_{k}},
y_{t_{k-1}},...,y_{t_0} \right]
\]</span> The Laplace filter produces <em>smoothed</em> state estimates.
This is a state estimate based observations at all time-points <span class="math inline">\(t_{0}\)</span> to <span class="math inline">\(t_{N}\)</span> where <span class="math inline">\(N\)</span> is the last index in the time-series.
These are denoted by <span class="math display">\[
\text{Smoothed:} \quad \mathrm{E}\left[ x_{t_k} \mid y_{N},
y_{N-1},...,y_{t_{k}},...,y_{t_1}, y_{t_0} \right]
\]<