Hi all,
I found that when using EKF in rats, the maximum likelihood estimation of parameters is not very good. I read the literature and found that the mcmc method is better for estimation, but I don't know how to combine the extended Kalman filter and MCMC. Also don't know how to implement it in rats, it would be great if anyone could provide any help
How to use Monte Carlo method to improve EKF est in rats
Re: How to use Monte Carlo method to improve EKF est in rats
What type of model do you have?
There are two main problems with maximum likelihood in KF models:
1. They can be far too optimistic about the quality of the estimates; sometimes the model provides very weak information about the states, and the point estimates choose one from a wide range of very different possibilities and it may not be clear that they aren't particularly representative.
2. If the model has too many free parameters (typically components with non-zero variances), the maximum likelihood estimates produce a priori unrealistic components. For instance, if you apply ML to all components to the unobserved components trend model which is used in the HP filter, the "trend" that it estimates is usually quite similar to the data---the HP filter "works" by restricting the ratio of the variances to force the trend to be fairly stiff rather than follow the data too closely.
MCMC methods can often help with problem #1, since that's what they are designed to do---produce a better idea of a distribution which isn't well-approximated by an asymptotic Normal. They can only help with problem #2 if you incorporate some form of restriction into the priors---#2 is a problem of the likelihood of the model with unrestricted parameters, and the methods of MCMC with basically unrestricted parameters won't fix that.
How MCMC even applies to an EKF model isn't all that clear. An EKF model is dependent upon linearization points so the likelihood is conditional on those, and it's not clear how MCMC would properly update those.
Both the EKF (and other non-linear state-space methods) and MCMC techniques for state-space models are covered in the State-Space and DSGE Models e-course.
There are two main problems with maximum likelihood in KF models:
1. They can be far too optimistic about the quality of the estimates; sometimes the model provides very weak information about the states, and the point estimates choose one from a wide range of very different possibilities and it may not be clear that they aren't particularly representative.
2. If the model has too many free parameters (typically components with non-zero variances), the maximum likelihood estimates produce a priori unrealistic components. For instance, if you apply ML to all components to the unobserved components trend model which is used in the HP filter, the "trend" that it estimates is usually quite similar to the data---the HP filter "works" by restricting the ratio of the variances to force the trend to be fairly stiff rather than follow the data too closely.
MCMC methods can often help with problem #1, since that's what they are designed to do---produce a better idea of a distribution which isn't well-approximated by an asymptotic Normal. They can only help with problem #2 if you incorporate some form of restriction into the priors---#2 is a problem of the likelihood of the model with unrestricted parameters, and the methods of MCMC with basically unrestricted parameters won't fix that.
How MCMC even applies to an EKF model isn't all that clear. An EKF model is dependent upon linearization points so the likelihood is conditional on those, and it's not clear how MCMC would properly update those.
Both the EKF (and other non-linear state-space methods) and MCMC techniques for state-space models are covered in the State-Space and DSGE Models e-course.