Bayesian Estimation Research Articles

Comparison of some Bayesian estimation methods for type-I generalized extreme value distribution with simulation

The Weibull distribution is considered one of the Type-I Generalized Extreme Value (GEV) distribution, and it plays a crucial role in modeling extreme events in various fields, such as hydrology, finance, and environmental sciences. Bayesian methods play a strong, decisive role in estimating the parameters of the GEV distribution due to their ability to incorporate prior knowledge and handle small sample sizes effectively. In this research, we compare several shrinkage Bayesian estimation methods based on the squared error and the linear exponential loss functions. They were adopted and compared by the Monte Carlo simulation method. The performance of these methods is assessed based on their accuracy and computational efficiency in estimating the scale parameter of the Weibull distribution. To evaluate their performance, we generate simulated datasets with different sample sizes and varying parameter values. A technique for pre-estimation shrinkage is suggested to enhance the precision of estimation. Simulation experiments proved that the Bayesian shrinkage estimator and shrinkage pre-estimation under the squared loss function method are better than the other methods because they give the least mean square error. Overall, our findings highlight the advantages of shrinkage Bayesian estimation methods for the proposed distribution. Researchers and practitioners in fields reliant on extreme value analysis can benefit from these findings when selecting appropriate Bayesian estimation techniques for modeling extreme events accurately and efficiently.

Today’s positive affect predicts tomorrow’s experience of meaningful coincidences: a cross-lagged multilevel analysis

ABSTRACT The perception of meaningful patterns in random arrangements and unrelated events takes place in our everyday lives, coined apophenia, synchronicity, or the experience of meaningful coincidences. However, we do not know yet what predicts this phenomenon. To investigate this, we re-analyzed a combined data set of two daily diary studies with a total of N = 169 participants (mean age 29.95 years; 54 men). We investigated if positive or negative affect (PA, NA) predicts the number of meaningful coincidences on the following day (or vice versa). By means of a cross-lagged multilevel modelling approach (Bayesian estimation) we evaluated with which of two theoretical assumptions the data are more in line. First, if meaningful coincidences are facilitated by a broader and more flexible thinking style, PA should positively predict meaningful coincidences at the following day. However, if the experience of meaningful coincidences signifies a strategy to cope with negative feeling states, NA should predict the experience of meaningful coincidences during the following day. In favour of a more flexible thinking style, we found that PA predicted the number of perceived coincidences the following day. We did not find any effect for NA, and therefore, no evidence arguing for the coping mechanism hypothesis of meaningful coincidences.

Bayesian estimation of sensitivity and specificity of IgG and serum total protein cutoff values for the diagnosis of failed transfer of passive immunity in neonatal beef calves

Calves with failed transfer of passive immunity (FTPI) possess increased risk of preweaning disease. Common tests for FTPI such as radial immunodiffusion (RID) and optical refractom­etry (OR) have inherent imprecision. The objective of this study was to estimate test sensitivity and specificity for various IgG concentrations and serum total protein values (STP) to classify neonatal beef calves with FTPI.

Population Pharmacokinetics and Limited Sampling Strategy of Mycophenolate Mofetil for Indian Patients with Lupus Nephritis

Background: Mycophenolic acid is widely used to treat lupus nephritis (LN). However, it exhibits complex pharmacokinetics with large interindividual variability. This study aimed to develop a population pharmacokinetic (popPK) model and a 3-sample limited sampling strategy (LSS) to optimize therapeutic drug monitoring in Indian patients with LN. Methods: Five blood samples from each LN patient treated with mycophenolic acid were collected at steady-state predose and 1, 2, 4, and 6 hours postdose. Demographic parameters were tested as covariates to explain interindividual variability. PopPK analysis was performed using Monolix and the stochastic approximation expectation-maximization algorithm. An LSS was derived from 500 simulated pharmacokinetic (PK) profiles using maximum a posteriori Bayesian estimation to estimate individual PK parameters and area under the curve (AUC). The LSS-calculated AUC was compared with the AUC calculated using the trapezoidal rule and all the simulated samples. Results: A total of 51 patients were included in this study. Based on the 245 mycophenolic acid concentrations, a 1-compartmental model with double absorption using gamma distributions best fitted the data. None of the covariates improved the model significantly. The model was internally validated using diagnostic plots, prediction-corrected visual predictive checks, and bootstrapping. The best LSS included samples at 1, 2, and 4 hours postdose and exhibited good performances in an external dataset (root mean squared error, 21.9%; mean bias, −4.20%). Conclusions: The popPK model developed in this study adequately estimated the PK of mycophenolic acid in adult Indian patients with LN. This simple LSS can optimize TDM based on the AUC in routine practice.

