Bayesian inference under qualitative assumptions and its applications

在定性假设下估计函数的问题有很长的历史，至少可以追溯到过去的五十年。形状约束回归模型对于分析具有特定形状响应的数据非常有用，例如（单调的）剂量-反应曲线，风险规避决策者的（凹的）效用函数，儿童身高随时间的（递增的）增长曲线，这些在医学、经济和流行病学研究中特别常见。由于理论和实践的原因，形状和平滑约束下的回归和密度估计都引起了相当大的兴趣。从理论上讲，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，在许多情况下，有必要考虑到一些关于函数形状的先验知识来进行估计。局部支持基函数的b样条nice特性使我们能够对约束进行局部化，或者考虑约束的组合，例如在区间[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，我已经进行了这样的约束组合。最近，我通过考虑局部形状优化的随机模拟退火方法，对光滑样条上下文进行了扩展，并将其思想推广到自由结自适应贝叶斯推理。现在，我已经开始推导基于贝叶斯因子标准的渐近分析和模型选择，以配合该方法。我刚刚提出了我所谓的拉普拉斯-贝叶斯因子它很好地选择了应该进入模型的最佳约束。

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.