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

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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.