Benchmarking Bayesian quantum estimation

Abstract The quest for precision in parameter estimation is a fundamental task in different scientific areas. The relevance of this problem thus provided the motivation to develop methods for the application of quantum resources to estimation protocols. Within this context, Bayesian estimation offers a complete framework for optimal quantum metrology techniques, such as adaptive protocols. However, the use of the Bayesian approach requires extensive computational resources, especially in the multiparameter estimations that represent the typical operational scenario for quantum sensors. Hence, the requirement to characterize protocols implementing Bayesian estimations can become a significant challenge. This work focuses on the crucial task of robustly benchmarking the performances of these protocols in both single and multiple-parameter scenarios. By comparing different figures of merits, evidence is provided in favor of using the median of the quadratic error in the estimations in order to mitigate spurious effects due to the numerical discretization of the parameter space, the presence of limited data, and numerical instabilities. These results, providing a robust and reliable characterization of Bayesian protocols, find natural applications to practical problems within the quantum estimation framework.

Video-based sequential Bayesian homography estimation for soccer field registration

A novel Bayesian framework is proposed, which explicitly relates the homography of one video frame to the next through an affine transformation while explicitly modelling keypoint uncertainty. The literature has previously used differential homography between subsequent frames, but not in a Bayesian setting. In cases where Bayesian methods have been applied, camera motion is not adequately modelled, and keypoints are treated as deterministic. The proposed method, Bayesian Homography Inference from Tracked Keypoints (BHITK), employs a two-stage Kalman filter and significantly improves existing methods. Existing keypoint detection methods may be easily augmented with BHITK. It enables less sophisticated and less computationally expensive methods to outperform the state-of-the-art approaches in most homography evaluation metrics. Furthermore, the homography annotations of the WorldCup and TS-WorldCup datasets have been refined using a custom homography annotation tool that has been released for public use. The refined datasets are consolidated and released as the consolidated and refined WorldCup (CARWC) dataset.

Bayesian inference of graph-based dependencies from mixed-type data

Mixed data comprise measurements of different types, with both categorical and continuous variables, and can be found in various areas, such as in life science or industrial processes. Inferring conditional independencies from the data is crucial to understand how these variables relate to each other. To this end, graphical models provide an effective framework, which adopts a graph-based representation of the joint distribution to encode such dependence relations. This framework has been extensively studied in the Gaussian and categorical settings separately; on the other hand, the literature addressing this problem in presence of mixed data is still narrow. We propose a Bayesian model for the analysis of mixed data based on the notion of Conditional Gaussian (CG) distribution. Our method is based on a canonical parameterization of the CG distribution, which allows for posterior inference of parameters indexing the (marginal) distributions of continuous and categorical variables, as well as expressing the interactions between the two types of variables. We derive the limiting Gaussian distributions, centered on the correct unknown value and with vanishing variance, for the Bayesian estimators of the canonical parameters expressing continuous, discrete and mixed interactions. In addition, we implement the proposed method for structure learning purposes, namely to infer the underlying graph of conditional independencies. When compared to alternative frequentist methods, our approach shows favorable results both in a simulation setting and in real-data applications, besides allowing for a coherent uncertainty quantification around parameter estimates.

Equivalence between constrained optimal smoothing and Bayesian estimation

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) by adding convex constraints to the problem. Through a sequence of approximating Hilbertian subspaces and a discretised model, we prove that the Maximum a posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalisation of the paper Kimeldorf and Wahba (1970), where it is proved that the optimal smoothing solution is the mean of the posterior distribution. Synthetic and real data studies confirm the correspondence established in this paper.

