Essential and incidental measurement error: Bayesian estimation and inference when sample measurements are random-variable-valued

背景：这笔发现补助金将支持进一步发展随机变量值测量(RVMs)理论的长期目标，这是我作为博士后研究员和新的助理教授在过去5年里一直在开发的经典和Berkson测量误差模型的新推广，并将该理论应用于解决现代生态学和濒危物种保护问题。传统的测量误差概念认为这是一个错误分类或错误测量的问题。然而，在应用的数据收集系统中，当样本测量不能简单地产生实值随机变量的样本实例化时，经常会出现这样的情况。相反，我们可能会发现，测量过程生成的样本数据本身就是新的随机变量，这些变量描述了在每次抽样调查中观察到目标现象的特定值或值范围的机会。目标：在拨款期间要解决的最相关的理论问题是：(1)将其他流行的测量误差模型推广到RVVM设置的不同校准条件的特征；(2)从RVVM样本数据导出的最大似然估计器的研究；(3)推广到算子理论框架，以及与希尔伯特空间上作为线性算子的观测值的数学物理方法的联系。此外，RVVM框架的下列应用将得到进一步发展：(1)用于鸟类环志作业中种群建模和数据收集的多项式测量和协议的简易R实施；(2)通过eBird和相关平台轻松实施源自公民科学的鸟类种群数据的多项式测量和正常值测量；(3)使RVVM框架适应时空建模场景，用于难以调查的城市筑巢鸟类物种的嵌套识别；(4)将RVVM框架整合到来自海鸟群体空中图像的自动目标分类中。影响：这项工作的理论发展可以提高在数学、统计和决策科学中对测量的科学理解，而其应用可以在生态和环境科学中产生立竿见影的影响，特别是在解决与濒危物种保护有关的真实和困难问题方面。这项工作将产生各种理论和应用论文和技术演示文稿，以及为应用实践者创建几个R软件包，以在各种研究方案中实施这些方法。该项目将培养数学、统计和生态科学方面的2名本科生、4名硕士和2名博士生，并为他们在数据科学、统计、生态和保护领域的职业生涯做好准备。

BACKGROUND: This Discovery Grant will support the long term goal of further developing a theory of random-variable-valued measurements (RVVMs), a novel generalization of classical and Berkson measurement error modelling that I have been developing for the past 5 years as a postdoctoral researcher and new Assistant Professor, and applying this theory to solve modern problems in ecology and conservation of species-at-risk. Traditional notions of measurement error treat the problem as one of misclassification or mismeasurement. However, there are often instances that arise in an applied data collection system when sample measurements do not generate simply sample instantiations of a real-valued random variable. Instead, we may find ourselves in a situation where the measurement process generates sample data that are themselves new random variables that describe the chance of observing a particular value, or range of values, for the target phenomenon in question on each sample draw. OBJECTIVES: The most pertinent theoretical matters to be addressed over the duration of the grant are: (1) A characterization of different calibration conditions that generalize other popular measurement error models to the RVVM setting; (2) A study of maximum likelihood estimators derived from RVVM sample data; (3) Generalization to an operator theoretic framework and connections to the mathematical physics approach of observables as linear operators on a Hilbert space. In addition, the following applications of the RVVM framework will be further developed: (1) Easy R implementation of multinomial-valued measurements and protocols for population modelling and data collection in bird-banding operations; (2) Easy R implementation of multinomial-valued and normal-valued measurements for citizen-science-sourced population data of avian species via eBird and related platforms; (3) Adaptation of RVVM framework to spatio-temporal modelling scenarios for application to unsure nest identification of difficult-to-survey urban-nesting avian species; (4) Integration of RVVM framework into automated object classification from aerial imagery of seabird colonies. IMPACT: This work's theoretical developments can improve the scientific understanding of measurement in the mathematical, statistical, and decision sciences, while its applications can have immediate impact in the ecological and environmental sciences, particularly with regards to solving real and difficult problems related to conservation of species-at-risk. A variety of theoretical and applied papers and technical presentations will result from this work, as well as the creation of several R software packages for applied practitioners to implement these methods in various research scenarios. This program will train 2 undergraduates, 4 MSc and 2 PhD students in the mathematical, statistical, and ecological sciences and prepare them for careers in data science, statistics, ecology, and conservation.