Markov Chain Monte Carlo Techniques Research Articles

Spatial-temporal Bayesian accelerated failure time models for survival endpoints with applications to prostate cancer registry data

Prostate cancer is the most common cancer after non-melanoma skin cancer and the second leading cause of cancer deaths in US men. Its incidence and mortality rates vary substantially across geographical regions and over time, with large disparities by race, geographic regions (i.e., Appalachia), among others. The widely used Cox proportional hazards model is usually not applicable in such scenarios owing to the violation of the proportional hazards assumption. In this paper, we fit Bayesian accelerated failure time models for the analysis of prostate cancer survival and take dependent spatial structures and temporal information into account by incorporating random effects with multivariate conditional autoregressive priors. In particular, we relax the proportional hazards assumption, consider flexible frailty structures in space and time, and also explore strategies for handling the temporal variable. The parameter estimation and inference are based on a Monte Carlo Markov chain technique under a Bayesian framework. The deviance information criterion is used to check goodness of fit and to select the best candidate model. Extensive simulations are performed to examine and compare the performances of models in different contexts. Finally, we illustrate our approach by using the 2004-2014 Pennsylvania Prostate Cancer Registry data to explore spatial-temporal heterogeneity in overall survival and identify significant risk factors.

Statistical inference of unified hybrid censoring scheme for generalized inverted exponential distribution with application to COVID-19 data

Using a unified hybrid censoring scheme, this study explores statistical inferences for products with lifetimes following the generalized inverted exponential distribution. The model’s parameters are estimated using the maximum likelihood approach. In addition, likelihood functions and asymptotic theories are employed to generate approximate confidence intervals. Moreover, Bayesian estimates based on classical likelihood functions are investigated, considering both asymmetric and symmetric loss functions with prior information. It is recommended to approximate the Bayes estimates using Gibbs sampling, which utilizes the Markov chain Monte Carlo technique to establish credible intervals for the parameters. A numerical example is provided to demonstrate the effectiveness of the proposed methods. Furthermore, a simulation study is conducted to illustrate the confidence intervals and the statistical characteristics of the parameters.

A modified lognormal approximation of the Lyman-α forest: comparison with full hydrodynamic simulations at 2 ≤ z ≤ 2.7

Observations of the Lyman-α forest in distant quasar spectra with upcoming surveys are expected to provide significantly larger and higher-quality datasets. To interpret these datasets, it is imperative to develop efficient simulations. One such approach is based on the assumption that baryonic densities in the intergalactic medium (IGM) follow a lognormal distribution. We extend our earlier work to assess the robustness of the lognormal model of the Lyman-α forest in recovering the parameters characterizing IGM state, namely, the mean-density IGM temperature (T 0), the slope of the temperature-density relation (γ), and the hydrogen photoionization rate (Γ12), by comparing with high-resolution Sherwood SPH simulations across the redshift range 2 ≤ z ≤ 2.7. These parameters are estimated through a Markov Chain Monte Carlo (MCMC) technique, using the mean and power spectrum of the transmitted flux. We find that the usual lognormal distribution of IGM densities cannot recover the parameters of the SPH simulations. This limitation arises from the fact that the SPH baryonic density distribution cannot be described by a simple lognormal form. To address this, we extend the model by scaling the linear density contrast by a parameter ν. While the resulting baryonic density is still lognormal, the additional parameter gives us extra freedom in setting the variance of density fluctuations. With this extension, values of T 0 and γ implied in the SPH simulations are recovered at ∼ 1 - σ (≲ 10%) of the median (best-fit) values for most redshifts bins. However, this extended lognormal model cannot recover Γ12 reliably, with the best-fit value discrepant by ≳ 3 - σ for z > 2.2. Despite this limitation in the recovery of Γ12, whose origins we explain, we argue that the model remains useful for constraining cosmological parameters.

Joint analysis of multivariate longitudinal, imaging, and time-to-event data

Abstract Alzheimer’s (AD) is a progressive neurodegenerative disease frequently associated with memory deficits and cognitive decline. Despite its irreversible once onset, some discoveries revealed the existence of a certain percentage of people who are non-susceptible to AD. This study proposes a joint analysis of multivariate longitudinal data, survival data with a non-susceptible fraction, and ultrahigh-dimensional imaging data. The proposed model comprises three major components. The first component is a mixture proportional hazards cure model with images to examine the potential predictors of the non-susceptible probability and hazards of interest. The second component is a dynamic factor analysis model with images to characterize group-specific latent factors through multiple observed variables. The last component is a semiparametric trajectory model to reveal the change patterns of the dynamic latent factors in the ‘non-susceptible’ and ‘susceptible’ groups. A two-stage approach is developed for statistical inference. The first stage manages the imaging data through high-dimensional functional principal component analysis. The second stage develops a Bayesian approach coupled with penalized splines, data augmentation, and Markov chain Monte Carlo techniques to perform estimation. The application to the Alzheimer’s Disease Neuroimaging Initiative dataset sheds new insight into the pathology of AD.

