The paper characterizes the invariant filtering measures resulting from Kalman filtering with intermittent observations in which the observation arrival is modeled as a Bernoulli process with packet arrival probability . Our prior work showed that, for , the sequence of random conditional error covariance matrices converges weakly to a unique invariant distribution . This paper shows that, as approaches one, the family satisfies a moderate deviations principle with good rate function : 1) as , the family converges weakly to the Dirac measure concentrated on the fixed point of the associated discrete time Riccati operator; 2) the probability of a rare event (an event bounded away from ) under decays to zero as a power law of as ; and, 3) the best power law decay exponent is obtained by solving a deterministic variational problem involving the rate function . For specific scenarios, the paper develops computationally tractable methods that lead to efficient estimates of rare event probabilities under .