  We consider the following communication scenario. An encoder causally
observes the Wiener process and decides when and what to transmit about it. A
decoder makes real-time estimation of the process using causally received
codewords. We determine the causal encoding and decoding policies that jointly
minimize the mean-square estimation error, under the long-term communication
rate constraint of $R$ bits per second. We show that an optimal encoding policy
can be implemented as a causal sampling policy followed by a causal compressing
policy. We prove that the optimal encoding policy samples the Wiener process
once the innovation passes either $\sqrt{\frac{1}{R}}$ or
$-\sqrt{\frac{1}{R}}$, and compresses the sign of the innovation (SOI) using a
1-bit codeword. The SOI coding scheme achieves the operational distortion-rate
function, which is equal to $D^{\mathrm{op}}(R)=\frac{1}{6R}$. Surprisingly,
this is significantly better than the distortion-rate tradeoff achieved in the
limit of infinite delay by the best non-causal code. This is because the SOI
coding scheme leverages the free timing information supplied by the zero-delay
channel between the encoder and the decoder. The key to unlock that gain is the
event-triggered nature of the SOI sampling policy. In contrast, the
distortion-rate tradeoffs achieved with deterministic sampling policies are
much worse: we prove that the causal informational distortion-rate function in
that scenario is as high as $D_{\mathrm{DET}}(R) = \frac{5}{6R}$. It is
achieved by the uniform sampling policy with the sampling interval
$\frac{1}{R}$. In either case, the optimal strategy is to sample the process as
fast as possible and to transmit 1-bit codewords to the decoder without delay.
