  When a sensor has continuous measurements but sends limited messages over a
data network to a supervisor which estimates the state, the available packet
rate fixes the achievable quality of state estimation. When such rate limits
turn stringent, the sensor's messaging policy should be designed anew. What are
the good causal messaging policies ? What should message packets contain ? What
is the lowest possible distortion in a causal estimate at the supervisor ? Is
Delta sampling better than periodic sampling ? We answer these questions under
an idealized model of the network and the assumption of perfect measurements at
the sensor. For a scalar, linear diffusion process, we study the problem of
choosing the causal sampling times that will give the lowest aggregate squared
error distortion. We stick to finite-horizons and impose a hard upper bound on
the number of allowed samples. We cast the design as a problem of choosing an
optimal sequence of stopping times. We reduce this to a nested sequence of
problems each asking for a single optimal stopping time. Under an unproven but
natural assumption about the least-square estimate at the supervisor, each of
these single stopping problems are of standard form. The optimal stopping times
are random times when the estimation error exceeds designed envelopes. For the
case where the state is a Brownian motion, we give analytically: the shape of
the optimal sampling envelopes, the shape of the envelopes under optimal Delta
sampling, and their performances. Surprisingly, we find that Delta sampling
performs badly. Hence, when the rate constraint is a hard limit on the number
of samples over a finite horizon, we should should not use Delta sampling.
