Editing Kalman filter (section)

=== Discriminative Kalman filter ===
When the observation model <math>p(\mathbf{z}_k\mid\mathbf{x}_k)</math> is highly non-linear and/or non-Gaussian, it may prove advantageous to apply [[Bayes' rule]] and estimate
:<math>
p(\mathbf{z}_k\mid\mathbf{x}_k) \approx 
\frac{p(\mathbf{x}_k\mid\mathbf{z}_k)}{p(\mathbf{x}_k)}
</math>
where <math>p(\mathbf{x}_k\mid\mathbf{z}_k) \approx \mathcal{N}(g(\mathbf{z}_k),Q(\mathbf{z}_k))</math> for nonlinear functions <math>g,Q</math>. This replaces the generative specification of the standard Kalman filter with a [[discriminative model]] for the latent states given observations.

Under a [[Stationary process|stationary]] state model
:<math>
\begin{align}
p(\mathbf{x}_1) &= \mathcal{N}(0, \mathbf{T}), \\
p(\mathbf{x}_k\mid\mathbf{x}_{k-1}) &= 
\mathcal{N}(\mathbf{F}\mathbf{x}_{k-1}, \mathbf{C}),
\end{align}
</math>
where <math>\mathbf{T}= \mathbf{F}\mathbf{T}\mathbf{F}^\intercal + \mathbf{C}</math>, if 
:<math>
p(\mathbf{x}_k\mid\mathbf{z}_{1:k}) \approx \mathcal{N}(\hat{\mathbf{x}}_{k|k-1}, \mathbf{P}_{k|k-1}),
</math>
then given a new observation <math>\mathbf{z}_k</math>, it follows that<ref name="Bur20">{{cite journal |last1=Burkhart |first1=Michael C. |last2=Brandman |first2=David M. |last3=Franco |first3=Brian |last4=Hochberg |first4=Leigh |last5=Harrison |first5=Matthew T. |title=The Discriminative Kalman Filter for Bayesian Filtering with Nonlinear and Nongaussian Observation Models |journal=Neural Computation |date=2020 |volume=32 |issue=5 |pages=969–1017 |doi=10.1162/neco_a_01275 |pmid=32187000 |pmc=8259355 |s2cid=212748230 |access-date=26 March 2021 | url=https://direct.mit.edu/neco/article/32/5/969/95592/The-Discriminative-Kalman-Filter-for-Bayesian}}</ref>
:<math>
p(\mathbf{x}_{k+1}\mid\mathbf{z}_{1:k+1}) \approx \mathcal{N}(\hat{ \mathbf{x}}_{k+1|k}, \mathbf{P}_{k+1|k})
</math>
where
:<math>
\begin{align}
\mathbf{M}_{k+1} &= \mathbf{F}\mathbf{P}_{k|k-1}\mathbf{F}^\intercal + \mathbf{C}, \\
\mathbf{P}_{k+1|k} &= (\mathbf{M}_{k+1}^{-1} + Q(\mathbf{z}_k)^{-1} - \mathbf{T}^{-1})^{-1}, \\
\hat{\mathbf{x}}_{k+1|k} &= \mathbf{P}_{k+1|k} (\mathbf{M}_{k+1}^{-1}\mathbf{F}\hat{\mathbf{x}}_{k|k-1} + \mathbf{P}_{k+1|k}^{-1}g(\mathbf{z}_k) ).
\end{align}
</math>
Note that this approximation requires <math> Q(\mathbf{z}_k)^{-1} - \mathbf{T}^{-1} </math> to be positive-definite; in the case that it is not, 
:<math>
\mathbf{P}_{k+1|k} = (\mathbf{M}_{k+1}^{-1} + Q(\mathbf{z}_k)^{-1})^{-1}
</math>
is used instead. Such an approach proves particularly useful when the dimensionality of the observations is much greater than that of the latent states<ref name="Bur19">{{cite thesis |last1=Burkhart |first1=Michael C. |title=A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding |date=2019 |publisher=Brown University |location=Providence, RI, USA |doi=10.26300/nhfp-xv22 }}</ref> and can be used build filters that are particularly robust to nonstationarities in the observation model.<ref name="Bra18">{{cite journal |last1=Brandman |first1=David M. |last2=Burkhart |first2=Michael C. |last3=Kelemen |first3=Jessica |last4=Franco |first4=Brian |last5=Harrison |first5=Matthew T. |last6=Hochberg |first6=Leigh R. |title=Robust Closed-Loop Control of a Cursor in a Person with Tetraplegia using Gaussian Process Regression |journal=Neural Computation |date=2018 |volume=30 |issue=11 |pages=2986–3008 |doi=10.1162/neco_a_01129 |pmid=30216140 |pmc=6685768 |url=https://direct.mit.edu/neco/article/30/11/2986/8418/Robust-Closed-Loop-Control-of-a-Cursor-in-a-Person |access-date=26 March 2021}}</ref>