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Description
In this work, the author analyzes quantitative and structural aspects of Bayesian inference using Markov kernels, Wasserstein metrics, and Kantorovich monads. In particular, the author shows the following main results: first, that Markov kernels can be viewed as Borel measurable maps with values in a Wasserstein space; second, that the Disintegration Theorem can be interpreted as a literal equality of integrals using an original theory of integration for Markov kernels; third, that the Kantorovich monad can be defined for Wasserstein metrics of any order; and finally, that, under certain assumptions, a generalized Bayes’s Law for Markov kernels provably leads to convergence of the expected posterior distribution in the Wasserstein metric. These contributions provide a basis for studying further convergence, approximation, and stability properties of Bayesian inverse maps and inference processes using a unified theoretical framework that bridges between statistical inference, machine learning, and probabilistic programming semantics.
ContributorsEikenberry, Keenan (Author) / Cochran, Douglas (Thesis advisor) / Lan, Shiwei (Thesis advisor) / Dasarathy, Gautam (Committee member) / Kotschwar, Brett (Committee member) / Shahbaba, Babak (Committee member) / Arizona State University (Publisher)
Created2023
Description
Linear-regression estimators have become widely accepted as a reliable statistical tool in predicting outcomes. Because linear regression is a long-established procedure, the properties of linear-regression estimators are well understood and can be trained very quickly. Many estimators exist for modeling linear relationships, each having ideal conditions for optimal performance. The differences stem from the introduction of a bias into the parameter estimation through the use of various regularization strategies. One of the more popular ones is ridge regression which uses ℓ2-penalization of the parameter vector. In this work, the proposed graph regularized linear estimator is pitted against the popular ridge regression when the parameter vector is known to be dense. When additional knowledge that parameters are smooth with respect to a graph is available, it can be used to improve the parameter estimates. To achieve this goal an additional smoothing penalty is introduced into the traditional loss function of ridge regression. The mean squared error(m.s.e) is used as a performance metric and the analysis is presented for fixed design matrices having a unit covariance matrix. The specific problem setup enables us to study the theoretical conditions where the graph regularized estimator out-performs the ridge estimator. The eigenvectors of the laplacian matrix indicating the graph of connections between the various dimensions of the parameter vector form an integral part of the analysis. Experiments have been conducted on simulated data to compare the performance of the two estimators for laplacian matrices of several types of graphs – complete, star, line and 4-regular. The experimental results indicate that the theory can possibly be extended to more general settings taking smoothness, a concept defined in this work, into consideration.
ContributorsSajja, Akarshan (Author) / Dasarathy, Gautam (Thesis advisor) / Berisha, Visar (Committee member) / Yang, Yingzhen (Committee member) / Arizona State University (Publisher)
Created2022
Description
The problem of multiple object tracking seeks to jointly estimate the time-varying cardinality and trajectory of each object. There are numerous challenges that are encountered in tracking multiple objects including a time-varying number of measurements, under varying constraints, and environmental conditions. In this thesis, the proposed statistical methods integrate the use of physical-based models with Bayesian nonparametric methods to address the main challenges in a tracking problem. In particular, Bayesian nonparametric methods are exploited to efficiently and robustly infer object identity and learn time-dependent cardinality; together with Bayesian inference methods, they are also used to associate measurements to objects and estimate the trajectory of objects. These methods differ from the current methods to the core as the existing methods are mainly based on random finite set theory.
The first contribution proposes dependent nonparametric models such as the dependent Dirichlet process and the dependent Pitman-Yor process to capture the inherent time-dependency in the problem at hand. These processes are used as priors for object state distributions to learn dependent information between previous and current time steps. Markov chain Monte Carlo sampling methods exploit the learned information to sample from posterior distributions and update the estimated object parameters.
The second contribution proposes a novel, robust, and fast nonparametric approach based on a diffusion process over infinite random trees to infer information on object cardinality and trajectory. This method follows the hierarchy induced by objects entering and leaving a scene and the time-dependency between unknown object parameters. Markov chain Monte Carlo sampling methods integrate the prior distributions over the infinite random trees with time-dependent diffusion processes to update object states.
The third contribution develops the use of hierarchical models to form a prior for statistically dependent measurements in a single object tracking setup. Dependency among the sensor measurements provides extra information which is incorporated to achieve the optimal tracking performance. The hierarchical Dirichlet process as a prior provides the required flexibility to do inference. Bayesian tracker is integrated with the hierarchical Dirichlet process prior to accurately estimate the object trajectory.
The fourth contribution proposes an approach to model both the multiple dependent objects and multiple dependent measurements. This approach integrates the dependent Dirichlet process modeling over the dependent object with the hierarchical Dirichlet process modeling of the measurements to fully capture the dependency among both object and measurements. Bayesian nonparametric models can successfully associate each measurement to the corresponding object and exploit dependency among them to more accurately infer the trajectory of objects. Markov chain Monte Carlo methods amalgamate the dependent Dirichlet process with the hierarchical Dirichlet process to infer the object identity and object cardinality.
Simulations are exploited to demonstrate the improvement in multiple object tracking performance when compared to approaches that are developed based on random finite set theory.
The first contribution proposes dependent nonparametric models such as the dependent Dirichlet process and the dependent Pitman-Yor process to capture the inherent time-dependency in the problem at hand. These processes are used as priors for object state distributions to learn dependent information between previous and current time steps. Markov chain Monte Carlo sampling methods exploit the learned information to sample from posterior distributions and update the estimated object parameters.
The second contribution proposes a novel, robust, and fast nonparametric approach based on a diffusion process over infinite random trees to infer information on object cardinality and trajectory. This method follows the hierarchy induced by objects entering and leaving a scene and the time-dependency between unknown object parameters. Markov chain Monte Carlo sampling methods integrate the prior distributions over the infinite random trees with time-dependent diffusion processes to update object states.
The third contribution develops the use of hierarchical models to form a prior for statistically dependent measurements in a single object tracking setup. Dependency among the sensor measurements provides extra information which is incorporated to achieve the optimal tracking performance. The hierarchical Dirichlet process as a prior provides the required flexibility to do inference. Bayesian tracker is integrated with the hierarchical Dirichlet process prior to accurately estimate the object trajectory.
The fourth contribution proposes an approach to model both the multiple dependent objects and multiple dependent measurements. This approach integrates the dependent Dirichlet process modeling over the dependent object with the hierarchical Dirichlet process modeling of the measurements to fully capture the dependency among both object and measurements. Bayesian nonparametric models can successfully associate each measurement to the corresponding object and exploit dependency among them to more accurately infer the trajectory of objects. Markov chain Monte Carlo methods amalgamate the dependent Dirichlet process with the hierarchical Dirichlet process to infer the object identity and object cardinality.
Simulations are exploited to demonstrate the improvement in multiple object tracking performance when compared to approaches that are developed based on random finite set theory.
ContributorsMoraffah, Bahman (Author) / Papandreou-Suppappola, Antonia (Thesis advisor) / Bliss, Daniel W. (Committee member) / Richmond, Christ D. (Committee member) / Dasarathy, Gautam (Committee member) / Arizona State University (Publisher)
Created2019