This paper investigates a distributed ISEA system under a Bayesian framework, focusing on incorporating task-relevant priors to maximize inference performance. At the sensing level, an RWB estimator with a GM prior is designed. By weighting class-conditional posterior means with responsibilities, RWB effectively denoises features and outperforms ML at low SNR. At the communication level, two theoretical proxies are introduced: the computation-optimal and decision-optimal proxies. Optimal transceiver designs in terms of closed-form power allocation are derived for both TDM and FDM settings, revealing threshold-based and dual-decomposition structures. Results show that the discriminant-aware allocation yields additional inference gains.