scholarly journals Local solutions of maximum likelihood estimation in quantum state tomography

2012 ◽  
Vol 12 (9&10) ◽  
pp. 775-790
Author(s):  
Douglas S. Goncalves ◽  
Marcia A. Gomes-Ruggiero ◽  
Carlile Lavor ◽  
Osvaldo J. Farias ◽  
P. H. Souto Ribeiro
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Density Matrix ◽  
Quantum State ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Local Minima ◽  
Quantum State Tomography ◽  
Log Likelihood ◽  
State Tomography

Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.

Using Maximum Likelihood Estimation to Estimate Kriging Model Parameters

2007 ◽  
Author(s):  
Jay D. Martin
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Kriging Model ◽  
Model Parameters ◽  
Spatial Correlation Function ◽  
Log Likelihood ◽  
Gradient Based ◽  
Computationally Expensive

A kriging model can be used as a surrogate to a more computationally expensive model or simulation. It is capable of providing a continuous mathematical relationship that can interpolate a set of observations. One of the major issues with using kriging models is the potentially computationally expensive process of estimating the best model parameters. One of the most common methods used to estimate model parameters is Maximum Likelihood Estimation (MLE). MLE of kriging model parameters requires the use of numerical optimization of a continuous but possibly multi-modal log-likelihood function. This paper presents some enhancements to gradient-based methods to make them more computationally efficient and compares the potential reduction in computational burden. These enhancements include the development of the analytic gradient and Hessian for the log-likelihood equation of a kriging model that uses a Gaussian spatial correlation function. The suggested algorithm is very similar to the Scoring algorithm traditionally used in statistics, a Newton-Raphson gradient-based optimization method.

Maximum likelihood estimation for symmetric α-stable Ornstein–Uhlenbeck processes

2020 ◽  
pp. 2150018
Author(s):  
Zhifen Chen ◽  
Xiaopeng Chen
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Transition Function ◽  
Closed Form Expression ◽  
Discrete Observations ◽  
Form Expression ◽  
Probability Density Function Pdf ◽  
Accuracy And Stability

In this paper, we consider the maximum likelihood estimation for the symmetric [Formula: see text]-stable Ornstein–Uhlenbeck (S[Formula: see text]S-OU) processes based on discrete observations. Since the closed-form expression of maximum likelihood function is hard to obtain in the Lévy case, we choose a mixture of Cauchy and Gaussian distribution to approximate the probability density function (PDF) of the S[Formula: see text]S distribution. By means of transition function and Laplace transform, we construct an explicit approximate sequence of likelihood function, which converges to the likelihood function of S[Formula: see text]S distribution. Based on the approximation of likelihood function we give an algorithm for computing maximum likelihood estimation. We also numerically simulate some experiments which demonstrate the accuracy and stability of the proposed estimator.

scholarly journals Maximum Likelihood Estimation for Three-Parameter Weibull Distribution Using Evolutionary Strategy

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Fan Yang ◽  
Hu Ren ◽  
Zhili Hu
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Weibull Distribution ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Estimation Procedure ◽  
Evolutionary Strategy ◽  
Likelihood Method ◽  
Conventional Algorithm

The maximum likelihood estimation is a widely used approach to the parameter estimation. However, the conventional algorithm makes the estimation procedure of three-parameter Weibull distribution difficult. Therefore, this paper proposes an evolutionary strategy to explore the good solutions based on the maximum likelihood method. The maximizing process of likelihood function is converted to an optimization problem. The evolutionary algorithm is employed to obtain the optimal parameters for the likelihood function. Examples are presented to demonstrate the proposed method. The results show that the proposed method is suitable for the parameter estimation of the three-parameter Weibull distribution.

scholarly journals The maximum likelihood degree of Fermat hypersurfaces

2015 ◽  
Vol 6 (2) ◽  
Author(s):  
Daniele Agostini ◽  
Davide Alberelli ◽  
Francesco Grande ◽  
Paolo Lella
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Projective Space ◽  
Critical Points ◽  
Projective Variety ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Topological Invariant ◽  
Fermat Curve ◽  
Computational Comparison

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section. 

scholarly journals Global convergence of diluted iterations in maximum-likelihood quantum tomography

2014 ◽  
Vol 14 (11&12) ◽  
pp. 966-980
Author(s):  
Doglas S. Goncalves ◽  
Marcia A. Gomes-Ruggiero ◽  
Carlile Lavor
Keyword(s):  
Maximum Likelihood ◽  
Global Convergence ◽  
Density Matrix ◽  
Quantum State ◽  
Maximum Likelihood Estimate ◽  
Quantum Tomography ◽  
Practical Implementation ◽  
Globally Convergent ◽  
State Tomography ◽  
New Interpretation

In this paper we address convergence issues of the Diluted $R \rho R$ algorithm \cite{rehacek2007}, used to obtain the maximum likelihood estimate for the density matrix in quantum state tomography. We give a new interpretation to the diluted $R \rho R$ iterations that allows us to prove the global convergence under weaker assumptions. Thus, we propose a new algorithm which is globally convergent and suitable for practical implementation.

scholarly journals A scalable maximum likelihood method for quantum state tomography

2013 ◽  
Vol 15 (12) ◽  
pp. 125004 ◽  
Author(s):  
T Baumgratz ◽  
A Nüßeler ◽  
M Cramer ◽  
M B Plenio
Keyword(s):  
Maximum Likelihood ◽  
Quantum State ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Quantum State Tomography ◽  
State Tomography

scholarly journals Comparison of Recent Acceleration Techniques for the EM Algorithm in One- and Two-Parameter Logistic IRT Models

Psych ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 209-252
Author(s):  
Marie Beisemann ◽  
Ortrud Wartlick ◽  
Philipp Doebler
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Em Algorithm ◽  
Binary Data ◽  
Likelihood Estimation ◽  
Acceleration Techniques ◽  
Irt Models ◽  
The Em Algorithm ◽  
Log Likelihood ◽  
Two Parameter

The expectation–maximization (EM) algorithm is an important numerical method for maximum likelihood estimation in incomplete data problems. However, convergence of the EM algorithm can be slow, and for this reason, many EM acceleration techniques have been proposed. After a review of acceleration techniques in a unified notation with illustrations, three recently proposed EM acceleration techniques are compared in detail: quasi-Newton methods (QN), “squared” iterative methods (SQUAREM), and parabolic EM (PEM). These acceleration techniques are applied to marginal maximum likelihood estimation with the EM algorithm in one- and two-parameter logistic item response theory (IRT) models for binary data, and their performance is compared. QN and SQUAREM methods accelerate convergence of the EM algorithm for the two-parameter logistic model significantly in high-dimensional data problems. Compared to the standard EM, all three methods reduce the number of iterations, but increase the number of total marginal log-likelihood evaluations per iteration. Efficient approximations of the marginal log-likelihood are hence an important part of implementation.

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