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Kalman Filter
We assume the current state can be modeled as a Gaussian distribution
We assume neural observations can also be modeled a Gaussian distribution
We also assume abase case
Thus the model parameters are:
We aim to maximize the joint likelihood of the state and observed date
Suppose
the minimize is achieved when the derivative vanishes
In the testing phase, we aim to compute
. At each time step, we apply two sub-steps, a one-step prediction and then a measurement update.
- One step Prediction
- Measurement updateWe’ll use the following notation for mean and convenience:One step PredictionWe assume that we have the meanand covariancefrom the previous iteration. We need to computeand. Because
Measurement update
We take advantage of the property of the Gaussian distributions If
, then
is Gaussian with
Because
, Then
. Similarly
Now we have the joint distribution
We can write the measurement updates with the Kalman gain
where
- Maximum likelihood for Multivariant Gaussian https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/other-readings/chapter13.pdf
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