Estimation and comparison of Weibull-Normal distribution with some other probability models using Bayesian method of estimation

In statistical applications the Normal distribution is adjudged to be the best. Recent studies Terna (2017) using classical method indicated that Weibull-Normal distribution outperformed the Normal distribution. In this study we used the non-classical Bayesian method of estimation to estimate and compare the Weibull-Normal distribution with some other distributions including Normal and Gamma-Normal distributions. This study derived explicit expressions for basic statistical properties such as moments, moment generating function, the characteristic function, reliability analysis and the distribution of order statistics. It looks at estimation of confidence intervals for the parameters of the Weibull-Normal distribution and estimated the parameters of the new distribution using a non-classical approach for the purpose of theoretical comparisons. The two other distributions whose parameters were also estimated by using Bayesian estimation are the normal distribution and gamma distribution as well as the combination Gamma-Normal distribution. Based on the analyses and interpretations of the results obtained it was found that the parameters and other general properties of Normal distribution gives a better fit than other distributions. R-software was used; the models were written as an R code in R program using the rjags library, the distribution parameters were obtained from a Gibbs sampling of a Bayesian Fit for data set I and data set II. The criteria used in R for comparisons were the negative log-likelihood, AIC (Akaike information criterion), CAIC (Consistent Akaike Information Criterion) and BIC (Bayesian information Criterion).

Dissecting Bayes: Using influence measures to test normative use of probability density information derived from a sample test.

Bayesian decision theory (BDT) is frequently used to model normative performance in perceptual, motor, and cognitive decision tasks where the possible outcomes of actions are associated with rewards or penalties. The resulting normative models specify how decision makers should encode and combine information about uncertainty and value-step by step-in order to maximize their expected reward. When prior, likelihood, and posterior are probabilities, the Bayesian computation requires only simple arithmetic operations: addition, etc. We focus on visual cognitive tasks where Bayesian computations are carried out not on probabilities but on (1) probability density functions and (2) these probability density functions are derived from samples. We break the BDT model into a series of computations and test human ability to carry out each of these computations in isolation. We test three necessary properties of normative use of pdf information derived from a sample-accuracy, additivity and influence. Influence measures allow us to assess how much weight each point in the sample is assigned in making decisions and allow us to compare normative use (weighting) of samples to actual, point by point. We find that human decision makers violate accuracy and additivity systematically but that the cost of failure in accuracy or additivity would be minor in common decision tasks. However, a comparison of measured influence for each sample point with normative influence measures demonstrates that the individual's use of sample information is markedly different from the predictions of BDT. We will show that the normative BDT model takes into account the geometric symmetries of the pdf while the human decision maker does not. An alternative model basing decisions on a single extreme sample point provided a better account for participants' data than the normative BDT model.

An investigation of self-interstitial diffusion in α-zirconium by an on-the-fly machine learning force field

The on-the-fly machine learning force field approach, based on the Gaussian approximation potential and Bayesian error estimation, was used to study the diffusion of self-interstitial atoms in α-zirconium. Ab initio molecular dynamics simulations of lattice vibration and interstitial diffusion at different temperatures were employed to develop the force field. The radial and angular descriptors of the potential were further optimized to achieve better agreement with first-principles results. Subsequent long-term diffusion simulations were performed to assess the diffusion behavior based on the obtained force field. Tracer diffusion coefficients and diffusion anisotropy were studied at temperatures of 600–1200 K, and the Bayesian errors were estimated throughout the diffusion simulations. The mean and maximum estimated Bayesian errors of atomic force were approximately twice as large as those observed during the learning period. The basal diffusion was greatly favored compared to the interstitial diffusion along the c-axis, consistent with previous simulations based on first-principles results and classical potentials. The accuracy and applicability of the current on-the-fly machine learning approach were critically evaluated.

Bayesian Estimate of Mean Proper Scores for Diversity-Enhanced Active Learning.

