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Abstract:
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The daily temperature data measured at different weather stations show commonalities in records when their locations are geographically close. Likewise, in many modern applications, data are observed in high-resolution, essentially continuously in time at units having intrinsic relations among themselves that are best described by the nodes of a large graph. Furthermore, the data at each unit may be best described as taking values in a space of functions. It is often sensible to think that the underlying signals in these functional observations vary smoothly over the graph, in that neighboring nodes have similar underlying signals. We attach a precise meaning to the notion through the graph Laplacian operator. We propose a novel Bayesian method that allows borrowing of strengths over neighboring nodes, leading to a more accurate inference. In this work, we consider a model with Gaussian functional observations. We show that an appropriate prior constructed from the graph Laplacian can attain the minimax lower bound. Using a mixture prior, the minimax rate up to a logarithmic factor can be attained simultaneously for all possible values of functional and graphical smoothness.
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