We discuss the representation of knowledge and of belief from the viewpoint of decision theory. While the Bayesian approach enjoys general-purpose applicability and axiomatic foundations, it suffers from several drawbacks. In particular, it does not model the belief formation process, and does not relate beliefs to evidence. We survey alternative approaches, and focus on formal model of casebased prediction and case-based decisions. A formal model of belief and knowledge representation needs to address several questions. The most basic ones are: (i) how do we represent knowledge?
We present a theoretical analysis of Gaussian-binary restricted Boltzmann machines (GRBMs) from the perspective of density models. The key aspect of this analysis is to show that GRBMs can be formulated as a constrained mixture of Gaussians, which gives a much better insight into the model's capabilities and limitations. We show that GRBMs are capable of learning meaningful features both in a two-dimensional blind source separation task and in modeling natural images. Further, we show that reported difficulties in training GRBMs are due to the failure of the training algorithm rather than the model itself. Based on our analysis we are able to propose several training recipes, which allowed successful and fast training in our experiments. Finally, we discuss the relationship of GRBMs to several modifications that have been proposed to improve the model.
We extend the Chow-Liu algorithm for general random variables while the previous versions only considered finite cases. In particular, this paper applies the generalization to Suzuki's learning algorithm that generates from data forests rather than trees based on the minimum description length by balancing the fitness of the data to the forest and the simplicity of the forest. As a result, we successfully obtain an algorithm when both of the Gaussian and finite random variables are present.
Most generative models for clustering implicitly assume that the number of data points in each cluster grows linearly with the total number of data points. Finite mixture models, Dirichlet process mixture models, and Pitman--Yor process mixture models make this assumption, as do all other infinitely exchangeable clustering models. However, for some applications, this assumption is inappropriate. For example, when performing entity resolution, the size of each cluster should be unrelated to the size of the data set, and each cluster should contain a negligible fraction of the total number of data points. These applications require models that yield clusters whose sizes grow sublinearly with the size of the data set. We address this requirement by defining the microclustering property and introducing a new class of models that can exhibit this property. We compare models within this class to two commonly used clustering models using four entity-resolution data sets.
Many real-world problems, including inference in Bayes Nets, can be reduced to #SAT, the problem of counting the number of models of a propositional theory. This has motivated the need for efficient #SAT solvers. Currently, such solvers utilize a modified version of DPLL that employs decomposition and caching, techniques that significantly increase the time it takes to process each node in the search space. In addition, the search space is significantly larger than when solving SAT since we must continue searching even after the first solution has been found. It has previously been demonstrated that the size of a DPLL search tree can be significantly reduced by doing more reasoning at each node. However, for SAT the reductions gained are often not worth the extra time required. In this paper we verify the hypothesis that for #SAT this balance changes. In particular, we show that additional reasoning can reduce the size of a #SAT solver's search space, that this reduction cannot always be achieved by the already utilized technique of clause learning, and that this additional reasoning can be cost effective.