Differences Between Classical and Bayesian Estimates for Mixed Logit Models: A Replication Study

The frequentist (local) quantum parameter estimation based on the Cramer-Rao bound is an asymptotic theory and therefore requires an infinite number of samples for the measurements. On the other hand, by introducing the prior probability distribution, Bayesian parameter estimation approach relaxes this requirement and can be successfully applied under realistic assumption of possessing a limited prior information about the parameter value under investigation. Meanwhile, recent advances in ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$-symmetric quantum mechanics, both theoretically and experimentally, provide an opportunity to significantly enhance the parameter estimation precision by manipulating the Hilbert space of the system. Taking advantage of these, we develop the Bayesian parameter estimation approach for the two-level system possessing ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$ symmetry. We consider both options available in ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$-symmetric quantum mechanics, namely the ${{{\mathcal{C}}}}{{{\mathcal{P}}}}{{{\mathfrak{T}}}}$ and Hermitian measurements, for encoding the probe state and manipulation of the quantum Fisher information of the system and take into account possible decoherence effects inherent for the practical implementations. Our approach should be particularly relevant for applications where the accuracy of quantum parameter estimation is crucial, such as quantum randomness sources and the quantum key distribution. Finally, to bridge the gap with the practical implementations, we consider the relation of our scheme with the current implementation of ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$ symmetry on the IBM quantum processor and formulate our approach as Bayesian parameter estimation employing an ancilla qubit.

Model occurrence and model selection in panel data sets

This paper concentrates on Bayesian state estimation approach in the presence of slow-rate integrated measurements. In chemical process, some quality variables, in the sense of measuring, often have an important characteristic resulting from the time taken for samples collection. These kinds of measurements are obtained based on sample of materials that are collected in a period of time. So, the measurement indicates the average property of the measurand in the period of samples collection, which is called Slow-Rate inTegrated Measurement (SRTM). In this paper, our goal is to estimate the fast-rate instantaneous states using available SRTM. Bayesian estimation approach is reformulated to acquire this goal. First, new Bayesian formulation is provided which ends to some complex integral formula. Then, with using the idea of Monte Carlo sampling method a numerical solution is presented for this problem. The advantage of the proposed algorithm is that it can deal with integrated measurement problem in a broader range of models with nonlinearity and non-Gaussian noise. The effectiveness of the proposed methodology is verified and demonstrated through simulation and empirical experiment on a level-flow laboratory-scale plant.

In classical estimation usually unbiased estimators are used. This is mainly because the bias term in classical biased estimators in general depends on the parameter to be estimated. However, recently a considerable amount of research has been spent on improving unbiased estimators by introducing a bias, e.g. based on a minimax optimization strategy. In this work we follow this idea of introducing a bias, but we describe a different strategy for optimizing the estimators' performance. Although we stick to classical estimation, we show that the Bayesian linear minimum mean square error estimator can be brought into the same algebraic form as the resulting biased estimator improving the best linear unbiased estimator. This not only emphasizes the fact that this approach leads to betters estimators than the minimax approach on average over all parameters, but also can be seen as another way of reducing the gap between classical and Bayesian estimation.

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

Bayesian model selection and parameter estimation is attracting a lot of interest in the astronomical community because of its power and logical consistency. Markov chain Monte Carlo provides the computational power for Bayesian parameter estimation problems in large parameter spaces but needs to be supported with other numerical techniques for efficient exploration of multi-modal probability distributions. Bayesian model selection is easy in concept but remains a difficult challenge for large parameter spaces. My comments here on the paper by Roberto Trotta are based on lessons learned from developing a controlled statistical fusion approach to some of these issues.KeywordsMarkov Chain Monte CarloMarginal LikelihoodProposal DistributionMarkov Chain Monte Carlo AlgorithmMarkov Chain Monte Carlo SampleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

