Non-Homogeneous Poisson Processes Applied to Count Data: A Bayesian Approach Considering Different Prior Distributions

In the decades since Markov chain Monte Carlo methods were first introduced, they have revolutionised Bayesian approaches to statistical inference. Each new advance in MCMC methodology produces near immediate benefits for Bayesian practitioners, expanding the range of problems they can feasibly solve. In this paper, we explore ways in which Bayesian approaches can return something of the debt owed to MCMC, by using explicitly Bayesian concepts to aid in the design of MCMC samplers. The art of efficient MCMC sampling lies in designing a Markov process that (a) has the required limiting distribution, (b) has good convergence and mixing properties and (c) can be implemented in a computationally efficient manner. In this paper, we explore the idea that the selection of an appropriate process, and in particular the tuning of the parameters of the process to achieve the above goals, can be regarded as a problem of estimation. As such, it is amenable to a conventional Bayesian approach, in which a prior distribution for optimal parameters of the sampler is specified, data relevant to sampler performance is obtained and a posterior distribution for optimal parameters is formed. Sampling from this posterior distribution can then be incorporated into the MCMC sampler to produce an adaptive method. We present a new MCMC algorithm for Bayesian adaptive Metropolis-Hasting sampling (BAMS), using an explicitly Bayesian inference to update the proposal distribution. We show that author Keith’s earlier Bayesian adaptive independence sampler (BAIS) and a new Bayesian adaptive random walk sampler (BARS) emerge as instances. More important than either of these instances, BAMS provides a general framework within which to explore adaptive schemes that are guaranteed to converge to the required limiting distribution.

In this study we compared the efficiency of frequentist approach with Bayesian approach by carrying out extreme value analysis of Annual Maximum Daily Rainfall (AMDR). For frequentist frequency analysis of AMDR, we used the data of one station i.e. Lahore in Punjab province, Pakistan while for Bayesian analysis we used the data of three other neighboring stations as prior information. During frequentist approach, Generalized Extreme Value (GEV) was found to be a best-fit distribution on the data. In frequentist method, the parameters of GEV distribution were estimated using Maximum Likelihood Estimation (MLE), while in the Bayesian framework the Markov Chain Monte Carlo (MCMC) simulation technique along with Metropolis-Hasting algorithm and Gibbs sampler were implemented. Findings of this study indicate that despite the asymptotic requirements of the MLE, its performance can be enhanced by adopting the Bayesian MCMC paradigm. In order to acquire the posterior densities of GEV parameters, non-informative and informative priors based on the historical data of surrounding weather stations were employed. The result of Bayesian MCMC might be affected by the choice of priors. In addition, the performance of the parameters estimation methods was verified by employing several robustness measures. Robustness measures results proved that the Bayesian MCMC method performed better than MLE in estimating GEV parameters and future return levels. Therefore, the findings of these analyses could be helpful in adopting proper flood protection measures and designing infrastructures of culverts, buildings, bridges and dams in the region.

This paper proposed a Bayesian method-based structural equation model (SEM) of miners’ work injury for an underground coal mine in India. The environmental and behavioural variables for work injury were identified and causal relationships were developed. For Bayesian modelling, prior distributions of SEM parameters are necessary to develop the model. In this paper, two approaches were adopted to obtain prior distribution for factor loading parameters and structural parameters of SEM. In the first approach, the prior distributions were considered as a fixed distribution function with specific parameter values, whereas, in the second approach, prior distributions of the parameters were generated from experts’ opinions. The posterior distributions of these parameters were obtained by applying Bayesian rule. The Markov Chain Monte Carlo sampling in the form Gibbs sampling was applied for sampling from the posterior distribution. The results revealed that all coefficients of structural and measurement model parameters are statistically significant in experts’ opinion-based priors, whereas, two coefficients are not statistically significant when fixed prior-based distributions are applied. The error statistics reveals that Bayesian structural model provides reasonably good fit of work injury with high coefficient of determination (0.91) and less mean squared error as compared to traditional SEM.

A Markov Chain Monte Carlo technique for parameter estimation and inference in pesticide fate and transport modeling

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

The aim of the research is to implement Markov Chain Monte Carlo (MCMC) simulation method to price the futures contract of cocoa commodities. The result shows that MCMC is more flexible than Standard Monte Carlo (SMC) simulation method because MCMC method uses hit-and-run sampler algorithm to generate proposal movements that are subsequently accepted or rejected with a probability that depends on the distribution of the target that we want to be achieved. This research shows that MCMC method is suitable to be used to simulate the model of cocoa commodity price movement. The result of this research is a simulation of future contract prices for the next three months and future contract prices that must be paid at the time the contract expires. Pricing future contract by using MCMC method will produce the cheaper contract price if it compares to Standard Monte Carlo simulation.