The effectiveness of active learning largely depends on the sampling efficiency of the acquisition function. Expected Loss Reduction (ELR) focuses on a Bayesian estimate of the reduction in classification error, and more general costs fit in the same framework. We propose Bayesian Estimate of Mean Proper Scores (BEMPS) to estimate the increase in strictly proper scores such as log probability or negative mean square error within this framework. We also prove convergence results for this general class of costs. To facilitate better experimentation with the new acquisition functions, we develop a complementary batch AL algorithm that encourages diversity in the vector of expected changes in scores for unlabeled data. To allow high-performance classifiers, we combine deep ensembles, and dynamic validation set construction on pretrained models, and further speed up the ensemble process with the idea of Monte Carlo Dropout. Extensive experiments on both texts and images show that the use of mean square error and log probability with BEMPS yields robust acquisition functions and well-calibrated classifiers, and consistently outperforms the others tested. The advantages of BEMPS over the others are further supported by a set of qualitative analyses, where we visualise their sampling behaviour using data maps and t-SNE plots.

Bayesian Estimation of the Time-Varying Reproduction Number for Pulmonary Tuberculosis in Iran: A Registry-Based Study from 2018 to 2022 Using New Smear-Positive Cases

IntroductionTuberculosis (TB) is one of the most prevalent infectious diseases in the world, causing major public health problems in developing countries. The rate of TB incidence in Iran was estimated to be 13 per 100,000 in 2021. This study aimed to estimate the reproduction number and serial interval for pulmonary tuberculosis in Iran. Material and MethodsThe present national historical cohort study was conducted from March 2018 to March 2022 based on data from the National Tuberculosis and Leprosy Registration Center of Iran's Ministry of Health and Medical Education (MOHME). The study included 30,762 tuberculosis cases and 16,165 new smear-positive pulmonary tuberculosis patients in Iran. We estimated the reproduction number of pulmonary tuberculosis in a Bayesian framework, which can incorporate uncertainty in estimating it. Statistical analyses were accomplished in R software. ResultsThe mean age at diagnosis of patients was 52.3±21.2 years, and most patients were in the 35–63 age group (37.1%). Among the data, 9,121 (56.4%) cases were males, and 7,044 (43.6%) were females. Among patients, 7,459 (46.1%) had a delayed diagnosis between 1 and 3 months. Additionally, 3,039 (18.8%) cases were non-Iranians, and 2,978 (98%) were Afghans. The time-varying reproduction number for pulmonary tuberculosis disease was calculated at an average of 1.06±0.05 (95% Crl 0.96–1.15). ConclusionsIn this study, the incidence and the time-varying reproduction number of pulmonary tuberculosis showed the same pattern. The mean of the time-varying reproduction number indicated that each infected person is causing at least one new infection over time, and the chain of transmission is not being disrupted.

Bayesian estimation of nonlinear Hawkes processes

Bayesian estimation of nonlinear Hawkes processes

Estimation of Stress - Strength Reliability for Parallel Redundant System Via Burr of type X Distribution

     Estimation for the reliability of a parallel redundant system with independent stress and strength Burr distribution (type  ) probability density functions is considered. Estimation of the reliability parameters is conducted according to the non-Bayesian estimation methods, namely the maximum likelihood and shrinkage methods. The Bayesian estimation is also considered using the Jeffery and Gamma priors with squared error, quadratic and weighted loss functions. Finally, the reliability estimate is calculated and the best method for estimation for each case is given using the mean squared error criteria.

Efficiency comparison of MCMC and Transport Map Bayesian posterior estimation for structural health monitoring

In this paper, an alternative to solving Bayesian inverse problems for structural health monitoring based on a variational formulation with so-called transport maps is examined. The Bayesian inverse formulation is a widely used tool in structural health monitoring applications. While Markov Chain Monte Carlo (MCMC) methods are often implemented in these settings, they come with the problem of using many model evaluations, which in turn can become quite costly. We focus here on recent developments in the field of transport theory, where the problem is formulated as finding a deterministic, invertible mapping between some easy to evaluate reference density and the posterior. The resulting variational formulation can be solved with integration and optimization methods. We develop a general formulation for the application of transport maps to vibration-based structural health monitoring. Further, we study influences of different integration approaches on the efficiency and accuracy of the transport map approach and compare it to the Transitional MCMC algorithm, a widely used method for structural identification. Both methods are applied to a lower-dimensional dynamic model with uni- and multi-modal properties, as well as to a higher-dimensional neural network surrogate system of an airplane structure. We find that transport maps have a significant increase in accuracy and efficiency, when used in the right circumstances.

