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We establish the no trade principle, ie, the no trade theorem and its converse, for any dual pair of bet and extended belief spaces, defined on a given measurable space. A key condition is that, except perhaps one of the agents, everyone else has (weak*) compact sets of beliefs. We find out that in most of the models of uncertainty adopted in the economic literature, roughly speaking, the epistemic statement that an agent has a compact set of beliefs is equivalent to the economic statement that he has an open cone of positive bets. This improves our understanding of what compactness actually means within an economic context.

We develop an expectation-maximization algorithm with local adaptivity for image segmentation and classification. The key idea of our approach is to combine global statistics extracted from the Gaussian mixture model or other proper statistical models with local statistics and geometrical information, such as local probability distribution, orientation, and anisotropy. The combined information is used to design an adaptive local classification strategy that improves the robustness of the algorithm and also keeps fine features in the image. The proposed methodology is flexible and can be easily generalized to deal with other inferred information/quantities and statistical methods/models.

When people in a society want to make inference about some parameter, each person may want to use data collected by other people. Information (data) exchange in social networks is usually costly, so to make reliable statistical decisions, people need to weigh the benefits and costs of information acquisition. Conflicts of interests and coordination problems will arise in the process. Classical statistics does not consider peoples incentives and interactions in the data-collection process. To address this imperfection, this work explores multi-agent Bayesian inference problems with a game theoretic social network model. Motivated by our interest in aggregate inference at the societal level, we propose a new concept, <i>finite population learning</i>, to address whether with high probability, a large fraction of people in a given finite population network can make good inference. Serving as a foundation, this concept enables

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures and small Seifert fibred structures on closed 3-manifolds.