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We consider the problem of jointly decoding the correlated data picked up and transmitted by the nodes of a large-scale sensor network. Assuming that each sensor node uses a very simple encoder (a scalar quantizer and a modulator), we focus on decoding algorithms that exploit the correlation structure of the sensor data to produce the best possible estimates under the minimum mean-square error (MMSE) criterion. Our analysis shows that a standard implementation of the optimal MMSE decoder is unfeasible for large-scale sensor networks, because its complexity grows exponentially with the number of nodes in the network. Seeking a scalable alternative, we use factor graphs to obtain a simplified model for the correlation structure of the sensor data. This model allows us to use the sum-product decoding algorithm, whose complexity can be made to grow linearly with the size of the network. Considering large sensor networks with arbitrary topologies, we focus on factor trees and give an exact characterization of the decoding complexity, as well as mathematical tools for factorizing Gaussian sources and optimization algorithms for finding optimal factor trees under the Kullback-Leibler criterion.