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In this correspondence, the problem of distributed Bayesian estimation is considered in the context of a wireless sensor network. The Bayesian estimation performance is analyzed in terms of the expected Fisher information normalized by the transmission rate of the sensors. The sensors use a communication scheme known as the type-based random access (TBRA) scheme. Under a constraint on the expected transmission energy, an optimal spatio-temporal allocation scheme that maximizes the performance metric is characterized. It is shown that the performance metric is crucially dependent on the fading parameter known as the channel coherence index. For channels with low coherence indices, sensor transmissions tend to cancel each other, and there exists an optimal finite mean transmission rate that maximizes the performance metric. On the other hand, for channels with high coherence indices, there should be as many simultaneous transmissions as allowed by the network. The presence of a critical coherence index where the change from one behavior to another occurs is established.