Currently I dig deep in the fields of stochastic theory. More precisely, I read a lot about Sequential Monte Carlo methods (SMCs) and Particle Filters in general. For non-mathematicians like me, it’s really tough stuff. But I hopefully can handle it with a lot of efforts.
I just finished reading the work by Villaverde and Ramirez called “Estimating dynamic equilibrium economies: linear versus nonlinear likelihood” which originates more from the economical field of research. However, I consider this work as a good step for getting the very basics of SMCs in general. I really learned a lot from this work. The authors distinguish linear (Kalman) and non-linear (SMCs) filters and discuss the advantages for estimating dynamic equilibrium models. Sounds crazy and it quite is – especially the evaluation algorithms are difficult to understand.
For the better understanding of the methodology, I visualized the state model which is similar to the one used in the paper:
This figure shows the context of the different states. The transition function is referred to as g() and the measurement function is named h(). Obviously, the measurement of the state vector x is influenced by the noise vector v. Therefore, the observation z is only an approximation of the state vector x. Any comments so far?
The intention to use SMCs is to get information about the real state vector – not the noise-corrupted measurement vector. So SMC methods have the intention to estimate the current unknown probability density of the state space for derivating the most probable state of a dynamic system. This means, SMC methods can be used to estimate the state in a dynamic process in which only the crucial disturbance variables are known. In most of the cases, observations can only be performed on a partly basis which means that there are a lot of hidden states within the system.
It is now possible to make assumptions about the real forensic state vector with the help of SMCs. PF methods make use of large sets of random samples, so called particles, for approximating the sequence of probability distributions.
Finally, I have to admit that it’s kind of black magic to handle such stuff. You cannot understand this topic with common sense – or at least this is not possible for me. I still try to find the “golden thread” in this field of research. There are so many parameters which have to be taken into account – so feel free to correct me – I would really appreciate it!