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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.
The climate scenarios that form the basis for current climate risk assessments have no assigned probabilities, and this impedes the analysis of future climate risks. This paper proposes an approach to estimate the probability of carbon dioxide (CO 2 ) concentration scenarios used in key climate change modeling experiments. It computes the CO 2 emissions compatible with the concentrations prescribed by Coupled Model Intercomparison Project Phase 5 (CMIP5) and CMIP6 experiments. The distribution of these compatible cumulative emissions is interpreted as the likelihood of future emissions given a concentration pathway. Using Bayesian analysis, the probability of each pathway can be estimated from a probabilistic sample of future emissions. The approach is demonstrated with five probabilistic CO 2 emission simulation ensembles from four Integrated Assessment Models (IAM), leading to independent estimates of the likelihood of the CO 2 concentration of Representative Concentration Pathways (RCP) and Shared Socioeconomic Pathways (SSP). Results suggest that SSP5-8.5 is unlikely for the second half of the 21st century, but offer no clear consensus on which of the remaining scenarios is most likely. Estimates of likelihoods of CO 2 concentrations associated with RCP and SSP scenarios are affected by sampling errors, differences in emission sources simulated by the IAMs, and a lack of a common experimental framework for IAM simulations. These shortcomings, along with a small IAM ensemble size, limit the applicability of the results presented here. Novel joint IAM and the Earth System Model experiments are needed to deliver actionable probabilistic climate risk assessments.
Here we present a new Planck CMB lensing-CMB temperature cross-correlation likelihood that can be used to constrain cosmology via the integrated Sachs-Wolfe (ISW) effect. CMB lensing is an excellent tracer of ISW, and we use the latest PR4 Planck data maps and lensing reconstruction to produce the first public Planck likelihood to constrain this signal. We demonstrate the likelihood by constraining the CMB background temperature from Planck data alone, where the ISW-lensing cross-correlation is a powerful way to break the geometric degeneracy, substantially improving constraints from the CMB and lensing power spectra alone.
PSL is a software framework for localizing radiation point sources in a 3D environment using a freely moving radiation detector system. The framework assumes a single point source of radiation is in the environment and uses known detector positions and orientations along with their radiation measurements to find the most likely location of the point source. This approach to the problem enables analyses that improve upon the capability of a standard Maximum Likelihood Expectation Maximization (MLEM) approach, such as setting a source detection threshold, finding a likelihood interval for the position of the source, finding a likelihood interval for the activity (intensity) of the source, determining whether more than one point source may be present, and generating a map of Minimum Detectable Activity (MDA) for a given source type. Applications include radiological and nuclear security, consequence management, decontamination and decommissioning of nuclear facilities, and radiological emergency response.
We propose an intuitive, machine-learning approach to multiparameter inference, dubbed the InferoStatic Networks (ISN) method, to model the score and likelihood ratio estimators in cases when the probability density can be sampled but not computed directly. The ISN uses a backend neural network that models a scalar function called the inferostatic potential \varphi φ . In addition, we introduce new strategies, respectively called Kernel Score Estimation (KSE) and Kernel Likelihood Ratio Estimation (KLRE), to learn the score and the likelihood ratio functions from simulated data. We illustrate the new techniques with some toy examples and compare to existing approaches in the literature. We mention en passant some new loss functions that optimally incorporate latent information from simulations into the training procedure.
We propose an intuitive, machine-learning approach to multiparameter inference, dubbed the InferoStatic Networks (ISN) method, to model the score and likelihood ratio estimators in cases when the probability density can be sampled but not computed directly. The ISN uses a backend neural network that models a scalar function called the inferostatic potential $\varphi$. In addition, we introduce new strategies, respectively called Kernel Score Estimation (KSE) and Kernel Likelihood Ratio Estimation (KLRE), to learn the score and the likelihood ratio functions from simulated data. We illustrate the new techniques with some toy examples and compare to existing approaches in the literature. We mention en passant some new loss functions that optimally incorporate latent information from simulations into the training procedure.
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.
When adjusting nuclear data with integral experiments, care must be taken that spurious adjustments are not made by assimilating poorly characterized integral parameters. If there are unaccounted for biases or poorly estimated uncertainties in the calculated and experimental values for an integral parameter, the Bayesian data assimilation may adjust the nuclear data in a manner that does not reflect the physics of the integral parameter. To identify and lessen the impact of these inconsistent integral parameters, in this study we present a Marginal Likelihood Optimization algorithm. In a data-driven way, the marginalized likelihood is used to modulate hyperparameter terms that decrease the influence of inconsistent integral parameters on the adjustment. The advantage of this approach over other methods in the literature is that it incorporates correlation information and does not remove an integral parameter from the adjustment. Herein, we present and motivate the algorithm, and apply it to an integral data assimilation case study.
Here, we present a likelihood-free probabilistic inversion method based on normalizing flows for high-dimensional inverse problems. The proposed method is composed of two complementary networks: a summary network for data compression and an inference network for parameter estimation. The summary network encodes raw observations into a fixed-size vector of summary features, while the inference network generates samples of the approximate posterior distribution of the model parameters based on these summary features. The posterior samples are produced in a deep generative fashion by sampling from a latent Gaussian distribution and passing these samples through an invertible transformation. We construct this invertible transformation by sequentially alternating conditional invertible neural network and conditional neural spline flow layers. The summary and inference networks are trained simultaneously. We apply the proposed method to an inversion problem in groundwater hydrology to estimate the posterior distribution of the log-conductivity field conditioned on spatially sparse time-series observations of the system’s hydraulic head responses. The conductivity field is represented with 706 degrees of freedom in the considered problem. Comparison with the likelihood-based iterative ensemble smoother PEST-IES method demonstrates that the proposed method accurately estimates the parameter posterior distribution and the observations’ predictive posterior distribution at a fraction of the inference time of PEST-IES.
