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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635795
• 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.

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。