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In this paper, the distributed filtering problem is investigated for a class of nonlinear systems. Each individual sensing node provides the state estimate by using not only its own measurements but also its neighbors' information propagated according to the communication topology. With the purpose of mitigating the effects from possible abnormal data during the signal transmission, an innovation constraint with adaptively determined threshold is imposed on the transmitted innovation during the filter process. The aim of the addressed problem is to design a distributed filtering algorithm which is capable of 1) confining all the estimation errors within certain ellipsoidal regions with prescribed probability; and 2) achieving the required average disturbance attenuation specification. By virtue of convex optimization method, sufficient conditions are derived for the existence of the requested filtering algorithm and the desired filtering parameters are then obtained by iteratively solving the corresponding matrix inequalities. Within the established framework, two optimization problems are put forward to seek locally optimal filtering parameters. Finally, an illustrative numerical example is provided to demonstrate the applicability of the proposed filtering paradigm.