Model Uncertainty in Prediction, Variable Selection and Related Decision Problems

DMS 9626135克莱德 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，并且可能会由于未将模型选择的不确定性纳入最终答案而显著低估预测区间。 模型的不确定性往往超过其他来源的不确定性的问题，但往往被忽视。贝叶斯方法提供了一个非常有效和概念上有吸引力的替代方案：预测和推断可以基于一组模型，而不是单个模型;每个模型的贡献与它从观察数据中获得的支持成比例。本研究涉及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，并为每个模型分配权重（近似后验概率），用于基于混合的预测或其他效用计算。 该方法的例子包括应用小波和广义加法模型：校准和预测光谱使用小波包;确定颗粒物对死亡率的影响调整其他协变量的存在模型的不确定性;和变量的选择和预测的二元回归模型幼苗存活。 利用重要性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。 这种方法被扩展到选择变量的变换，子空间选择和小波阈值，以及上述应用中的广义加性模型。 方法抽样模型的概率成比例的预期效用也被开发出来。 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解释变量集非常敏感。 基于选择解释变量的子集来选择特定模型并使用该模型进行预测，可能会导致风险更高的决策，因为没有将模型选择的不确定性纳入最终答案。 模型的不确定性往往超过其他来源的不确定性的问题，但通常被忽略。 在这项研究中，预测和推断可以基于一组模型，而不是单个模型;每个模型对决策的贡献与其从观察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，挑战在于找到有效的方法来探索模型空间，选择合理的模型，并为每个模型分配权重，以进行加权预测或其他决策。方法学的发展是由以下应用驱动的：光谱学中的校准和预测;以及当预测模型中应包括哪些变量存在不确定性时，确定颗粒物对死亡率的影响，并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. ***