Model Uncertainty in Prediction, Variable Selection and Related Decision Problems

基于复杂模型的统计预测可能对建模假设非常敏感，例如协变量的选择。因此，选择单一模型可能不会导致令人满意的预测，并且由于没有将模型选择的不确定性纳入最终答案，可能会大大低估预测间隔。在问题中，模型的不确定性往往超过其他不确定性来源，但往往被忽略。贝叶斯方法提供了一个非常有效和概念上吸引人的选择：预测和推断可以基于一组模型而不是单一模型；每个模型的贡献与它从观测数据中得到的支持成比例。本研究涉及到随机搜索高维模型空间的贝叶斯方法。由于模型的数量非常大，因此面临的挑战是找到有效的方法来探索模型的空间，选择合理的模型，并为每个模型赋予一个权重（近似后验概率），用于基于混合的预测或其他效用计算。该方法的实例包括小波和广义加性模型的应用：在光谱学中使用小波包进行校准和预测；在存在模式不确定性的情况下，调整其他协变量，确定颗粒物对死亡率的影响；幼苗成活率二元回归模型的变量选择与预测。利用重要抽样对高维模型空间进行模型平均是一种有效的解决方案。在上述应用中，将该方法扩展到变量变换的选择、子空间的选择和小波的阈值化，以及广义加性模型。采用概率与期望效用成正比的抽样模型的方法也被开发出来。寻找和使用模型来描述数据是统计学和科学中的一个基本问题。统计预测可能对模型中包含的一组解释变量非常敏感。在选择解释变量子集的基础上选择一个特定的模型，并使用该模型进行预测，由于没有将模型选择的不确定性纳入最终答案，可能会导致风险更大的决策。在问题中，模型的不确定性往往超过其他不确定性来源，但通常被忽略。在这项研究中，预测和推论可以基于一组模型而不是单一模型；每个模型对决策的贡献与它从观测数据中获得的支持成比例。由于可能的模型数量非常大，因此挑战在于找到探索模型空间的有效方法，选择合理的模型，并为加权预测或其他决策赋予每个模型的权重。以下应用推动了方法学的发展：光谱学中的校准和预测；确定颗粒物对死亡率的影响，在不确定应将哪些变量纳入预测模型时，对其他气象变量进行调整。＊＊＊

DMS 9626135 Clyde Statistical predictions based on complex models may be very sensitive to modeling assumptions, such as choice of covariates. As a result, choosing a single model may not lead to satisfactory predictions and may significantly underestimate prediction intervals due to not incorporating uncertainty about the model choice into the final answer. Model uncertainty often outweighs other sources of uncertainty in problems, but is often ignored. Bayesian methods offer a very effective and conceptually appealing alternative: predictions and inferences can be based on a set of models rather than a single model; each model contributes proportionally to the support it receives from the observed data. This research involves Bayesian methods for stochastically searching high dimensional model spaces. As the number of models is very large, the challenge is therefore that of finding efficient ways of exploring the space of models, selecting plausible ones, and attributing to each of them a weight (approximating the posterior probability) for the mixing-based prediction or other utility calculations. Examples for the methodology include applications in wavelets and generalized additive models: calibration and prediction in spectroscopy using wavelet packets; determining the influence of particulate matter on mortality adjusting for other covariates in the presence of model uncertainty; and variable selection and prediction in binary regression models for seedling survival. Model averaging using importance sampling to sample from high dimensional model spaces is an effective solution. This approach is extended to selecting transformations of variables, subspace selection and thresholding in wavelets, and generalized additive models in the applications described above. Methods for sampling models with a probability proportional to their expected utility are also developed. %%% Finding and using models to describe data is a fundamental problem in both statis tics and the sciences. Statistical predictions may be very sensitive to the set of explanatory variables included in a model. Selecting a particular model based on selecting a subset of the explanatory variables and using this model for prediction, may lead to riskier decisions due to not incorporating uncertainty about model choice into the final answer. Model uncertainty often outweighs other sources of uncertainty in problems, but is usually ignored. In this research, predictions and inferences can be based on a set of models rather than a single model; each model contributes to the decision proportionally to the support it receives from the observed data. As the number of possible models is very large, the challenge is therefore that of finding efficient ways of exploring the space of models, selecting plausible ones, and attributing to each of them a weight for the weighted prediction or other decisions. The methodological developments are driven by the following applications: calibration and prediction in spectroscopy; and determining the influence of particulate matter on mortality adjusting for other meteorological variables when there is uncertainty about which variables should be included in the prediction model. ***