coincidence was found.
Neutron stars provide a unique opportunity to study strongly interacting matter under extreme density conditions. The intricacies of matter inside neutron stars and their equation of state are not directly visible, but determine bulk properties, such as mass and radius, which affect the star's thermal X-ray emissions. However, the telescope spectra of these emissions are also affected by the stellar distance, hydrogen column, and effective surface temperature, which are not always well-constrained. Uncertainties on these nuisance parameters must be accounted for when making a robust estimation of the equation of state. In this study, we develop a novel methodology that, for the first time, can infer the full posterior distribution of both the equation of state and nuisance parameters directly from telescope observations. This method relies on the use of neural likelihood estimation, in which normalizing flows use samples of simulated telescope data to learn the likelihood of the neutron star spectra as a function of these parameters, coupled with Hamiltonian Monte Carlo methods to efficiently sample from the corresponding posterior distribution. Our approach surpasses the accuracy of previous methods, improves the interpretability of the results by providing access to the full posterior distribution, and naturally scales to a growing number of neutron star observations expected in the coming years.
There have been many applications of deep neural networks to detector calibrations and a growing number of studies that propose deep generative models as automated fast detector simulators. We show that these two tasks can be unified by using maximum likelihood estimation (MLE) from conditional generative models for energy regression. Unlike direct regression techniques, the MLE approach is prior independent and non-Gaussian resolutions can be determined from the shape of the likelihood near the maximum. Using an ATLAS-like calorimeter simulation, we demonstrate this concept in the context of calorimeter energy calibration. Published by the American Physical Society 2025
Here, we present the one-dimensional Lyman-α forest power spectrum measurement derived from the data release 1 (DR1) of the Dark Energy Spectroscopic Instrument (DESI). The measurement of the Lyman-α forest power spectrum along the line of sight from high-redshift quasar spectra provides information on the shape of the linear matter power spectrum, neutrino masses, and the properties of dark matter. In this work, we use a Fast Fourier Transform (FFT)-based estimator, which is validated on synthetic data in a companion paper. Compared to the FFT measurement performed on the DESI early data release, we improve the noise characterization with a cross-exposure estimator and test the robustness of our measurement using various data splits. We also refine the estimation of the uncertainties and now present an estimator for the covariance matrix of the measurement. Furthermore, we compare our results to previous high-resolution and eBOSS measurements. In another companion paper, we present the same DR1 measurement using the Quadratic Maximum Likelihood Estimator (QMLE). These two measurements are consistent with each other and constitute the most precise one-dimensional power spectrum measurement to date, while being in good agreement with results from the DESI early data release.
We establish non-asymptotic efficiency guarantees for tensor decomposition–based inference in count data models. Under a Poisson framework, we consider two related goals: (i) parametric inference , the estimation of the full distributional parameter tensor, and (ii) multiway analysis , the recovery of its canonical polyadic (CP) decomposition factors. Our main result shows that in the rank-one setting, a rank-constrained maximum-likelihood estimator achieves multiway analysis with variance matching the Cramér–Rao Lower Bound (CRLB) up to absolute constants and logarithmic factors. This provides a general framework for studying “near-efficient” multiway estimators in finite-sample settings. For higher ranks, we illustrate that our multiway estimator may not attain the CRLB; nevertheless, CP-based parametric inference remains nearly minimax optimal, with error bounds that improve on prior work by offering more favorable dependence on the CP rank. Numerical experiments corroborate near-efficiency in the rank-one case and highlight the efficiency gap in higher-rank scenarios.
The one-dimensional power spectrum P 1D of Lyα forest offers rich insights into cosmological and astrophysical parameters, including constraints on the sum of neutrino masses, warm dark matter models, and the thermal state of the intergalactic medium. We present the measurement of P 1D using the optimal quadratic maximum likelihood estimator applied to over 300,000 Lyα quasars from Data Release 1 (DR1) of the Dark Energy Spectroscopic Instrument (DESI) survey. This sample represents the largest to date for P 1D measurements and is larger than the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) by a factor of 1.7. We conduct a meticulous investigation of instrumental and analysis systematics and quantify their impact on P 1D . This includes the development of a cross-exposure estimator that eliminates the need to model the pipeline noise and has strong potential for future P 1D measurements. We also present new insights into metal contamination through the 1D correlation function. Using a fitting function we measure the evolution of the Lyα forest bias with high precision: b F (z) = (-0.218 ± 0.002) × ((1 + z)/4) 2.96±0.06 . In a companion validation paper, we substantially extend our previous suite of CCD image simulations to quantify the pipeline's exquisite performance accurately. In another companion paper, we present DR1 P 1D measurements using the Fast Fourier Transform (FFT) approach to power spectrum estimation. These two measurements produce a forest bias parameter that differs by 2.2 sigma. However, our model is simplistic, so this disagreement will be investigated in future work.