Analysis of the effect of surface coating and texturization on the estimation of effective thermophysical properties

This article aims to investigate the influence of surface modifications on the effective thermophysical properties of textured carbon steel and coated hard metal. The novelty of this work lies in the simultaneous estimation of thermal diffusivity and thermal conductivity via measurements using one active heating surface and Bayesian inference to solve the inverse heat conduction problem. To achieve this, an experimental setup was developed where samples were partially heated on an active surface, and the temperature was measured at two different points on the surface to estimate the thermal properties. The method was applied to two different thermal models using the same set of experimental data. Thus, the thermophysical properties were determined with and without the Markov Chain Monte Carlo technique using the Metropolis-Hastings sampling algorithm. Both techniques proved suitable for estimating the thermophysical properties. It is noteworthy that texturization and coating did not significantly modify the effective thermal diffusivity of the analyzed samples. However, there were considerable changes in the effective thermal conductivity, with an increase of approximately 13 % in the textured carbon steel sample and a reduction of approximately 11 % in the coated hard metal sample. The expanded uncertainty of the estimated effective thermophysical properties was less than 14 %. These results reflect low dispersion and good accuracy of the experimental technique used.

Estimation of stress-strength reliability in s-out-of-k system for new flexible exponential distribution under progressive type-II censoring

This article addresses stress-strength reliability estimation in s-out-of-k systems under progressive type-II censoring.  The system consists of components facing common stress, where stress and strength variable follows independent new flexible exponential (NFE) distributions, sharing a common parameter. The estimation procedure is carried out using MLE, Bayesian estimation and uniformly minimum variance unbiased estimation (UMVUE) techniques. MLE is derived for known and unknown common parameters. Bayesian estimates are obtained using Lindley’s approximation and Markov chain Monte Carlo techniques due to the complexity of obtaining the exact Bayesian estimates, when the common parameter is unknown. Moreover, UMVUE is derived when the common parameter is known. The 95% asymptotic confidence interval, highest posterior density credible intervals, and Bayesian credible intervals, are constructed. An extensive simulation study is carried out to compare the performance of various censoring schemes and estimation methods. Finally, two progressively censored real datasets are analyzed for illustration.

Evaluating the lifetime performance index of omega distribution based on progressive type-II censored samples

Besides achieving high quality products, statistical techniques are applied in many fields associated with health such as medicine, biology and etc. Adhering to the quality performance of an item to the desired level is a very important issue in various fields. Process capability indices play a vital role in evaluating the performance of an item. In this paper, the larger-the-better process capability index for the three-parameter Omega model based on progressive type-II censoring sample is calculated. On the basis of progressive type-II censoring the statistical inference about process capability index is carried out through the maximum likelihood. Also, the confidence interval is proposed and the hypothesis test for estimating the lifetime performance of products. Gibbs within Metropolis–Hasting samplers procedure is used for performing Markov Chain Monte Carlo (MCMC) technique to achieve Bayes estimation for unknown parameters. Simulation study is calculated to show that Omega distribution's performance is more effective. At the end of this paper, there are two real-life applications, one of them is about high-performance liquid chromatography (HPLC) data of blood samples from organ transplant recipients. The other application is about real-life data of ball bearing data. These applications are used to illustrate the importance of Omega distribution in lifetime data analysis.

An Analytical Solution for Variable Viscosity Flow in Fractured Media: Development and Comparative Analysis With Numerical Simulations

AbstractExplicit fracture models often use analytical solutions for predicting flow in fractured media, usually assuming uniform fluid viscosity for simplicity. This assumption, however, can be inaccurate as fluid viscosity varies due to factors like composition, temperature, and dissolved substances. Our study, recognizing these discrepancies, abandons this uniform viscosity assumption for a more realistic model of variable viscosity flow, focusing on viscous displacement scenarios. This includes instances like injecting viscous surfactants for hydrocarbon recovery in fractured reservoirs or soil decontamination. This presents a significant challenge, enhancing our understanding of transport within fractures, mainly governed by advection. Our study centers on a low‐permeability rock matrix intersected by two fractures with variable apertures. We employ two methods: an analytical approach with a new solution and numerical simulations with two distinct in‐house codes, discretizing both the rock matrix and fractures with two‐dimensional triangular elements. The first code uses a Discontinuous Galerkin finite element method, while the second utilizes a finite‐volume method, allowing a comprehensive comparison of solutions. Additionally, we investigate parameter identifiability, like fracture apertures and viscosity ratios, using breakthrough curves from our analytical solution, applying the Markov Chain Monte Carlo technique.

