Model
Particle methods assume and the observations can be modeled in this form:
- is a first order Markov process such that
and with an initial distribution .
- The observations are conditionally independent provided that are known
- In other words, each only depends on
One example form of this scenario is
where both and are mutually independent and identically distributed sequences with known probability density functions and and are known functions. These two equations can be viewed as state space equations and look similar to the state space equations for the Kalman filter. If the functions and are linear, and if both and are Gaussian, the Kalman filter finds the exact Bayesian filtering distribution. If not, Kalman filter based methods are a first-order approximation (EKF) or a second-order approximation (UKF in general, but if probability distribution is Gaussian a third-order approximation is possible). Particle filters are also an approximation, but with enough particles they can be much more accurate.
Read more about this topic: Particle Filter
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