Bayesian Nonparametric Regression Research Articles

Novel machine learning technique further clarifies unrelated donor selection to optimize transplantation outcomes

AbstractWe investigated the impact of donor characteristics on outcomes in allogeneic hematopoietic cell transplantation (HCT) recipients using a novel machine learning approach, the Nonparametric Failure Time Bayesian Additive Regression Trees (NFT BART). NFT BART models were trained on data from 10 016 patients who underwent a first HLA-A, B, C, and DRB1 matched unrelated donor HCT between 2016 and 2019, reported to the Center for International Blood and Marrow Transplant Research, then validated on an independent cohort of 1802 patients. The NFT BART models were adjusted based on recipient, disease, and transplant variables. We defined a clinically meaningful impact on overall survival (OS) or event-free survival (EFS; survival without relapse, graft failure, or moderate to severe chronic graft-versus-host disease) as >1% difference in predicted outcome at 3 years. Characteristics with <1% impact (within a zone of indifference) were not considered to be clinically relevant. Donor cytomegalovirus, parity, HLA-DQB1, and HLA-DPB1 T-cell epitope matching fell within the zone of indifference. The only significant donor factor that associated with OS was age, in which, compared with 18-year-old donors, donors aged ≥31 years old were associated with lower OS. Both donor age (≤32 years) and use of a male donor, regardless of recipient sex, improved EFS. We, therefore, recommend selecting the earliest available donor within the 18 to 30 years age range for HCT to optimize OS. If several donors in the 18 to 30 years age range are available, a male donor may be chosen to optimize EFS.

Machine learning revealing overlooked conjunction of working volume and mixing intensity in anammox optimization

Extensive studies on improving anammox performance have taken place for decades with particular focuses on its operational and environmental factors, but such parameter-based optimization is difficult, because of the sheer number of possible combinations and multidimensional arrays of these factors. Utilizing machine-learning algorithm and published anammox data, Bayesian nonparametric general regression (BNGR) was applied to identify the possible governing variable(s) from among 11 operating and environmental parameters: reactor type, mixing type, working volume, hydraulic retention time, temperature, influent pH, nitrite, ammonium, nitrate concentration, nitrogen loading rate, and organic concentration. The results showed that working volume is a key but oft-overlooked governing parameter. By integrating the BNGR findings with computational fluid dynamics simulation, which assessed mixing properties, it became feasible to conclude that working volume and mixing intensity co-regulated flow fields in reactors and had a significant influence on anammox performance. Furthermore, this study experimentally validated how mixing intensity affected performance, and specific input power (x), a parameter that conjugates both working volume and mixing intensity, was used to establish the relationship with ammonium removal rate (NH4+-N RR, y) y = 49.90x+1.97 (R2 = 0.94). With specific input power increased from 3.4 × 10−4 to 2.6 × 10−2 kW m−3, the ammonium removal rate exhibited a rise from 1.8 to 3.2 mg L−1h−1. Following, a relationship among input power-working volume-nitrogen removal rate was also established with a view to determining the design variables for anammox reactor. Consequently, the study highlighted the necessity to consider the working volume-mixing intensity correlation when optimizing the anammox process.

Application of non parametric Bayesian methods in high dimensional data

With the development of technology and the widespread collection of data, high-dimensional data analysis has become a research hotspot in many fields. Traditional parameter methods often face problems such as dimensional disasters in high-dimensional data analysis. Non parametric methods have broad application prospects in high-dimensional data because they do not rely on specific parameter distribution assumptions. The Bayesian rule is more suitable for dealing with noise and outliers in high-dimensional data because it takes uncertainty into account. Therefore, it is of great significance to combine non parametric methods with Bayesian methods for application research in high-dimensional data analysis. In this paper, the nonparametric Bayesian method was applied to the analysis of high-dimensional data, and the Dirichlet process Mixture model was used to cluster high-dimensional data. The regression analysis of high-dimensional data was carried out through the prediction model of nonparametric Bayesian regression. In this paper, the nonparametric Bayesian method based on Bayesian sparse linear model was used for feature selection of high-dimensional data. In order to determine the superiority of nonparametric Bayesian methods in high-dimensional data analysis, this paper conducted experiments on nonparametric Bayesian methods and traditional parametric methods in high-dimensional data analysis from five aspects of cluster analysis, classification analysis, regression analysis, feature selection and anomaly detection, and evaluated them through multiple indicators. This article explored the application of non parametric Bayesian methods in high-dimensional data analysis from these aspects through simulation experiments. The experimental results show that the clustering accuracy of the non parametric Bayesian clustering algorithm was 0.93, and the accuracy of the non parametric Bayesian classification algorithm was between 0.93 and 0.99; the coefficient of determination of nonparametric Bayesian regression algorithm was 0.98; the F1 values of non parametric Bayesian methods in anomaly detection ranged from 0.86 to 0.91, which was superior to traditional methods. Non parametric Bayesian methods have broad application prospects in high-dimensional data analysis, and can be applied in multiple fields such as clustering, classification, regression, etc.

