Browsing by Subject "Hypothesis testing"

Inspired by various applications including ad auctions, matching markets, and voting, mechanism design deals with the problem of designing algorithms that take inputs from strategic agents and return an outcome optimizing a given objective, while taking the strategic behavior from the agents into account.

The focus of this thesis is to design mechanisms in dynamic environments that take into account rich constraints (e.g., budget constraints), features (e.g., robustness and credibility), and different types of agents (e.g., utility-maximizing agents and learning agents). Two main reasons why dynamic mechanism design is hard compared to mechanism design in a static environment are the need to make decisions in an online manner while the future might be unpredictable or even be chosen by an adversary arbitrarily, and the need to cope with strategic agents, who aim to maximize their cumulative utilities by looking into the future.

We propose a framework to design dynamic mechanisms with simple structures for utility-maximizing agents without losing any optimality, which facilitates both the design for the designer and the participation for the agents. Our framework enables the design of mechanisms achieving non-trivial performance guarantees relative to the optimal mechanism that has access to all future information in advance, even though our mechanisms are not equipped with any knowledge about the future. We further develop a class of dynamic mechanisms that are robust against estimation errors in agents' valuation distributions, a class of dynamic mechanisms that are credible so that the designer is incentivized to follow the rules, and a class of dynamic mechanisms for learning agents. In addition to dynamic mechanism design frameworks, we develop statistical tools to test whether a dynamic mechanism has correctly aligned the agents' incentives, and to measure the extent of the misalignment if it exists.

We consider unnormalized models in which the probability density function contains an unknown normalization constant. This term normalizes the model so that its probability density function integrates to one. The computation of this normalization term can be NP-hard and intractable in high-dimensional settings. Classical statistical analysis, e.g. hypothesis testing and quickest change detection, require the exact likelihood estimation of the data generating distribution. They may not be feasible for unnormalized models because of the complexity of computing the explicit probability density function. In this dissertation, we address this difficulty by replacing the likelihood estimation with a new estimation procedure, named score matching, and developing new approaches to the analysis of unnormalized models and applications.

This dissertation is organized into three parts. We first derive a new test statistic for hypothesis testing of unnormalized models. The test statistic is designed by the difference between Hyv\"arinen scores of the null and alternative distributions. Under some reasonable conditions, we provide the asymptotic distribution of this test statistic under the null hypothesis. When this distribution cannot be expressed in a closed form, we outline a bootstrap approach to learn the critical values and provide consistency guarantees.

Next we consider sequential analysis. We develop a new variant of the classical Cumulative Sum (CUSUM) algorithm for the quickest change detection. This variant is again based on the Hyv\"arinen score and is called the Score-based CUSUM (SCUSUM) algorithm. The asymptotic optimality of the proposed algorithm is investigated by deriving expressions for average detection delay and average running length to a false alarm. We further extend the SCUSUM to a robust scenario, where we introduce a notion of the ``least favorable'' distribution in the sense of Fisher divergence. Accordingly, we derive the asymptotic analysis of detection delay and false alarms for the robust SCUSUM.

Finally, we study the application of score-based generative methods to cross-subject mapping of neural activity. The objective is to obtain a task-specific representation of the source subject's neural signals in the feature space of the destination subject. In principle, the mapping function can be assumed to be purely deterministic. Alternatively, we propose to adopt a probabilistic approach, where we learn a conditional probability distribution of destination features given source features. Specifically, we consider learning the Restricted Boltzmann Machine with Gaussian inputs and Bernoulli hidden units (Gauss-Bernoulli RBM). We derive the closed-form gradient to learn Gauss-Bernoulli RBM by minimizing the Fisher divergence, and the well-learned RBM generates task-specific representations of source subjects into the feature space of the destination subject.

Each chapter of the contributions is accompanied by thorough numerical results demonstrating the potentials and the limits of the proposed approach with other benchmarks in various scenarios.

Technological advances in protein mass spectrometry (MS), aka proteomics, haveenabled high-throughput quantification of spatially-resolved, subcellular-specific proteomes. Biological insight in these experiments depends upon sound statistical analysis. Despite the myriad of existing proprietary and open-source software solutions for statistical analysis of proteomics data, these tools suffer a drawback inherent in any general solution: a loss of specificity. These tools often fail to be easily adapted to analyze experiment-specific designs. I present a flexible, linear mixed-effects model framework for assessing differential abundance in protein mass spectrometry experiments. Combined with methods to identify communities of proteins in biological networks, I extend this framework to perform inference at the level of protein groups or modules. Using these software tools, I demonstrate how module-level insight in proximity and spatial proteomics generates hypotheses that identify foci of biological function and dysfunction which may underlie the neuropathology of disease.