Equations for particle filter

Question

The particle filter is based on the state and observation model equations

$x_{t+1}=f_t(x_t, v_t)$

$y_t=h_t(x_t, u_t)$

The idea is to randomly generate some particles then propagate them through the equations, resample, and normalize to get an estimate of the state $x_t$.

My question is: we need to know the state and observation model equations so why do we need the particle filter? I hope I am missing something because it seems like if this knowledge is required then why bother with particle filter, can't we just do ML estimation?

Answer 1

why do we need the particle filter?

When your model is non-linear there is no analytical solution to obtain the PDF of the state. The Kalman filter/smoother can't be applied in the sense that it is not optimal. The particle filter is one method to obtain an estimate of state PDF in a sequential setup.

can't we just do ML estimation

Dependent on the model you may find that even an ML estimator can't be found without going through some approximations. But the particle filter is also used in a Bayesian contexts where other estimators are relevant.

Answer 2

There is a strict difference between KF and ML. Former provides density of the state while the latter provides a point estimate. So, they are not alternative of each others exactly. But, sure you might use ML instead of KF if you need point estimate.