Direct Bayesian linear regression for distribution-valued covariates.

In this manuscript, we study scalar-on-distribution regression; that is, instances where subject-specific distributions or densities are the covariates, related to a scalar outcome via a regression model. In practice, only repeated measures are observed from those covariate distributions and common approaches first use these to estimate subject-specific density functions, which are then used as covariates in standard scalar-on-function regression. We propose a simple and direct method for linear scalar-on-distribution regression that circumvents the intermediate step of estimating subject-specific covariate densities. We show that one can directly use the observed repeated measures as covariates and endow the regression function with a Gaussian process prior to obtain a closed form or conjugate Bayesian inference. Our method subsumes the standard Bayesian non-parametric regression using Gaussian processes as a special case, corresponding to covariates being Dirac-distributions. The model is also invariant to any transformation or ordering of the repeated measures. Theoretically, we show that, despite only using the observed repeated measures from the true density-valued covariates that generated the data, the method can achieve an optimal estimation error bound of the regression function. The theory extends beyond i.i.d. settings to accommodate certain forms of within-subject dependence among the repeated measures. To our knowledge, this is the first theoretical study on Bayesian regression using distribution-valued covariates. We propose numerous extensions including a scalable implementation using low-rank Gaussian processes and a generalization to non-linear scalar-on-distribution regression. Through simulation studies, we demonstrate that our method performs substantially better than approaches that require an intermediate density estimation step especially with a small number of repeated measures per subject. We apply our method to study association of age with activity counts.

Is the Youngest Donor Always the Best Choice to Optimize Outcomes for Matched Unrelated Allogeneic Transplant? Improving Precision Using Novel Statistical Methodology

INTRODUCTION Our recently published research has shown younger donors have superior survival for recipients of allogeneic hematopoietic stem cell transplant (HSCT). But, previously, we had not explored the donor age impact on other outcomes, or the impact of smaller age differences (&amp;lt;4 years) when balanced against other factors. We report results using our recently developed machine learning survival methodology, Nonparametric Failure Time Bayesian Additive Regression Trees (NFT BART). Here we study patient-specific optimization of donor selection for overall survival (OS) and event-free survival (EFS including events of death, relapse, graft failure/rejection, or moderate/severe chronic graft vs. host disease, GVHD). METHODS The patient data was provided by the Center for International Blood and Marrow Transplant Research. Each recipient underwent matched unrelated donor HSCT from 2016 to 2019 and was followed for OS and EFS outcomes up to three years. We trained our NFT BART model on 10,016 patients to provide predictions for OS and EFS. The model was validated on 1,802 patients among whom 699 had archived unrelated donor search records available from the National Marrow Donor Program. We examined donor age, sex/parity, CMV status, DPB1 nonpermissive mismatching, and DQB1 matching, while adjusting for relevant patient characteristics. We further restricted donor factors to those with a &amp;gt;1% positive impact on 3-year OS and/or EFS. Using archived search data for each patient, we examined three optimal strategies for donor selection: OS only, EFS only and a 2:1 weighted average between OS and EFS (denoted OS2:EFS1). The OS2:EFS1 effectively prioritizes optimizing OS while still allowing EFS consideration when the difference in OS predictions between two potential donors is minimal. We calculated predicted OS and EFS outcomes for each patient under all three optimal donor strategies. These predicted outcomes were benchmarked against observed practice by subtracting the predicted outcomes from the actual donor used for the transplant. RESULTS Among donor features, only age (for OS and EFS) and sex (for EFS) had clinically important impacts on outcomes. For OS, the youngest donor provided the best result regardless of patient characteristics: see the waterfall plot presented in Figure 1. As shown, the older the donor, the greater the impact of selection by age. Compared to donors 34+, use of an 18 year-old donor is associated with a substantial increase in OS where all of the recipient OS differentials are above 0.01. Compared to donors 22 to 30, the impact of an 18 year-old donor is negligible. For EFS, donor sex was the dominant factor, with males having a median 4% improvement in EFS at 3 years compared to females, while the difference for 42 vs. 18 was &amp;lt;2%. Optimizing for OS led to selection of the youngest donor regardless of sex, while optimizing EFS led to selection of the youngest male donor. Optimizing OS2:EFS1 generally picked the youngest donor except for selections of a male donor that may be older than the female when the age difference is small (median is 1 year older: Q1=0, Q3=2). In said cases, the OS detriment was minimal (median=-0.0005, Q1=-0.0008, Q3=-0.0003) with a substantial improvement in EFS (median=0.038, Q1=0.031, Q3=0.042) gained by switching to the male donor. Individual patient differences (OS2:EFS1 vs. actual donor) in OS vs. EFS are plotted in Figure 2. CONCLUSION Using our novel NFT BART methodology, we showed that donor selection to optimize HSCT outcomes can be improved by considering both OS and EFS. This weighted optimization strategy balances outcomes, prioritizing OS while also allowing for consideration of EFS when OS is likely similar between donors. Additionally, this supports previous research showing that choosing among donors from 18 to 30 with resultant minimal differences in outcomes that are negligible allowing for more flexibility in donor selection.