Multivariate random effects meta-analysis (MRMA) is an appropriate way for synthesizing data from studies reporting multiple correlated outcomes. In a Bayesian framework, it has great potential for integrating evidence from a variety of sources. In this paper, we propose a Bayesian model for MRMA of mixed outcomes, which extends previously developed bivariate models to the trivariate case and also allows for combination of multiple outcomes that are both continuous and binary. We have constructed informative prior distributions for the correlations by using external evidence. Prior distributions for the within-study correlations were constructed by employing external individual patent data and using a double bootstrap method to obtain the correlations between mixed outcomes. The between-study model of MRMA was parameterized in the form of a product of a series of univariate conditional normal distributions. This allowed us to place explicit prior distributions on the between-study correlations, which were constructed using external summary data. Traditionally, independent ‘vague’ prior distributions are placed on all parameters of the model. In contrast to this approach, we constructed prior distributions for the between-study model parameters in a way that takes into account the inter-relationship between them. This is a flexible method that can be extended to incorporate mixed outcomes other than continuous and binary and beyond the trivariate case. We have applied this model to a motivating example in rheumatoid arthritis with the aim of incorporating all available evidence in the synthesis and potentially reducing uncertainty around the estimate of interest. © 2013 The Authors. Statistics inMedicine Published by John Wiley & Sons, Ltd.

An introduction of Markov chain Monte Carlo method to geochemical inverse problems: Reading melting parameters from REE abundances in abyssal peridotites

저수분석(low flow analysis)은 수자원공학에서 중요한 분야 중 하나이며, 특히 저수량 빈도분석(low flow frequency analysis)의 결과는 저수(貯水)용량의 설계, 물 수급계획, 오염원의 배치 및 관개와 생태계의 보존을 위한 수량과 수질의 관리에 중요하게 사용된다. 그러므로 본 연구에서는 저수량 빈도분석을 위한 점 빈도분석을 수행하였으며, 특히 빈도분석에 있어서의 불확실성을 탐색하기 위하여 Bayesian 방법을 적용하고 그 결과를 기존에 사용되던 불확실성 탐색방법과 비교하였다. 본 논문의Ⅰ편에서는 Bayesian 방법 중 사전분포(prior distribution)와 우도함수(likelihood function)의 복잡성에 상관없이 계산이 가능한 Bayesian MCMC(Bayesian Markov Chain Monte Carlo) 방법과 Metropolis-Hastings 알고리즘을 사용하기 위한 여러 과정의 이론적 배경과 Bayesian 방법에서 가장 중요한 요소인 사전분포를 구축하고 이를 비교 및 평가하였다. 고려된 사전분포는 자료에 기반하지 않은 사전분포와 자료에 기반한 사전분포로써 두 사전분포를 이용하여 Metropolis-Hastings 알고리즘을 수행하고 그 결과를 비교하여 저수량 빈도분석에 합리적인 사전분포를 선정하였다. 또한 알고리즘의 수행과정에서 필요한 제안분포(proposal distribution)를 적용하여 그에 따른 알고리즘의 효율성을 채택률(acceptance rate)을 산정하여 검증해 보았다. 사전분포의 분석 결과, 자료에 기반한 사전분포가 자료에 기반하지 않은 사전분포보다 정확성 및 불확실성의 표현에 있어서 우수한 결과를 제시하는 것을 확인할 수 있었고, 채택률을 이용한 알고리즘의 효용성 역시 기존 연구자들이 제시하였던 만족스러운 범위를 가지는 것을 알 수 있었다. 최종적으로 선정된 사전분포는 본 연구의 II편에서 Bayesian MCMC방법의 사전분포로 이용되었으며, 그 결과를 기존 불확실성의 추정방법의 하나인 2차 근사식을 이용한 최우추정(maximum likelihood estimation)방법의 결과와 비교하였다. The low flow analysis is an important part in water resources engineering. Also, the results of low flow frequency analysis can be used for design of reservoir storage, water supply planning and design, waste-load allocation, and maintenance of quantity and quality of water for irrigation and wild life conservation. Especially, for identification of the uncertainty in frequency analysis, the Bayesian approach is applied and compared with conventional methodologies in at-site low flow frequency analysis. In the first manuscript, the theoretical background for the Bayesian MCMC (Bayesian Markov Chain Monte Carlo) method and Metropolis-Hasting algorithm are studied. Two types of the prior distribution, a non-data- based and a data-based prior distributions are developed and compared to perform the Bayesian MCMC method. It can be suggested that the results of a data-based prior distribution is more effective than those of a non-data-based prior distribution. The acceptance rate of the algorithm is computed to assess the effectiveness of the developed algorithm. In the second manuscript, the Bayesian MCMC method using a data-based prior distribution and MLE(Maximum Likelihood Estimation) using a quadratic approximation are performed for the at-site low flow frequency analysis.