A readily parallelized Maximum Likelihood Estimation (MLE) algorithm with linear computational complexity is demonstrated in real time using only measurements from an extreme ultraviolet (EUV) diagnostic to control the horizontal position of a tokamak plasma. A set of trial emissivity profiles are parameterized by the control quantity of interest (R m ), and the MLE is identified from the profile which minimizes the signal reconstruction residual. The algorithm depends on an empirically determined likelihood function with exponential form. EUV emission (λ ≈ 15eV-1keV) is captured in a poloidal plane by four 16-channel AXUV diodes mounted at different poloidal angles with radial and angular resolution sufficient to discern plasma equilibrium evolution in HBT-EP. Calculations of the plasma major radius by the system are consistent within diagnostic uncertainty for the majority of the discharge with those of: a weighted average of vertical soft X-ray or EUV chords, magnetic sensors, and an equilibrium reconstruction. The feedback system corrects for a horizontal displacement of the major radius equal to 20% of the plasma minor radius by adjusting the vertical field produced from 40 in-vessel control coils in real time. The MLE calculation is performed on a GPU in a 15 μs cycle, with similar performance in this application to a simple weighted average of vertical chords. Finally, results demonstrate horizontal position control using magnetic actuators and an optical observer.
The three-parameter generalized extreme value distribution arises from classical univariate extreme value theory and is in common use for analysing the far tail of observed phenomena, yet important asymptotic properties of likelihood-based estimation under this standard model have not been established. In this paper, we prove that the maximum likelihood estimator is global and unique. An interesting secondary result entails the uniform consistency of a class of limit relations in a tight neighbourhood of the true shape parameter.
Intensity-based likelihood functions in crystallographic applications have the potential to enhance the quality of structures derived from marginal diffraction data. Their usage, however, is complicated by the ability to efficiently compute these target functions. Here, a numerical quadrature is developed that allows the rapid evaluation of intensity-based likelihood functions in crystallographic applications. By using a sequence of change-of-variable transformations, including a nonlinear domain-compression operation, an accurate, robust and efficient quadrature is constructed. The approach is flexible and can incorporate different noise models with relative ease.
Abstract Correcting spatial orientations of groups of high-dimensional data sets such that they are all in a consistent coordinate system is often a time-consuming and error-prone process. Automation of this process can be accomplished by using Generalized Procrustes Analysis to estimate the relative orientations among a population of high-dimensional data sets. A least squares Procrustes solution is applied through a maximum likelihood estimation and random sample consensus framework for robustness. The likelihood model is comprised of a mixture distribution where inliers are modeled using t -distribution and outliers from a uniform distribution. Applications will focus on a synthetic data set that emulates triaxial acceleration data and also real shock data from a population of triaxial accelerometers. Outliers represent either non-rigid body responses, environmental noise, and/or sensor and data acquisition issues. The intended application for the methodology is to robustly automate the rotation of populations of experimentally collected triaxial accelerometer data sets to a single global coordinate system.
The interiors of neutron stars reach densities and temperatures beyond the limits of terrestrial experiments, providing vital laboratories for probing nuclear physics. While the star's interior is not directly observable, its pressure and density determine the star's macroscopic structure which affects the spectra observed in telescopes. The relationship between the observations and the internal state is complex and partially intractable, presenting difficulties for inference. Previous work has focused on the regression from stellar spectra of parameters describing the internal state. We demonstrate a calculation of the full likelihood of the internal state parameters given observations, accomplished by replacing intractable elements with machine learning models trained on samples of simulated stars. Our machine-learning-derived likelihood allows us to perform maximum a posteriori estimation of the parameters of interest, as well as full scans. We demonstrate the technique by inferring stellar mass and radius from an individual stellar spectrum, as well as equation of state parameters from a set of spectra. Our results are more precise than pure regression models, reducing the width of the parameter residuals by 11.8% in the most realistic scenario. The neural networks will be released as a tool for fast simulation of neutron star properties and observed spectra.
We present FlameNEST, a framework providing explicit likelihood evaluations in noble element particle detectors using data-driven models from the Noble Element Simulation Technique. FlameNEST provides a way to perform statistical analyses on real data with no dependence on large, computationally expensive Monte Carlo simulations by evaluating the likelihood on an event-by-event basis using analytic probability elements convolved together in a single TensorFlow multiplication. Furthermore, this robust framework creates opportunities for simple inter-collaboration analyses which will be fundamental for the future of experimental dark matter physics.
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.
ABSTRACT Strong gravitational lensing has emerged as a promising approach for probing dark matter (DM) models on sub-galactic scales. Recent work has proposed the subhalo effective density slope as a more reliable observable than the commonly used subhalo mass function. The subhalo effective density slope is a measurement independent of assumptions about the underlying density profile and can be inferred for individual subhaloes through traditional sampling methods. To go beyond individual subhalo measurements, we leverage recent advances in machine learning and introduce a neural likelihood-ratio estimator to infer an effective density slope for populations of subhaloes. We demonstrate that our method is capable of harnessing the statistical power of multiple subhaloes (within and across multiple images) to distinguish between characteristics of different subhalo populations. The computational efficiency warranted by the neural likelihood-ratio estimator over traditional sampling enables statistical studies of DM perturbers and is particularly useful as we expect an influx of strong lensing systems from upcoming surveys.