Active learning aided Bayesian nonparametric general regression for model updating using modal data

In this paper, we propose an active learning aided Bayesian nonparametric general regression (ALBNGR) network for structural model updating using modal data. The proposed network provides an approximate, nonlinear and nonparametric mapping from the modal data to the structural parameters. This serves as a surrogate model for the natural frequencies and mode shapes of a finite element model. To further reduce the number of finite element model evaluations, the proposed method adopts an active learning aided sequential modeling strategy to improve the local accuracy of the surrogate model. Active learning assists in enriching the dataset using gradient descent, with the gradient vector calculated using analytical expressions. The training dataset is updated iteratively and then sequential surrogate models are constructed but only the model in the initial round is trained to obtain the optimal scaling parameter. It is reused in the subsequent rounds to refine the surrogate model along with the updated dataset. Therefore, the efficiency of the active learning aided sequential modeling process can be enhanced. The effectiveness and advantages of the proposed method are demonstrated through the application of two simulated examples and an experimental case of the Canton Tower.

Gaussian process regression and classification using International Classification of Disease codes as covariates

AbstractIn electronic health records (EHRs) data analysis, nonparametric regression and classification using International Classification of Disease (ICD) codes as covariates remain understudied. Automated methods have been developed over the years for predicting biomedical responses using EHRs, but relatively less attention has been paid to developing patient similarity measures that use ICD codes and chronic conditions, where a chronic condition is defined as a set of ICD codes. We address this problem by first developing a string kernel function for measuring the similarity between a pair of primary chronic conditions, represented as subsets of ICD codes. Second, we extend this similarity measure to a family of covariance functions on subsets of chronic conditions. This family is used in developing Gaussian process (GP) priors for Bayesian nonparametric regression and classification using diagnoses and other demographic information as covariates. Markov chain Monte Carlo (MCMC) algorithms are used for posterior inference and predictions. The proposed methods are tuning free, so they are ideal for automated prediction of biomedical responses depending on chronic conditions. We evaluate the practical performance of our method on EHR data collected from 1660 patients at the University of Iowa Hospitals and Clinics (UIHC) with six different primary cancer sites. Our method provides better sensitivity and specificity than its competitors in classifying different primary cancer sites and estimates the marginal associations between chronic conditions and primary cancer sites.

Bayesian nonparametric general regression with adaptive kernel bandwidth and its application to seismic attenuation

General Regression Neural Network (GRNN) possesses distinct function approximation capability and predictive power without the requirement of a prescribed functional form. However, its prediction accuracy relies on uniformly distributed input training data. If the input training data are non-uniformly distributed, considerable bias will occur. It is especially pronounced when the data points are sparsely distributed. Moreover, GRNN presumes a set of input variables to be included in the regression model so it remains an issue to determine the proper set of input variables. To address these issues, we propose the Bayesian Nonparametric General Regression with Adaptive Kernel Bandwidth (BNGR-AKB). First, it determines the bandwidth of the kernels adaptively so as to accommodate non-uniformly distributed input training data. Furthermore, it utilizes Bayesian inference to determine the input variables to be included in the regression model. To demonstrate the variable selection and regression capacity of the proposed method for non-uniformly distributed input training data, we present three simulated examples and one real data example using the ground motion records of Wenchuan earthquake.

