## 3 Results
### Bayesian Normal Form DA
When confronted with explicit non-Gaussian and nonlinear processes, adhering to Gaussian assumptions can not only result in suboptimal performance but can also lead to misleading results [187]. Against this backdrop, the importance of MCMC-based stochastic sampling methods becomes evident, offering a more flexible and effective solution. MCMC methods excel at drawing samples from complex posterior distributions, particularly demonstrating their strengths when dealing with problems involving non-normality and multimodality that are difficult to sample directly. By constructing a Markov chain and iteratively updating it according to predefined transition rules (such as the Metropolis-Hastings algorithm or Gibbs sampling), these methods ensure that the stationary distribution of the chain aligns with the target posterior probability distribution [188]. In theory, once the Markov chain reaches its stationary state, the sampling comprehensively covers the support region of the posterior distribution. However, in high-dimensional spaces, the Markov chain might require a long \"burn-in period\" and numerous iterations to converge, resulting in decreased sampling efficiency, commonly referred to as the \"curse of dimensionality\" [189]. To verify convergence to the target distribution, diagnostic indicators such as Gelman-Rubin statistic (R-hat) and effective sample size are usually employed [190]. In practice, combining MCMC with other methods or algorithmic improvements might be necessary to enhance efficiency and practicality. As shown in Figure 10, the basic steps of the Metropolis-Hastings MCMC method alongside the PF algorithm illustrate the operational framework of this methodology. Consider that if the PF is placed in a high-dimensional and long-running scene, most of the particles may have very little weight, and only a few particles have significant weights, resulting in the decrease of the effective coverage of the filter to the state space and the loss of particle diversity, which is the so-called \"particle degeneration\" problem [192; 193]. To  
Figure 10: Diagrams of Metropolis–Hastings MCMC and PF. Light gray shows the outline of the target distribution (near). The circle represents the combination of parameters in the algorithm. (**a**) the Metropolis–Hastings MCMC sampler proposes a new candidate value based on the last sampling value, and then accepts (green) or rejects (red) according to the ratio of the likelihood approximation of the reference point. (**b**) the sequential Monte Carlo sampler weights the point-by-point likelihood values from the initial set of parameter values and selects new candidates from the current set based on weight [191]. deal with this problem, researchers have proposed a series of strategies, such as increasing the number of particles, introducing dynamic particle proliferation or an elimination mechanism, and using adaptive particle number adjustment in order to maintain the diversity of particle systems [194]. However, although these methods can improve the uncertainty estimation, they also increase the computational burden accordingly. To reduce the computational overhead while maintaining the diversity of particles, the researchers also tried to dynamically adjust the resampling threshold, resampling method [195], or migration distribution according to the current particle distribution characteristics (such as entropy and effective number of particles). It is worth noting that the traditional resampling techniques (such as system sampling, importance sampling, etc.) may fail in high-dimensional complex environments [30; 124]. It is therefore necessary to employ more complex and computationally expensive strategies such as multivariate kernel density estimation and stratified sampling [196]. On the other hand, the performance of PF depends largely on the choice of initial particle distribution [197].