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Well-known debates among statistical inferential paradigms emerge from conflicting views on the notion of probability. One dominant view understands probability as a representation of sampling variability; another prominent view understands probability as a measure of belief. The former generally describes model parameters as fixed values, in contrast to the latter. We propose that there are actually two versions of a parameter within both paradigms: a fixed unknown value that generated the data and a random version to describe the uncertainty in estimating the unknown value. An inferential approach based on CDs deciphers seemingly conflicting perspectives on parameters and probabilities.

The increasing prevalence of wearable devices enables low-cost, long-term collection of health relevant data such as heart rate, exercise, and sleep signals. Currently these data are used to monitor short term changes with limited interpretation of their relevance to health. These data provide an untapped resource to monitor daily and long-term activity patterns. Changes and trends identified from such data can provide insights and guidance to the management of many chronic conditions that change over time. In this study we conducted a machine learning based analysis of longitudinal heart rate data collected over multiple years from Fitbit devices. We built a multi-resolutional pipeline for time series analysis, using model-free clustering methods inspired by statistical conformal prediction framework. With this method, we were able to detect health relevant events, their interesting patterns (e.g., daily routines, seasonal differences, and anomalies), and correlations to acute and chronic changes in health conditions. We present the results, lessons, and insights learned, and how to address the challenge of lack of labels. The study confirms the value of long-term heart rate data for health monitoring and surveillance, as complementary to extensive yet intermittent examinations by health care providers.

Active learning (AL) aims to improve model performance within a fixed labeling budget by choosing the most informative data points to label. Existing AL focuses on the single-domain setting, where all data come from the same domain (e.g., the same dataset). However, many real-world tasks often involve multiple domains. For example, in visual recognition, it is often desirable to train an image classifier that works across different environments (e.g., different backgrounds), where images from each environment constitute one domain. Such a multi-domain AL setting is challenging for prior methods because they (1) ignore the similarity among different domains when assigning labeling budget and (2) fail to handle distribution shift of data across different domains. In this paper, we propose the first general method, dubbed composite active learning (CAL), for multi-domain AL. Our approach explicitly considers the domain-level and instance-level information in the problem; CAL first assigns domain-level budgets according to domain-level importance, which is estimated by optimizing an upper error bound that we develop; with the domain-level budgets, CAL then leverages a certain instance-level query strategy to select samples to label from each domain. Our theoretical analysis shows that our method achieves a better error bound compared to current AL methods. Our empirical results demonstrate that our approach significantly outperforms the state-of-the-art AL methods on both synthetic and real-world multi-domain datasets. Code is available at https://github.com/Wang-ML-Lab/multi-domain-active-learning.