Utility-based Bayesian personalized treatment selection for advanced breast cancer.

A Bayesian method is proposed for personalized treatment selection in settings where data are available from a randomized clinical trial with two or more outcomes. The motivating application is a randomized trial that compared letrozole plus bevacizimab to letrozole alone as first-line therapy for hormone receptor positive advanced breast cancer. The combination treatment arm had larger median progression-free survival time, but also a higher rate of severe toxicities. This suggests that the risk-benefit trade-off between these two outcomes should play a central role in selecting each patient's treatment, particularly since older patients are less likely to tolerate severe toxicities. To quantify the desirability of each possible outcome combination for an individual patient, we elicited from breast cancer oncologists a utility function that varied with age. The utility was used as an explicit criterion for quantifying risk-benefit trade-offs when making personalized treatment selections. A Bayesian nonparametric multivariate regression model with a dependent Dirichlet process prior was fit to the trial data. Under the fitted model, a new patient's treatment can be selected based on the posterior predictive utility distribution. For the breast cancer trial dataset, the optimal treatment depends on the patient's age, with the combination preferable for patients 70 years or younger and the single agent preferable for patients older than 70.

Semi-parametric Bayes regression with network-valued covariates

Although there has been an explosive rise in network data in a variety of disciplines, there is very limited development of regression modeling approaches based on high-dimensional networks. The scarce literature in this area typically assume linear relationships between the outcome and the high-dimensional network edges that results in an inflated model plagued by the curse of dimensionality and these models are unable to accommodate non-linear relationships or higher order interactions. In order to overcome these limitations, we develop a novel two-stage Bayesian non-parametric regression modeling framework using high-dimensional networks as covariates, which first finds a lower dimensional node-specific representation for the networks, and then embeds these representations in a flexible Gaussian process regression framework along with supplemental covariates for modeling the continuous outcome variable. Moving from edge-level analysis to node-level model allows us to scale up to high-dimensional networks, and enables node selection via an extension of the Gaussian process framework that involves spike-and-slab priors on the lengthscale parameters. Extensive simulations show a distinct advantage of the proposed approach in terms of prediction, coverage, and node selection. The proposed model achieves considerable gains when predicting posttraumatic stress disorder (PTSD) resilience based on brain networks in our motivating neuroimaging applications, and also identifies important brain regions associated with PTSD. In contrast, existing non-linear approaches that employ the full-edge set or those that use other dimension reduction techniques on the network are not equipped for node selection and results in poor prediction and characterization of predictive uncertainty, while linear approaches using the edge-level features are overly inflated and typically result in poor performance.

Scalable multi-task Gaussian processes with neural embedding of coregionalization

Multi-task regression attempts to exploit the task similarity in order to achieve knowledge transfer across related tasks for improving the quality of prediction and alleviating the demand of big data. The application of Gaussian process (GP) in this scenario yields the non-parametric yet informative Bayesian multi-task regression paradigm. Multi-task GP (MTGP) provides not only the prediction mean but also the associated prediction variance to quantify uncertainty, thus gaining popularity in various scenarios. The linear model of coregionalization (LMC) is a well-known MTGP paradigm which exploits the dependency of tasks through linear combination of several independent and diverse GPs. The LMC however suffers from high model complexity and limited model capability when handling complicated multi-task cases. To this end, we develop the neural embedding of coregionalization that transforms the latent GPs into a high-dimensional latent space to induce rich yet diverse behaviors. Furthermore, we use advanced variational inference as well as sparse approximation to devise a tight and compact evidence lower bound (ELBO) for higher quality of scalable model inference. Extensive numerical experiments have been conducted to verify the higher prediction quality and better generalization of our model, named NSVLMC, on various real-world multi-task datasets and the cross-fluid modeling of unsteady fluidized bed.

Nonparametric Bayesian Regression and Classification on Manifolds, With Applications to 3D Cochlear Shapes.

Advanced shape analysis studies such as regression and classification need to be performed on curved manifolds, where often, there is a lack of standard statistical formulations. To overcome these limitations, we introduce a novel machine-learning method on the shape space of curves that avoids direct inference on infinite-dimensional spaces and instead performs Bayesian inference with spherical Gaussian processes decomposition. As an application, we study the shape of the cochlear spiral-shaped cavity within the petrous part of the temporal bone. This problem is particularly challenging due to the relationship between shape and gender, especially in children. Experimental results for both synthetic and real data show improved performance compared to state-of-the-art methods.

