The problem of maximizing the information flow through a sensor network tasked with an inference objective at the fusion center is considered. The sensor nodes take observations, compress, and send them to the fusion center through a network of relays. The network imposes capacity constraints on the rate of transmission in each connection and flow conservation constraints. It is shown that this rate-constrained inference problem can be cast as a Network Utility Maximization problem by suitably defining the utility functions for each sensor, and can be solved using existing techniques. Two practical settings are analyzed: multi-terminal parameter estimation and binary hypothesis testing. It is verified via simulations that using the proposed formulation gives better inference performance than the Max-Flow solution that simply maximizes the total bit-rate to the fusion center.