Browsing by Author "Long, Carlos A. Sing"

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    On dependent Dirichlet processes for general Polish spaces
    (2024) Iturriaga, Andres; Long, Carlos A. Sing; Jara, Alejandro
    We study Dirichlet process-based models for sets of predictor- dependent probability distributions, where the domain and predictor space are general Polish spaces. We generalize the definition of dependent Dirichlet processes, originally constructed on Euclidean spaces, to more general Polish spaces. We provide sufficient conditions under which dependent Dirichlet processes and dependent Dirichlet process mixture models have appealing properties regarding continuity (weak and strong), association structure, and support (under different topologies). The results can be easily extended to more general dependent stick -breaking processes.
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    Tikhonov regularization as a nonparametric method for uncertainty quantification in aggregate data problems
    (2024) Villalon, Elena; Yang, Qian; Long, Carlos A. Sing
    Dynamical systems are widely used in mathematical models in engineering and in the applied sciences. However, the parameters of these systems are usually unknown and they must be estimated from measured data. Performing parameter estimation and quantifying the uncertainty in these estimates becomes critical to realize the predictive power of dynamical systems. In some applications, only measurements about the states of an ensemble of systems are available, where each one evolves according to the same model but for different parameter values, leading to aggregate data problems. In this case, an approach proposed in the literature is to estimate a density that is consistent with some estimates of the expected value of some quantities of interest. To solve the problem in practice, the density is discretized and, to solve the resulting ill-posed problem, Tikhonov regularization is used. In this work, we propose a Bayesian model that shows this approach can be interpreted as a maximum a posteriori estimate for the density. We show that the infinite-dimensional problem defining the MAP can be reformulated as a finite-dimensional problem that does not require discretizing the density. In several cases of interest, this problem is convex, unconstrained, and the objective function is smooth. Thus, it can be solved using algorithms with optimal convergence rates. The trade -off is that the objective is defined by an integral. However, our results characterize the regularity of the integrand, allowing the use of tailored numerical schemes to approximate it. Furthermore, our theoretical results characterize the form of the optimal density, whereas our numerical results illustrate the performance of our method and confirm our theoretical findings.