A Nonparametric Tropical Cyclone Wind Speed Estimation Model Based on Dual-Polarization SAR Observations

The C-band synthetic aperture radar (SAR) observation is one of the most popular sources for high-resolution tropical cyclone (TC) wind speed estimation. The scarcity of high wind speed data with good quality restricts the inversion accuracy of high wind speed. It is a challenge to obtain a high-precision wind speed inversion model with sparse data points. In this paper, the TC wind speed is successfully estimated from the dual-polarization SAR signal. Firstly, the training dataset with a total of 327 data points is formed using the Sentinel-1A EW/IW mode images and their temporal-spatial matched Stepped Frequency Microwave Radiometer (SFMR) measurements. Then, a novel nonparametric TC wind speed estimation model (hereafter NWSE model) is proposed with this dataset by using the Bayesian nonparametric general regression method. The wind speed is interpreted as a function of the cross-polarized normalized radar cross-sections (VH-NRCS) and incident angle for the NWSE model. Moreover, the wind speed obtained from the co-polarized signal is used to improve the accuracy of NWSE model under low wind speed. Finally, the validation results show the excellent overall consistency between the model retrieved wind speed and the collocated SFMR and SMAP measurements. Specifically, considering all the wind speeds, the overall root-mean-square error and absolute bias of NWSE model are 2.85 m/s and 2.26 m/s compared with the SMAP wind speed. When considering the wind speeds larger than 30 m/s, the RMSE and bias of NWSE model are 3.75 m/s and 2.78 m/s, respectively.

제약조건이 있는 베이지안 비모수 회귀모형을 활용한 방사선 반응자료 분석

본 논문에서는 제약조건 (shape-restriction)이 있는 베이지안 비모수 회귀모형 (Bayesian non-parametric regression models)을 이용하여 방사선 반응자료 (radiation response data)를 분석한다. 특히, 식품에 방사선을 조사 (irradiation)하여 얻어진 반응 자료를 분석하고 베이지안 추론 (Bayesian inference)의 결과를 제시한다. 베이지안 추론을 위해 사후분포 (posterior distribution)는 가우시안 분포를 근사 분포로 하는 가수시안 변분 베이즈 (Gaussian variational Bayes) 근사 방법을 활용하여 근사하고, 해밀토니안 몬테카를로 (Hamiltonian Monte Carlo)에 기반한 마코프 체인 몬테 카를로 (Markov chain Monte Carlo)방법과의 비교를 통해 변분 추론 (variational inference)의 타당성을 제시한다.

Robust Gaussian process regression with a bias model

This paper presents a new approach to a robust Gaussian process regression, creating a non-parametric Bayesian regression estimate robust to outliers. Most existing approaches replace an outlier-prone Gaussian likelihood with a non-Gaussian likelihood induced from a heavy tail distribution, such as the Laplace distribution and Student-t distribution. However, the use of a non-Gaussian likelihood would incur the need for a computationally expensive Bayesian approximate computation in the posterior inferences. The proposed approach models an outlier as a noisy and biased observation of an unknown regression function, and accordingly, the likelihood contains bias terms to explain the degree of deviations from the regression function. We introduce two bias models that handle the bias terms differently, treating a bias as an unknown and fixed quantity or treating a bias as a random quantity. We entail how the biases can be estimated accurately with other hyperparameters by a regularized maximum likelihood estimation. Conditioned on the bias estimates, the robust GP regression can be reduced to a standard GP regression problem with analytical forms of the predictive mean and variance estimates. Therefore, the proposed approach is simple and very computationally attractive. It also gives a very robust and accurate GP estimate for many tested scenarios. For the numerical evaluation, we perform a comprehensive simulation study to evaluate the proposed approach with the comparison to the existing robust GP approaches under various simulated scenarios of different outlier proportions and different noise levels. The approach is applied to data from two measurement systems, where the predictors are based on robust environmental parameter measurements and the response variables utilize more complex chemical sensing methods that contain a certain percentage of outliers. The utility of the measurement systems and value of the environmental data are improved through the computationally efficient GP regression and bias model.

Bayesian nonparametric quantile process regression and estimation of marginal quantile effects.

