# Bayesian Estimation of Simultaneous Regression Quantiles Using Hamiltonian Monte Carlo
## 1 Introduction
The proposed estimation procedure, SBQR, for simultaneous Bayesian quantile regression, as presented in [27], guarantees the fundamental property of non-crossing, under the assumption that the Asymmetric Laplace distribution (ALD) is the underlying data distribution. In extending and enhancing this foundational approach, our primary objective is to present a unified computational methodology. This methodology is designed to facilitate the simultaneous estimation of regression quantiles, employing the advanced Hamiltonian Monte Carlo (HMC) algorithm. Building on the achievements of [27], our proposed methodology aims to overcome potential limitations and further streamline the quantile estimation process. By integrating the HMC algorithm, we seek to harness its efficiency in exploring parameter spaces, thus enhancing the precision of quantile estimates. To evaluate the performance of our algorithm, we compare the mean of the Bayesian estimators with the theoretical quantile values using the root mean square error. For this reason, our objective is to simultaneously estimate the Bayesian regression quantile using the HMC algorithm. The main contributions of the paper are then the following:  
* Conduct a simultaneous quantile estimation embedded with the HMC algorithm through Bayesian inference;
* Conduct a sensitivity analysis on the performance of the HMC algorithm in handling univariate linear quantiles and multivariate linear quantiles. This article serves as a comprehensive guide to the unified computational framework, providing researchers and practitioners with a robust tool for simultaneous regression quantile estimation. By addressing challenges associated with multiple quantile modeling in a cohesive manner, our methodology proves versatile across diverse fields when used with ALD data. Notably, applications in finance, where risk assessment demands a nuanced understanding of the distribution, benefit from the precision offered by simultaneous regression quantile estimation. In environmental science, the ability to capture extreme events accurately is crucial, while in healthcare, predicting patient outcomes relies on a comprehensive grasp of varying response scenarios. Additionally, in manufacturing,ensuring quality control is facilitated by an in-depth understanding of distributional patterns. This versatility aligns with the broader societal goals, contributing to Sustainable Development Goal 9: Industry, Innovation, and Infrastructure. By fostering innovation in statistical methodologies and decision-making processes, our approach seeks to enhance the well-being of societies and align with broader sustainability objectives. This paper introduces a novel approach by combining existing research with the innovative application of the HMC algorithm to simultaneous quantile estimation, offering significant advancements in the field. The paper is organized as follows: The ALD and its Gaussian mixture are presented in Section 2, in addition to the method proposed by the author in [27] for simultaneous Bayesian quantiles. A brief overview of HMC is presented in Section 3. The performance of the HMC algorithm is examined in cases of univariate linear quantiles and multivariate linear quantiles on a simulated data set in Section 4. Finally, Section 5 highlights the paper's contributions and provides a discussion of perspectives.