An autoregressive moving average (ARMA) is a time series model that is applied in everyday life for pattern recognition and forecasting. The ARMA model contains a noise which is assumed to have a specific distribution. The noise is often considered to have a Gaussian distribution. However in applications, the noise is sometimes found that does not have a Gaussian distribution. The first objective is to develop the ARMA model in which noise has a Laplacian distribution. The second objective is to estimate the parameters of the ARMA model. The ARMA model parameters include ARMA model orders, ARMA model coefficients, and noise variance. The parameter estimation of the ARMA model is carried out in the Bayesian framework. In the Bayesian framework, the ARMA model parameters are treated as a variable that has a prior distribution. The prior distribution for the ARMA model parameters is combined with the likelihood function for the data to get the posterior distribution for the parameter. The posterior distribution for parameters has a complex form so that the Bayes estimator cannot be determined analytically. The reversible jump Markov chain Monte Carlo (MCMC) algorithm was adopted to determine the Bayes estimator. The first result, the ARMA model can be developed by assuming Laplacian distribution noise. The second result, the performance of the algorithm was tested using simulation studies. The simulation shows that the reversible jump MCMC algorithm can estimate the parameters of the ARMA model correctly.

In the context of climate change, it is essential to quantify the uncertainty for effective design and risk management practices. In the present study, we have accessed the climate model and flood return level uncertainties over a river basin. Six high-resolution global climate models (GCMs) with two Representative Concentration Pathways (RCPs) are used to project the future climate change impact on streamflow of Wainganga River basin. Uncertainty associated with the use of high-resolution multiple GCM is treated with reliability ensemble average (REA) followed by bias correction. The bias-corrected weighted outputs are used as input to variable infiltration capacity (VIC) model, a physically based hydrological model. Calibration and validation are carried out for the hydrological model, and the parameters of VIC are fixed through trial-and-error method. The uncertainty in flood return level associated with the future projected flows is dealt with the Bayesian analysis and modelled through Markov Chain Monte Carlo (MCMC) simulation technique using Metropolis–Hastings algorithm with the non-informative prior distribution. The study provides a robust framework, which will help in effective decision-making and adaptation strategies over the river basin.

A new method is proposed based on the finite difference method (FDM), differential evolution algorithm and Markov Chain Monte Carlo (MCMC) simulation to identify water quality model parameters of an open channel in a long distance water transfer project. Firstly, this parameter identification problem is considered as a Bayesian estimation problem and the forward numerical model is solved by FDM, and the posterior probability density function of the parameters is deduced. Then these parameters are estimated using a sampling method with differential evolution algorithm and MCMC simulation. Finally this proposed method is compared with FDM-MCMC by a twin experiment. The results show that the proposed method can be used to identify water quality model parameters of an open channel in a long distance water transfer project under different scenarios better with fewer iterations, higher reliability and anti-noise capability compared with FDM-MCMC. Therefore, it provides a new idea and method to solve the traceability problem in sudden water pollution accidents.

Use of non-homogeneous Poisson process for the analysis of new cases, deaths, and recoveries of COVID-19 patients: A case study of Kuwait

Among the multi-trait models selected to study several traits and environments jointly, the Bayesian framework has been a preferred tool when constructing a more complex and biologically realistic model. In most cases, non-informative prior distributions are adopted in studies using the Bayesian approach. However, the Bayesian approach presents more accurate estimates when informative prior distributions are used. The present study was developed to evaluate the efficiency and applicability of multi-trait multi-environment (MTME) models within a Bayesian framework utilizing a strategy for eliciting informative prior distribution using previous data on rice. The study involved data pertaining to rice (Oryza sativa L.) genotypes in three environments and five crop seasons (2010/2011 until 2014/2015) for the following traits: grain yield (GY), flowering in days (FLOR) and plant height (PH). Variance components, genetic and non-genetic parameters were estimated using the Bayesian method. In general, the informative prior distribution in Bayesian MTME models provided higher estimates of individual narrow-sense heritability and variance components, as well as minor lengths for the highest probability density interval (HPD), compared to their respective non-informative prior distribution analyses. More informative prior distributions make it possible to detect genetic correlations between traits, which cannot be achieved with non-informative prior distributions. Therefore, this mechanism presented to update knowledge for an elicitation of an informative prior distribution can be efficiently applied in rice breeding programs.

Bayesian sensitivity analysis in elliptical linear regression models