Monitoring critical infrastructures is highly dependent on the accuracy of the installed sensors and the robustness of estimation algorithms. Data-injection attacks can degrade the operational reliability and security of any cyber-physical infrastructures. An attacker can compromise the integrity of the monitoring algorithms by hijacking a subset of sensor measurements and sending manipulated readings. Such approach can result in wide-area blackouts in power grids. This paper considers several cases of severe data-injections with high probabilities of information loss. To achieve an accurate supervision, a Bayesian-based approximated filter (BAF) has been proposed at each monitoring node using a distributed architecture. To maintain a reduced communication overhead and time complexity, upper and lower bound methods have been developed. The performance of the proposed technique has been demonstrated in a mature synchrophasor application known as the oscillation detection. Two test cases have been generated to examine the immunity of the proposed estimation scheme in New Zealand and Oman power grids. The tests were conducted in the presence of harsh data-injection attacks and multiple system disturbances. Results show the proposed BAF method can accurately extract the oscillatory parameters from the contaminated measurements.