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) introduces 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 can avoid accumulating bias at each filter step.

$$\underset{\theta ,\varphi }{max}{\mathcal{L}}_t(\theta ,\varphi )={\mathbb{E}}_{q_t^{\varphi }\left(x_{1:t}\right)}[\log \frac{p_{\theta }(x_{1:t},y^t)}{q_t^{\varphi }\left(x_{1:t}\right)}]$$

Our key factorization: $q_t^{\varphi }\left(x_{1:t}\right)=q_t^{\varphi }\left(x_t\right)q_t^{\varphi }(x_{t-1}\mid x_t)q_{t-1}^{\varphi }(x_{t-2}\mid x_{t-1})\dots q_2^{\varphi }(x_1\mid x_2)$

True factorization: $p_{\theta }(x_t\mid y^t)p_{\theta }(x_{t-1}\mid x_t,y^{t-1})p_{\theta }(x_{t-2}\mid x_{t-1},y^{t-2})\cdots p_{\theta }(x_1\mid x_2,y^1)$