Estimation and Optimal Censoring Plan for a New Unit Log-Log Model via Improved Adaptive Progressively Censored Data

To gather enough data from studies that are ongoing for an extended duration, a newly improved adaptive Type-II progressive censoring technique has been offered to get around this difficulty and extend several well-known multi-stage censoring plans. This work, which takes this scheme into account, focuses on some conventional and Bayesian estimation missions for parameter and reliability indicators, where the unit log-log model acts as the base distribution. The point and interval estimations of the various parameters are looked at from a classical standpoint. In addition to the conventional approach, the Bayesian methodology is examined to derive credible intervals beside the Bayesian point by leveraging the squared error loss function and the Markov chain Monte Carlo technique. Under varied settings, a simulation study is carried out to distinguish between the standard and Bayesian estimates. To implement the proposed procedures, two actual data sets are analyzed. Finally, multiple precision standards are considered to pick the optimal progressive censoring scheme.

Selective Internal Radiation Therapy Using Y-90 Resin Microspheres for Metastatic Colorectal Cancer: An Updated Systematic Review and Network Meta-Analysis.

This literature review and exploratory network meta-analysis (NMA) aimed to compare the clinical effectiveness and tolerability of selective internal radiation therapy (SIRT) using yttrium-90 (Y-90) resin microspheres, regorafenib (REG), trifluridine-tipiracil (TFD/TPI), and best supportive care (BSC) in adult patients with chemotherapy-refractory or chemotherapy-intolerant metastatic colorectal cancer (mCRC). In light of recently published data, the literature was searched to complement and update a review published in 2018. Studies up to December 2022 comparing two or more of the treatments and reporting overall survival (OS), progression-free survival (PFS), or incidence of adverse events (AE) were included. The NMA compared hazard ratios (HRs) for OS and PFS using Markov chain Monte Carlo techniques. Fifteen studies were included, with eight studies added (none addressing SIRT). All active treatments improved OS in relation to BSC. SIRT had the longest OS among all treatments, although without statistically significant differences (HR [95% credible interval] for SIRT, 0.48 [0.27, 0.87]; TFD/TPI, 0.62 [0.46, 0.83]; REG, 0.78 [0.57, 1.05]) in a fixed effects model. Information regarding SIRT was insufficient for PFS analysis, and TFD/TPI was the best intervention (HR 2.26 [1.6, 3.18]). One SIRT study reported radioembolization-induced liver disease in > 10% of the sample; this was symptomatically managed. Non-haematological AEs (hand-foot skin reaction, fatigue, diarrhoea, hypertension, rash or desquamation) were more common with REG, while haematological events (neutropoenia, leukopenia, and anaemia) were more common with TFD/TPI. Current evidence supports SIRT treatment in patients with chemotherapy-refractory or chemotherapy-intolerant mCRC compared to newer oral agents, with comparable OS and low incidence of AEs.

On Estimation of Shannon’s Entropy of Maxwell Distribution Based on Progressively First-Failure Censored Data

Shannon’s entropy is a fundamental concept in information theory that quantifies the uncertainty or information in a random variable or data set. This article addresses the estimation of Shannon’s entropy for the Maxwell lifetime model based on progressively first-failure-censored data from both classical and Bayesian points of view. In the classical perspective, the entropy is estimated using maximum likelihood estimation and bootstrap methods. For Bayesian estimation, two approximation techniques, including the Tierney-Kadane (T-K) approximation and the Markov Chain Monte Carlo (MCMC) method, are used to compute the Bayes estimate of Shannon’s entropy under the linear exponential (LINEX) loss function. We also obtained the highest posterior density (HPD) credible interval of Shannon’s entropy using the MCMC technique. A Monte Carlo simulation study is performed to investigate the performance of the estimation procedures and methodologies studied in this manuscript. A numerical example is used to illustrate the methodologies. This paper aims to provide practical values in applied statistics, especially in the areas of reliability and lifetime data analysis.

Statistical analysis of parameters and adsorption isotherm models.

