The distributed unscented information filter(DUIF) based on weighted average consensus has the problems of suboptimal estimation accuracy and low filter efficiency in the sparse wireless sensor network(WSN), therefore, a fast DUIF algorithm taking into account the correlation among prior estimate errors is proposed. The observation model is linearized by using the weighted statistical linear regression(WSLR) method. And the mutual information is taken as the input of the average consensus algorithm, so that the information of the prior estimate cross-covariance can be introduced into the results of the maximum posteriori estimation. Meanwhile, by designing the optimal weights of the communication edges and modifying the state weighted matrices adaptively, the convergence rate of the average consensus algorithm can be improved. The simulation results show that the proposed DUIF algorithm can efficiently track the target in the sparse WSN.