To analyze the performance of various estimators, this paper compares Bayesian and classical estimate approaches across a variety of loss functions. The work focuses on informative versus non-informative priors in Bayesian estimation. The calculations are based on a number of loss functions, including the Square Error Loss Function (SELF), Quadratic Loss Function (QLF), Precautionary Loss Function (PLF), and Entropy Loss Function (ELF). The methodology uses simulation techniques as well as real-world datasets to test the performance and robustness of the proposed estimators. The Bayesian estimators, classical estimators, and related Bayes hazards using empirical analysis and extensive simulations to computed. The findings show that the Minimum Mean Square Error estimator (MiniMSE) is more accurate and reliable than other estimators in a variety of settings. Furthermore, for both types of priors, it is demonstrated that the Bayes risk under the Quadratic Loss Function (QLF) is the lowest among all loss functions considered, including SELF, PLF, and ELF. This suggests that, when compared to other loss functions, QLF is a more effective and balanced criterion for making estimation decisions. The study’s findings provide vital new insights into how to choose the optimum estimate techniques and loss functions for statistical decision theory and practical data processing.

As an extension of the standard Weibull distribution, a new crucial distribution termed alpha power Weibull distribution has been presented. It can model decreasing, increasing, bathtub, and upside-down bathtub failure rates. This research investigates the estimation of model parameters and some of its reliability characteristics using progressively Type-II censored data. To get estimates of unknown parameters, reliability, and hazard rate functions, the maximum likelihood, and Bayesian estimation approaches are studied. To acquire estimated confidence intervals for unknown parameters and reliability characteristics, the maximum likelihood asymptotic properties are used. The Markov chain Monte Carlo approach is used in Bayesian estimation to provide Bayesian estimates under squared error and LINEX loss functions. Furthermore, the highest posterior density credible intervals of the parameters and reliability characteristics are determined. A Monte Carlo simulation study is used to investigate the accuracy of various point and interval estimators. In addition, various optimality criteria are used to choose the best progressive censoring schemes. Two real data from the engineering field are analyzed to demonstrate the applicability and significance of the proposed approaches. Based on numerical results, the Bayesian procedure for estimating the parameters and reliability characteristics of alpha power Weibull distribution is recommended. The analysis of two real data sets showed that the alpha power Weibull distribution is a good model to investigate engineering data in the presence of progressive Type-II censoring.

The chapter “Bayesian Statistics” gives a brief overview of the Bayesian approach to statistical analysis. It starts off by examining the difference between frequentist statistics and Bayesian statistics. Next, it introduces Bayes’ theorem and explains how the theorem is used in statistics and model selection, with the prosecutor’s fallacy given as a practice example. The chapter then goes on to discuss priors and Bayesian parameter estimation. It concludes with some final thoughts on Bayesian approaches. The chapter does not answer the question “Should ecologists become Bayesian?” However, to the extent that alternative models can be posed as alternative values of parameters, Bayesian parameter estimation can help assign probabilities to those hypotheses.

Bandwidth plays an important role in determining the performance of nonparametric estimators, such as the local constant estimator. In this article, we propose a Bayesian approach to bandwidth estimation for local constant estimators of time-varying coefficients in time series models. We establish a large sample theory for the proposed bandwidth estimator and Bayesian estimators of the unknown parameters involved in the error density. A Monte Carlo simulation study shows that (i) the proposed Bayesian estimators for bandwidth and parameters in the error density have satisfactory finite sample performance; and (ii) our proposed Bayesian approach achieves better performance in estimating the bandwidths than the normal reference rule and cross-validation. Moreover, we apply our proposed Bayesian bandwidth estimation method for the time-varying coefficient models that explain Okun’s law and the relationship between consumption growth and income growth in the U.S. For each model, we also provide calibrated parametric forms of the time-varying coefficients. Supplementary materials for this article are available online.

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.

A Bayesian approach for the inverse design of acoustic coatings for broadband sound absorption is presented. The acoustic coating comprises a soft viscoelastic material submerged in water and attached to a steel backing plate. The coating is embedded with layers of resonant inclusions corresponding to cavities and hard scatterers yielding monopole and dipole sound scattering, respectively. The acoustic performance of the coating is primarily governed by strong wave scattering at frequencies around the resonance frequency of the inclusions, and enhanced wave scattering due to resonance coupling between adjacent inclusions. Designing an acoustic coating for a targeted performance presents a challenge due to the competing and often conflicting influence of several coating parameters corresponding to the material properties of the coating, the geometric and material properties of the inclusions, and the number of inclusion layers. This challenge is addressed using a dual-level Bayesian inference approach for targeted performance based on sound absorption by the coating. Parameter estimation is employed to determine the geometric dimensions of the inclusions within each layer, while model selection identifies the number of inclusion layers and material of the inclusions within each layer. The dual-level Bayesian approach is validated against coating designs of increasing complexity. By exploiting the resonances associated with different designs, a physics informed design of a coating with high broadband sound absorption is proposed. • A dual-level Bayesian approach to design high performance resonant coatings is employed. • Coating sound absorption arises from strong wave scattering at inclusion resonances. • Multiple layers and inclusion material are inferred using Bayesian model selection. • Inclusion geometric values are determined using Bayesian parameter estimation. • High broadband absorption is achieved by exploiting coating inclusion resonances.

