Bayesian inference under qualitative assumptions and its applications

定性假设下的贝叶斯推理及其应用

• 批准号: RGPIN-2016-05931
• 负责人: Khadraoui, Khader
• 金额: $ 1.09万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615998
• 关键词: Bayesian; inference; under; qualitative; assumptions

The problem of estimating functions under qualitative assumptions has a long history dating back at least to the last five decades. Shape constrained regression models are useful for analyzing data with specific shape responses, such as (monotone) dose-response curves, the (concave) utility functions of a risk averse decision maker, the (increasing) growth curves of children's height through time, that are particularly common in medicine, economic and epidemiological studies. Regression and density estimation under shape and smoothness constraints are of considerable interest both for theoretical and practical reasons. From a theoretical point of view, shape constraints and smoothing usually reduce the variance of the estimators. From a practical point of view, there are many situations in which it is necessary to take into account some prior knowledge on the shape of the function to estimate. The B-spline nice property of locally supported basis functions enables us to localize the constraint or to consider a combination of constraints like being increasing on the interval [1,2] and concave on the interval [3,4]. Such combinations of constraints have been undertaken in my PhD thesis and my recent papers. Recently, I proposed an extension to the the smoothing spline context by considering a stochastic simulated annealing method for localized shapes optimization and we generalized the idea to the free-knot adaptive Bayesian inference. Now, I have started to derive asymptotic analysis and model selection based on Bayes factor criteria to go along with this method. I have just come up with what I have called the Laplace-Bayes factor that works well to choose the best constraint that should enter the model. Now my research program over the next few years builds upon these recent developments and consists in developing new tools to fit regression and density models to datasets with specific shape responses. The short term objectives will be to fine tune the Laplace-Bayes Factor criterion to increase its efficiency when dealing with small datasets and massive databases. One key advantage of our method is that computations can easily be implemented by simulated annealing scheme. The analysis of acidification curves and global warming data will be the main applications of the new methodologies in the short term. In the medium term, I would like with my students to adapt my new techniques to the density and the mixture estimation problems and to develop model selection procedures that will be implementable in this context. Industrial partners will likely get involved at this stage and should provide data and internship opportunities for students. R packages to implement the methods will be made publicly available. In the long term, we would like to undertake the asymptotic analysis (consistency and convergence rates) of our Bayesian procedures in the presence of qualitative assumptions.

在定性假设下估计函数的问题有着悠久的历史，至少可以追溯到过去的五十年。形状约束回归模型可用于分析具有特定形状响应的数据，例如（单调）剂量响应曲线、风险厌恶决策者的（凹）效用函数、儿童身高随时间的（增加）生长曲线，这些在医学、经济和流行病学研究中特别常见。形状和光滑度约束下的回归和密度估计在理论和实践上都有很大的意义。从理论的角度来看，形状约束和平滑通常会减少估计量的方差。从实际的角度来看，有许多情况下，有必要考虑一些先验知识的形状的函数估计。局部支撑基函数的B样条良好性质使我们能够局部化约束或考虑约束的组合，如在区间[1，2]上增加和在区间[3，4]上凹。在我的博士论文和最近的论文中，我已经进行了这种约束的组合。最近，我提出了一个扩展的光滑样条上下文考虑随机模拟退火方法局部形状优化，我们推广的想法，自由结自适应贝叶斯推理。现在，我已经开始推导渐近分析和模型选择的基础上贝叶斯因子准则去沿着与这种方法。我刚刚提出了一个我称之为拉普拉斯-贝叶斯因子的方法，它可以很好地选择应该进入模型的最佳约束。 现在，我在未来几年的研究计划建立在这些最新发展的基础上，包括开发新的工具来拟合回归和密度模型到具有特定形状响应的数据集。短期目标将是微调拉普拉斯-贝叶斯因子准则，以提高其处理小型数据集和大型数据库时的效率。我们的方法的一个关键优点是，计算可以很容易地实现模拟退火方案。对酸化曲线和全球变暖数据的分析将是新方法在短期内的主要应用。在中期，我想与我的学生，以适应我的新技术的密度和混合估计问题，并制定模型选择程序，将在这种情况下实施。工业合作伙伴可能会在这个阶段参与进来，并为学生提供数据和实习机会。实现这些方法的R包将公开提供。从长远来看，我们希望在存在定性假设的情况下对我们的贝叶斯方法进行渐进分析（一致性和收敛率）。