The present work intends to discuss parameter estimation and statistical analysis in adsorption. The Langmuir and Tóth isotherm models are compared for a set of carbon dioxide adsorption data on 13X zeolite from literature at different temperatures: 303, 323, 373, and 423 K. Statistical analyses were performed under frequentist and Bayesian perspectives. Under the frequentist statistical view, parameters were estimated using Maximum Likelihood estimation (MLE). Statistical analyses of parameters were performed by confidence regions in terms of elliptical approximation and likelihood region, while the evaluation of models was performed by chi-square statistics. The results showed that, for these nonlinear models, the elliptical region offers a poor approximation of the parameter estimates' confidence region, especially for the most correlated parameter pairs. Additionally, the four-parameter Tóth's equation yields less correlated parameters than the three-parameter Langmuir model. From a Bayesian perspective, the Markov chain Monte Carlo (MCMC) technique facilitated the reconstruction of the probability density functions of parameters as well as enabled the propagation of parametric uncertainties in the model responses. Finally, the accurate assessment of experimental uncertainty significantly influences the evaluation of models and their respective parameters.

Measures of Model Risk for Continuous-Time Finance Models

Abstract Measuring model risk is required by regulators in financial and insurance markets. We separate model risk into parameter estimation risk (PER) and model specification risk (MSR), and we propose expected shortfall type model risk measures applied to Lévy jump, affine jump-diffusion, and multifactor models. We investigate the impact of PER and MSR on the models’ ability to capture the joint dynamics of stock and option prices. Using Markov chain Monte Carlo techniques, we implement two methodologies to estimate parameters under the risk-neutral probability measure and the real-world probability measure jointly.

Differentiable stochastic halo occupation distribution

ABSTRACT In this work, we demonstrate how differentiable stochastic sampling techniques developed in the context of deep reinforcement learning can be used to perform efficient parameter inference over stochastic, simulation-based, forward models. As a particular example, we focus on the problem of estimating parameters of halo occupation distribution (HOD) models that are used to connect galaxies with their dark matter haloes. Using a combination of continuous relaxation and gradient re-parametrization techniques, we can obtain well-defined gradients with respect to HOD parameters through discrete galaxy catalogue realizations. Having access to these gradients allows us to leverage efficient sampling schemes, such as Hamiltonian Monte Carlo, and greatly speed up parameter inference. We demonstrate our technique on a mock galaxy catalogue generated from the Bolshoi simulation using a standard HOD model and find near-identical posteriors as standard Markov chain Monte Carlo techniques with an increase of ∼8× in convergence efficiency. Our differentiable HOD model also has broad applications in full forward model approaches to cosmic structure and cosmological analysis.

A Non‐parametric Estimation of Productivity with Idiosyncratic and Aggregate Shocks: The Role of Research and Development (R&D) and Corporate Tax

We develop a non‐parametric technique framework for estimating firm‐level Total Factor Productivity (TFP). Our paper has two major novelties: first, we propose a modelling of productivity with both firm‐idiosyncratic factors and aggregate shocks. Second, we apply the Bayesian Markov Chain Monte Carlo (MCMC) technique that offers a numerical integration of productivity outside the posterior overcoming the restrictive assumptions about the relationship between productivity and variable production inputs. We implement our methodology in a group of 4,286 manufacturing firms from France, Germany, Italy, and the UK (2001–14). The results show that: (i) aggregate shocks matter for firm TFP evolution. The global financial crisis of 2008 caused severe, albeit short, adverse effects on TFP; (ii) there is substantial heterogeneity across countries in the way firms react to changes in R&D and taxation. German and UK firms are more sensitive to fiscal changes than R&D, while the opposite is true for Italian firms. R&D and taxation effects are symmetrical for French firms; (iii) the UK productivity handicap continues for years after the financial crisis; and (iv) there are substantial knowledge spillovers among German and Italian firms.

Bayesian estimation : Number of species from Poisson mixed exponential-gamma distribution using objective priors

In this paper, we have considered the Poisson-exponential-gamma mixture model as a species problem. The classical estimation of the parameter has been proposed through profile and conditional likelihood methods. While in the Bayesian paradigm, Bernardo’s reference and Jeffery’s priors have been used to estimate the number of species. Although a Markov Chain Monte Carlo (MCMC) technique has been used for obtaining the Bayes estimates of the number of species parameter. The performances of the proposed estimators are compared in terms of their risks through the simulation study. The applicability, and suitability of the proposed model have been performed using Mount Kenya insect species data.