Flexible estimation of multiple conditional quantiles is of interest in numerous applications, such as studying the effect of pregnancy-related factors on low and high birth weight. We propose a Bayesian nonparametric method to simultaneously estimate noncrossing, nonlinear quantile curves. We expand the conditional distribution function of the response in I-spline basis functions where the covariate-dependent coefficients are modeled using neural networks. By leveraging the approximation power of splines and neural networks, our model can approximate any continuous quantile function. Compared to existing models, our model estimates all rather than a finite subset of quantiles, scales well to high dimensions, and accounts for estimation uncertainty. While the model is arbitrarily flexible, interpretable marginal quantile effects are estimated using accumulative local effect plots and variable importance measures. A simulation study shows that our model can better recover quantiles of the response distribution when the data are sparse, and an analysis of birth weight data ispresented.

Congestion in cities: evidence on the form and causality of the macroscopic fundamental relationship of traffic flow

This paper quantifies the technology driving congestion in urban road networks. To do so, we estimate macroscopic fundamental relationships for homogeneously congested sub-networks (reservoirs) in thirty-four cities worldwide. We adopt a causal approach based on non-parametric instrumental variables to estimate the form of the reservoir-level flow-density relationship using large-scale traffic sensor data. Specifically, we apply a Bayesian non-parametric spline-based regression model with instrumental variables to adjust for potential confounding/endogeneity biases due to simultaneity and omitted variables such as vehicle interactions and traffic controls. Our estimates suggest that the provision of vehicular travel in cities is subject to decreasing returns to density and network size. Importantly, we find that increasing road network capacity is not an efficient solution to manage congestion because average travel speed does not change substantially with increase in capacity. As a by-product of the estimation, we also deliver estimates of important traffic control inputs such as capacity and critical occupancy for these reservoirs. Our results have implications for traffic flow modelling used by both economists and traffic engineers.

Variable selection consistency of Gaussian process regression

Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped with stochastic variable selection, are known to also adapt to the unknown intrinsic dimension of a sparse true regression function. But it remains unclear if such extensions offer variable selection consistency, that is, if the true subset of important variables could be consistently learned from the data. It is shown here that variable consistency may indeed be achieved with such models at least when the true regression function has finite smoothness to induce a polynomially larger penalty on inclusion of false positive predictors. Our result covers the high-dimensional asymptotic setting where the predictor dimension is allowed to grow with the sample size. The proof utilizes Schwartz theory to establish that the posterior probability of wrong selection vanishes asymptotically. A necessary and challenging technical development involves providing sharp upper and lower bounds to small ball probabilities at all rescaling levels of the Gaussian process prior, a result that could be of independent interest.

A Bayesian nonparametric analysis for zero-inflated multivariate count data with application to microbiome study.

High-throughput sequencing technology has enabled researchers to profile microbial communities from a variety of environments, but analysis of multivariate taxon count data remains challenging. We develop a Bayesian nonparametric (BNP) regression model with zero inflation to analyse multivariate count data from microbiome studies. A BNP approach flexibly models microbial associations with covariates, such as environmental factors and clinical characteristics. The model produces estimates for probability distributions which relate microbial diversity and differential abundance to covariates, and facilitates community comparisons beyond those provided by simple statistical tests. We compare the model to simpler models and popular alternatives in simulation studies, showing, in addition to these additional community-level insights, it yields superior parameter estimates and model fit in various settings. The model's utility is demonstrated by applying it to a chronic wound microbiome data set and a Human Microbiome Project data set, where it is used to compare microbial communities present in different environments.

Bayesian analysis for quantile smoothing spline

In Bayesian quantile smoothing spline [Thompson, P., Cai, Y., Moyeed, R., Reeve, D., & Stander, J. (2010). Bayesian nonparametric quantile regression using splines. Computational Statistics and Data Analysis, 54, 1138–1150.], a fixed-scale parameter in the asymmetric Laplace likelihood tends to result in misleading fitted curves. To solve this problem, we propose a new Bayesian quantile smoothing spline (NBQSS), which considers a random scale parameter. To begin with, we justify its objective prior options by establishing one sufficient and one necessary condition of the posterior propriety under two classes of general priors including the invariant prior for the scale component. We then develop partially collapsed Gibbs sampling to facilitate the computation. Out of a practical concern, we extend the theoretical results to NBQSS with unobserved knots. Finally, simulation studies and two real data analyses reveal three main findings. Firstly, NBQSS usually outperforms other competing curve fitting methods. Secondly, NBQSS considering unobserved knots behaves better than the NBQSS without unobserved knots in terms of estimation accuracy and precision. Thirdly, NBQSS is robust to possible outliers and could provide accurate estimation.