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We consider the problem of decentralized estimation of a random-field under communication constraints in a Bayesian setting. The underlying system is composed of sensor nodes which collect measurements due to random variables they are associated with and which can communicate through finite-rate channels in accordance with a directed acyclic topology. After receiving the incoming messages if any, each node evaluates its local rule given its measurement and these messages, producing an estimate as well as outgoing messages to child nodes. A rigorous problem definition is achieved by constraining the feasible set through this structure in order to optimize a Bayesian risk function that captures the costs due to both communications and estimation errors. We adopt an iterative solution through a team decision theoretic treatment previously proposed for decentralized detection. However, for the estimation problem, the iterations contain expressions with integral operators that have no closed form solutions in general. We propose approximations to these expressions through Monte Carlo methods. The result is an approximate computational scheme for optimization of distributed estimation networks under communication constraints. In an example scenario, we increase the price of communications and present the degrading estimation performance of the converged rules.