Post-reionization H i 21-cm signal: a probe of negative cosmological constant

ABSTRACT In this study, we investigate a cosmological model involving a negative cosmological constant (AdS vacua in the dark energy sector). We consider a quintessence field on top of a negative cosmological constant and study its impact on cosmological evolution and structure formation. We use the power spectrum of the redshifted H i 21-cm brightness temperature maps from the post-reionization epoch as a cosmological probe. The signature of baryon acoustic oscillations (BAO) on the multipoles of the power spectrum is used to extract measurements of the angular diameter distance DA(z) and the Hubble parameter H(z). The projected errors on these are then subsequently employed to forecast the constraints on the model parameters ($H_0, \Omega _{m}, \Omega _\Lambda , w_0, w_a$) using Markov chain Monte Carlo techniques. We find that a negative cosmological constant with a phantom dark energy equation of state (EoS) and a higher value of H0 is viable from BAO distance measurements data derived from galaxy samples. We also find that BAO imprints on the 21-cm power spectrum obtained from a futuristic SKA-mid like experiment yield a 1σ error on a negative cosmological constant and the quintessence dark energy EoS parameters to be $\Omega _\Lambda =-1.030^{0.589}_{-1.712}$ and $w_0=-1.023^{0.043}_{-0.060}$, $w_a=-0.141^{0.478}_{-0.409}$ respectively.

Subgroup identification of targeted therapy effects on biomarker for time to event data.

The initiation biomarker-driven trials have revolutionized oncology drug development by challenging the traditional phased approach and introducing basket studies. Notable successes in non-small cell lung cancer (NSCLC) with ALK, ALK/ROS1, and EGFR inhibitors have prompted the need to expand this approach to other cancer sites. This study explores the use of dose response modeling and time-to-event algorithms on the biomarker molecular targeted agent (MTA). By simulating subgroup identification in MTA-related time-to-event data, the study aims to develop statistical methodology supporting biomarker-driven trials in oncology. A total of n patients are selected assigned for different doses. A dataset is prepared to mimic the situation on Subgroup Identification of MTA for time to event data analysis. The response is measured through MTA. The MTA value is also measured through ROC. The Markov Chain Monte Carlo (MCMC) techniques are prepared to perform the proposed algorithm. The analysis is carried out with a simulation study. The subset selection is performed through the Threshold Limit Value (TLV) by the Bayesian approach. The MTA is observed with range 12-16. It is expected that there is a marginal level shift of the MTA from pre to post-treatment. The Cox time-varying model can be adopted further as causal-effect relation to establishing the MTA on prolonging the survival duration. The proposed work in the statistical methodology to support the biomarker-driven trial for oncology research. This study extends the application of biomarker-driven trials beyond NSCLC, opening possibilities for implementation in other cancer sites. By demonstrating the feasibility and efficacy of utilizing MTA as a biomarker, the research lays the foundation for refining and validating biomarker use in clinical trials. These advancements aim to enhance the precision and effectiveness of cancer treatments, ultimately benefiting patients.

Bayesian Analysis of Unit Log-Logistic Distribution Using Non-Informative Priors

The unit log-logistic distribution is a suitable choice for modeling data enclosed within the unit interval. In this paper, estimating the parameters of the unit-log-logistic distribution is performed through a Bayesian approach with non-informative priors. Specifically, we use Jeffreys, reference, and matching priors, with the latter depending on the interest parameter. We derive the corresponding posterior distributions and validate their propriety. The Bayes estimators are then computed using Markov Chain Monte Carlo techniques. To assess the finite sample performance of these Bayes estimators, we conduct Monte Carlo simulations, evaluating their mean squared errors and their coverage probabilities of the highest posterior density credible intervals. Finally, we use these priors to obtain estimations and credible sets for the parameters in an example of a real data set for illustrative purposes.

Accelerating Markov Chain Monte Carlo sampling with diffusion models

Global fits of physics models require efficient methods for exploring high-dimensional and/or multimodal posterior functions. We introduce a novel method for accelerating Markov Chain Monte Carlo (MCMC) sampling by pairing a Metropolis-Hastings algorithm with a diffusion model that can draw global samples with the aim of approximating the posterior. We briefly review diffusion models in the context of image synthesis before providing a streamlined diffusion model tailored towards low-dimensional data arrays. We then present our adapted Metropolis-Hastings algorithm which combines local proposals with global proposals taken from a diffusion model that is regularly trained on the samples produced during the MCMC run. Our approach leads to a significant reduction in the number of likelihood evaluations required to obtain an accurate representation of the Bayesian posterior across several analytic functions, as well as for a physical example based on a global analysis of parton distribution functions. Our method is extensible to other MCMC techniques, and we briefly compare our method to similar approaches based on normalizing flows. A code implementation can be found at https://github.com/NickHunt-Smith/MCMC-diffusion.