Abstract We develop a new algorithm to estimate the temperature of a nonneutral plasma in a Penning-Malmberg trap. The algorithm analyzes data obtained by slowly lowering a voltage that confines one end of the plasma and collecting escaping charges, and is a maximum likelihood estimator based on a physically-motivated model of the escape protocol presented in (Beck in Measurement of the magnetic and temperature dependence of the electron-electron anisotropic temperature relaxation rate. PhD thesis, 1990). Significantly, our algorithm may be used on single-count data, allowing for improved fits with low numbers of escaping electrons. This is important for low-temperature plasmas such as those used in antihydrogen trapping. We perform a Monte Carlo simulation of our algorithm, and assess its robustness to intrinsic shot noise and external noise. The assumptions in this paper allow for a lower bound for measurable plasma temperatures of approximately $3\,\mathrm{K}$ 3 K for plasmas of length $1\,\mathrm{cm}$ 1 cm , with approximately 100 particle counts needed for an accuracy of $\pm 10 \%$ ± 10 % .
Accurate particle identification is crucial in any high-energy physics experiment, allowing scientists to understand the unique interactions and mechanisms at play in a detector. In this project, I develop and study a new particle identification (PID) algorithm for the Short-Baseline Near Detector, a likelihood-based approach, different from out current $\chi^2$ method. A likelihood estimation offers a more physically motivated strategy for PID. The distribution random energy losses of charged particles traveling through a medium are described by the Vavilov probability density function. By using this model, we can account for random energy losses and construct likelihood functions specific to each particle type, potentially enabling a more accurate method for PID.
Most of the fundamental, emergent, and phenomenological parameters of particle and nuclear physics are determined through parametric template fits. Simulations are used to populate histograms which are then matched to data. This approach is inherently lossy, since histograms are binned and low-dimensional. Deep learning has enabled unbinned and high-dimensional parameter estimation through neural likelihood(-ratio) estimation. We compare two approaches for neural simulation-based inference (NSBI): one based on discriminative learning (classification) and one based on generative modeling. These two approaches are directly evaluated on the same datasets, with a similar level of hyperparameter optimization in both cases. In addition to a Gaussian dataset, we study NSBI using a Higgs boson dataset from the FAIR Universe Challenge. We find that both the direct likelihood and likelihood ratio estimation are able to effectively extract parameters with reasonable uncertainties. For the numerical examples and within the set of hyperparameters studied, we found that the likelihood ratio method is more accurate and/or precise. Both methods have a significant spread from the network training and would require ensembling or other mitigation strategies in practice.
Sequential Monte Carlo (SMC) samplers are powerful tools for Bayesian inference but suffer from high computational costs due to their reliance on large particle ensembles for accurate estimates. We introduce persistent sampling (PS), an extension of SMC that systematically retains and reuses particles from all prior iterations to construct a growing, weighted ensemble. By leveraging multiple importance sampling and resampling from a mixture of historical distributions, PS mitigates the need for excessively large particle counts, directly addressing key limitations of SMC such as particle impoverishment and mode collapse. Crucially, PS achieves this without additional likelihood evaluations-weights for persistent particles are computed using cached likelihood values. This framework not only yields more accurate posterior approximations but also produces marginal likelihood estimates with significantly lower variance, enhancing reliability in model comparison. Furthermore, the persistent ensemble enables efficient adaptation of transition kernels by leveraging a larger, decorrelated particle pool. Experiments on high-dimensional Gaussian mixtures, hierarchical models, and non-convex targets demonstrate that PS consistently outperforms standard SMC and related variants, including recycled and waste-free SMC, achieving substantial reductions in mean squared error for posterior expectations and evidence estimates, all at reduced computational cost. PS thus establishes itself as a robust, scalable, and efficient alternative for complex Bayesian inference tasks.