Doxorubicin was given by brief i.v. infusion (doses ranging from 25 to 72 mg/m2) to 28 patients for 2-7 successive courses of chemotherapy (68 courses studied in all). A Bayesian approach was developed to determine the individual pharmacokinetic parameters of doxorubicin. Statistical characteristics of the population pharmacokinetic parameters were first evaluated for 19 patients and a total of 30 courses, which, when combined with 4 individual plasma concentrations of drug, led to a Bayesian estimation of individual pharmacokinetic parameters for the remaining 38 courses. The estimated parameters for the elimination phase (A3/V1 and t1/2 elimination) and the residual plasma level at 48 h as computed by Bayesian estimation on this reduced sub-optimal sampling protocol were compared with a maximal likelihood estimation of these parameters. No statistically significant differences were found. Performance of the developed methodology was evaluated by computing bias and precision. The mean errors were -0.0315 x 10(-4) l-1 for A3/V1, 0.0839 h for t1/2 elimination, and -0.22 ng/ml for c(48 h). The precision of the prediction of these three parameters (0.304 x 10(-5) l-1, 3.34 h, and 0.659 ng/ml, respectively) remained lower than the interindividual standard deviation (1.42 x 10(-4) l-1, 14.9 h, and 4.54 ng/ml, respectively). This procedure enables the estimation of individual pharmacokinetic parameters for doxorubicin at minimal cost and minimal disturbance of the patient.

Phenazone (antipyrine) 1g was given by short intravenous infusion to 62 study participants (10 healthy drug-free volunteers and 52 patients with chronic liver disease). A Bayesian approach was developed to determine the individual pharmacokinetic parameters of phenazone. Statistical characteristics of the population pharmacokinetic parameters were first evaluated for 30 patients. When combined with 1 plasma drug concentration from members of the second group, these led to a Bayesian estimation of individual pharmacokinetic parameters for the remaining 32 individuals. Total clearance computed by Bayesian estimation was compared with maximal likelihood estimation of this parameter, the classical procedure. No statistically significant differences were found. Performance of the developed methodology was evaluated by computing bias and precision. The mean error was 0.0477 L/h. The precision of the prediction of this parameter (0.155 L/h) remained lower than the interindividual standard deviation (0.765 L/h). This procedure enables the estimation of individual pharmacokinetic parameters for phenazone. In this study, numerous laboratory tests were performed. A highly significant correlation (p < 0.001) was found between phenazone clearance and the prothrombin time, albumin, gamma-globulin, factor V, antithrombin III, fibrinogen and total bilirubin. Discriminant analysis determined that protein, alkaline phosphatase, creatininaemia and gamma-globulin had more significant discriminating power and gave better prognostic results than those seen with the Child-Pugh test.

This article compares maximum likelihood and Bayesian estimation of the correlated trait–correlated method (CT–CM) confirmatory factor model for multitrait–multimethod (MTMM) data. In particular, Bayesian estimation with minimally informative prior distributions—that is, prior distributions that prescribe equal probability across the known mathematical range of a parameter—are investigated as a source of information to aid convergence. Results from a simulation study indicate that Bayesian estimation with minimally informative priors produces admissible solutions more often maximum likelihood estimation (100.00% for Bayesian estimation, 49.82% for maximum likelihood). Extra convergence does not come at the cost of parameter accuracy; Bayesian parameter estimates showed comparable bias and better efficiency compared to maximum likelihood estimates. The results are echoed via 2 empirical examples. Hence, Bayesian estimation with minimally informative priors outperforms enables admissible solutions of the CT–CM model for MTMM data.