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I'm taking a time series course and am learning about exchangeable time series form of dynamic linear models (DLMs). This is given by: \begin{align*} \mathbf{y}_t' &= \mathbf{F}_t'\boldsymbol{\Theta}_t + \boldsymbol{\nu}_t', \qquad \boldsymbol{\nu}_t \sim \mathcal{N}(0, v_t\mathbf{V})\\ \boldsymbol{\Theta}_t &= \mathbf{G}_t\boldsymbol{\Theta}_{t-1} + \boldsymbol{\Omega}_t, \qquad \boldsymbol{\Omega}_t \sim \mathcal{N}(0, \mathbf{W}_t, \mathbf{V}) \end{align*} $\boldsymbol{\Omega}_t$ comes from a matrix-variate normal distribution.

All parameters except $\mathbf{U}$ and $\mathbf{V}$ can be fit using forward filtering backward sampling (FFBS). In order to fit $\mathbf{U}$ and $\mathbf{V}$, we must do a Gibbs sampling step.

My question is: what is the difference between FFBS and the Kalman filter? What other ways are there to fit DLMs? For example, I have heard of the particle filter - is this something that can be used in this context? And finally, is it always possible to do Gibbs sampling (perhaps with Metropolis-Hastings if prior distributions are not conjugate) to fit a DLM?

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The Kalman filter is the "forward filtering" part of FFBS, while the "backward sampling" part provides a draw from the joint distribution for the $\Theta_t$ for all $t$.

All the other ways of performing statistical parameter estimation, e.g. maximum likelihood, can be used for DLMs. Yes a particle filter could be used here, but since the model is linear and Gaussian, i.e. a DLM, you won't be gaining anything. Particle filters are better when you have non-linear or non-Gaussian models and thus cannot perform FFBS.

Yes, it is possible to perform Metropolis-within-Gibbs for any Bayesian model.

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