# Particle filtering for Bayesian parameter estimation in a high dimensional state space model

## Summary (1 min read)

### I. INTRODUCTION

- Researchers in some of the most active fields of science have to deal with very large scale stochastic dynamic models of real world phenomena for which conventional prediction and estimation methods are not well suited.
- The authors investigate a nested particle filtering (PF) scheme for the Bayesian estimation of the dynamic variables and the static parameters of state space models.
- The latter displays the basic physical features of atmospheric dynamics and, for this reason, the deterministic version of this model is commonly used as a benchmark for data assimilation [5] and parameter estimation techniques [6] in meteorology and climate science.
- The authors illustrate the performance of the proposed scheme by means of computer simulations on a stochastic two-scale Lorenz 96 model [6] with 16 slow and 160 fast dynamic variables as well as several unknown parameters.
- Furthermore, the authors obtain explicit convergence rates that link the computational cost of the PF algorithm, the kernel bandwidth and the dimension of the parameter space.

### B. Proposed algorithm

- The authors have carried out 20 independent simulations with N = 200 particles in the outer filter and M = 600 particles in the inner filters used to compute the importance weights of the outer filter.
- For reference, the authors approximate them in each simulation run using the least squares estimator âLS = arg min.
- The authors observed how the actual value is closely tracked.
- This is the case for the remaining dynamic variables, X 2:J,t (not shown).
- The mean square error of the dynamic variable estimates over the 20 independent simulations (normalised w.r.t. the power of the signals) was ⇡ 0.0313.

### VI. CONCLUSIONS

- The authors have proposed a nested PF scheme to jointly track the dynamic variables and approximate the posterior pdf of the fixed parameters in state space models.
- The authors have proved a.s. convergence of the pdf approximations and displayed numerical results for a stochastic two-scale Lorenz 96 system.

Did you find this useful? Give us your feedback

##### Citations

65 citations

### Cites background from "Particle filtering for Bayesian par..."

...Thus, the state space is a high-dimensional variable which means it needs huge amount of particles to draw the joint distribution [21]....

[...]

50 citations

32 citations

##### References

1,869 citations

### "Particle filtering for Bayesian par..." refers background or methods in this paper

...It is a well known result [1, 2, 4] that the sequence of probability measures ξt,θ(xt)dxt can be recursively approximated using a standard PF algorithm, and hence the integral (gt , ξt,θ) can be numerically approximated as well....

[...]

..., the popular particle Markov chain Monte Carlo (pMCMC) [1] and sequential Monte Carlo square (SMC(2)) [2] algorithms, are batch techniques and, therefore, they are not well suited to the processing of long observation...

[...]

400 citations

### "Particle filtering for Bayesian par..." refers methods in this paper

...The latter displays the basic physical features of atmospheric dynamics and, for this reason, the deterministic version of this model is commonly used as a benchmark for data assimilation [5] and parameter estimation techniques [6] in meteorology and climate science....

[...]

352 citations

### "Particle filtering for Bayesian par..." refers background in this paper

...Although some recursive algorithms exist [3], they only yield maximum likelihood (point) estimates for the parameters, and hence they are subject to various convergence (and complexity) issues when the likelihood is multimodal, contains singularities or cannot be computed exactly....

[...]

258 citations

### "Particle filtering for Bayesian par..." refers background or methods in this paper

...It is a well known result [1, 2, 4] that the sequence of probability measures ξt,θ(xt)dxt can be recursively approximated using a standard PF algorithm, and hence the integral (gt , ξt,θ) can be numerically approximated as well....

[...]

..., the popular particle Markov chain Monte Carlo (pMCMC) [1] and sequential Monte Carlo square (SMC(2)) [2] algorithms, are batch techniques and, therefore, they are not well suited to the processing of long observation...

[...]

...1: (a)-(b) Estimates of μt,F(f) over the interval [2, 30]....

[...]