GausSN: Bayesian time-delay estimation for strongly lensed supernovae

ABSTRACT We present gausSN, a Bayesian semiparametric Gaussian Process (GP) model for time-delay estimation with resolved systems of gravitationally lensed supernovae (glSNe). gausSN models the underlying light curve non-parametrically using a GP. Without assuming a template light curve for each SN type, gausSN fits for the time delays of all images using data in any number of wavelength filters simultaneously. We also introduce a novel time-varying magnification model to capture the effects of microlensing alongside time-delay estimation. In this analysis, we model the time-varying relative magnification as a sigmoid function, as well as a constant for comparison to existing time-delay estimation approaches. We demonstrate that gausSN provides robust time-delay estimates for simulations of glSNe from the Nancy Grace Roman Space Telescope and the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (Rubin-LSST). We find that up to 43.6 per cent of time-delay estimates from Roman and 52.9 per cent from Rubin-LSST have fractional errors of less than 5 per cent. We then apply gausSN to SN Refsdal and find the time delay for the fifth image is consistent with the original analysis, regardless of microlensing treatment. Therefore, gausSN maintains the level of precision and accuracy achieved by existing time-delay extraction methods with fewer assumptions about the underlying shape of the light curve than template-based approaches, while incorporating microlensing into the statistical error budget. gausSN is scalable for time-delay cosmography analyses given current projections of glSNe discovery rates from Rubin-LSST and Roman.

Numerical Comparison by using Some Criteria to Estimate the Two Parameters of the Exponentiated Exponential Distribution

     The present paper shows a numerical comparison by using the number of the following criteria: mean squared, root mean Square, mean absolute, and relative mean squared error values between the methods, namely  maximum likelihood and Bayesian estimators for the two parameters of the exponentiated exponential distribution. In estimating the maximum likelihood, The equations cannot be solved directly, Then Newton-Raphson method is used. Because of Bayes estimators under scale invariant squared and weighted composite linear-exponential loss functions the ratios cannot be simplified in a closed form. So, we use Lindley approximation. MATLAB program is used to display the results.

MONETARY POLICY DYNAMICS IN NIGERIA: EMPIRICAL EVIDENCES FROM BAYESIAN VECTOR AUTOREGRESSION WITH STOCHASTIC VOLATILITY

In this paper, we examine the dynamics of monetary policy in Nigeria with Bayesian approach to a vector autoregression (VAR). We construct and estimate Bayesian Vector Autoregression with Stochastic Volatility (BVAR-SV) model and extract important policy inputs from the model. Nigeria economy is unstable and it is a known fact that changes to monetary policy affects performance of some macroeconomic variables. The BVAR has the ability to capture sudden changes and nonlinearities arising from the interaction among macroeconomic variables and associated shocks. The study uses monthly data during the period 2003M01 till 2023M12 with three macroeconomic variables namely; inflation rate, money supply, and interest rate. A Markov Chain Monte Carlo algorithm that allows for Bayesian estimation and prediction is employed. Results show that there is strong evidence of monetary policy playing a significant role in explaining the dynamics of interest rate while the impulse responses for the variables to a monetary policy shock do change significantly over time. Also, the monetary policy exert less significant influence in terms of money supply and inflation than interest rate in explaining the dynamics in of monetary policy. It is recommended that BVAR should be also be extended to other macroeconomic variables to examine the effects on monetary policy dynamics.

Bayes estimator for the inequality-constrained regression model*

This article studies the Bayesian estimation of the inequality-constrained regression model under a balanced loss function. The usual Bayes estimator is often involved in some complex integrals and has no explicit expression, which causes the corresponding properties be difficult to describe. Based on the idea of linear Bayes method suggested by Rao (1973), the present paper constructs a linear Bayes estimator for the regression parameters without specify their specific prior distributions. It is proved that the proposed linear Bayes estimator outperforms the corresponding least squares estimator, and numerical simulations show that when there is collinearity in design matrix, the linear Bayes estimator performs better than the usual Bayes estimator.