# Browsing by Subject "Finite mixture models"

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Item Open Access Advances in Choquet theories(2022) Caprio, MicheleChoquet theory and the theory of capacities, both initiated by French mathematician Gustave Choquet, share the heuristic notion of studying the extrema of a convex set in order to give interesting results regarding its elements. In this work, we put to use Choquet theory in the study of finite mixture models and the theory of capacities in studying severe uncertainty.

In chapter 2, we show how by combining a classical non-parametric density estimator based on a Pólya tree with techniques from Choquet theory, it is possible to retrieve the weights of a finite mixture model. We also give the rate of convergence of the Pólya tree posterior to the Dirac measure on the weights.

In chapter 3, we introduce dynamic probability kinematics (DPK), a method for an agent to mechanically update subjective beliefs in the presence of partial information. We then generalize it to dynamic imprecise probability kinematics (DIPK), which allows the agent to express their initial beliefs via a set of probabilities. We provide bounds for the lower probability associated with the updated probability sets, and we study the behavior of the latter, in particular contraction, dilation, and sure loss. Examples are provided to illustrate how the methods work. We also formulate in chapter 4 an ergodic theory for the limit of the sequence of successive DIPK updates. As a consequence, we formulate a strong law of large numbers.

Finally, in chapter 5 we propose a new, more general definition of extended probability measures ("probabilities" whose codomain is the interval [-1,1]). We study their properties and provide a behavioral interpretation. We use them in an inference procedure, whose environment is canonically represented by a probability space, when both the probability measure and the composition of the state space are unknown. We develop an ex ante analysis - taking place before the statistical analysis requiring knowledge of the state space - in which we progressively learn its true composition. We describe how to update extended probabilities in this setting, and introduce the concept of lower extended probabilities. We provide two examples in the fields of ecology and opinion dynamics.

Item Open Access The Global Epidemic of Childhood Obesity and Its Non-medical Costs(2015) Fu, QiangThis dissertation consists of three parts of empirical analyses investigating temporal patterns and consequences of (childhood) overweight and obesity, mainly in the United States and the People's Republic of China. Based on the China Health and Nutrition Survey, the first part conducts hierarchical age-period-cohort analyses of childhood overweight in China and finds a strong cohort effect driving the overweight epidemic. Results from the growth-curve models show that childhood overweight and underweight are related such that certain socio-economic groups with higher levels of childhood overweight also exhibit lower levels of childhood underweight. The second part situates the discussion on childhood obesity in a broader context. It compares temporal patterns of childhood overweight in China with these of adulthood overweight and finds that the salient cohort component is absent in rising adulthood overweight, which is dominated by strong period effects. A positive association between human development index and overweight/obesity prevalence across countries is also documented. Using multiple waves of survey data from the National Longitudinal Study of Adolescent Health, the third part analyzes the (latent) trajectory of childhood overweight/obesity in the United States. It finds that individuals with obesity growth trajectories are less likely to avoid mental depression, tend to have higher levels of neuroticism and lower levels of agreeableness/conscientiousness, and show less delinquent behaviors.