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 (gamma) over bar. Our prior work showed that, for (gamma) over bar > 0, the sequence of random conditional error co-variance matrices converges weakly to a unique invariant distribution mu((gamma) over bar). This paper shows that, as (gamma) over bar approaches one, the family {mu((gamma) over bar)}((gamma) over bar >0) satisfies a moderate deviations principle with good rate function I(.): 1) as (gamma) over bar up arrow 1, the family {mu((gamma) over bar)}converges weakly to the Dirac measure delta(P*) concentrated on the fixed point of the associated discrete time Riccati operator; 2) the probability of a rare event (an event bounded away from P*) under mu((gamma) over bar)decays to zero as a power law of (1 - (gamma) over bar) as (gamma) over bar up arrow 1; and, 3) the best power law decay exponent is obtained by solving a deterministic variational problem involving the rate function I(.). For specific scenarios, the paper develops computationally tractable methods that lead to efficient estimates of rare event probabilities under mu((gamma) over bar).