Variational state filtering

A placeholder to discuss state filtering and parameter estimation where the unobserved state is quantified by variationally-learned distributions.

Campbell et al. (2021) introduce an elegant method which also performs system identification. I would like to have time to go into more detail about this but for now I will present the key insight without adequate explanation for my own benefit. The neat trick is that the variational approximation is in a sense global, in that it all telescopes into one big variational approximation, rather than a sequence of successive approximations, each of which accumulates a greater error inside the ELBO. Intuitively this gives us more hope that we are can avoid accumulating bias at each filter step.

\[ \max _{\theta, \phi} \mathcal{L}_{t}(\theta, \phi)=\mathbb{E}_{q_{t}^{\phi}\left(x_{1: t}\right)}\left[\log \frac{p_{\theta}\left(x_{1: t}, y^{t}\right)}{q_{t}^{\phi}\left(x_{1: t}\right)}\right] \]

Our key factorization: \(q_{t}^{\phi}\left(x_{1: t}\right)=q_{t}^{\phi}\left(x_{t}\right) q_{t}^{\phi}\left(x_{t-1} \mid x_{t}\right) q_{t-1}^{\phi}\left(x_{t-2} \mid x_{t-1}\right) \ldots q_{2}^{\phi}\left(x_{1} \mid x_{2}\right)\)

True factorization: \(p_{\theta}\left(x_{t} \mid y^{t}\right) p_{\theta}\left(x_{t-1} \mid x_{t}, y^{t-1}\right) p_{\theta}\left(x_{t-2} \mid x_{t-1}, y^{t-2}\right) \cdots p_{\theta}\left(x_{1} \mid x_{2}, y^{1}\right)\)