VARIATIONAL BAYES AND NONLINEAR STATE-SPACE MODELS

Variational Bayesian learning is based on approximating the posterior distribution $p(\boldsymbol{\theta}\vert \boldsymbol{X})$ with a tractable approximation $q(\boldsymbol{\theta}\vert \boldsymbol{\xi})$, where $\boldsymbol{X}$ is the data, $\boldsymbol{\theta}$ are the unknown variables (including both the parameters of the model and the latent variables), and $\boldsymbol{\xi}$ are the (variational) parameters of the approximation. The approximation is fitted by maximizing a lower bound on marginal log-likelihood

$$\mathcal{B}(q(\boldsymbol{\theta}\vert \boldsymbol{\xi})) = \left\langle \log \frac{p(\boldsymbol{X}, \boldsymbol{\theta})}{q(\boldsymbol{\theta}\vert \boldsymbol{\xi})} \right\rangle$$ (22)

$$= \log p(\boldsymbol{X}) - D_ (q(\boldsymbol{\theta}\vert \boldsymbol{\xi}) \Vert p(\boldsymbol{\theta}\vert \boldsymbol{X})),$$

where $\langle \cdot \rangle$ denotes expectation over $q$. This is equivalent to minimizing the Kullback-Leibler divergence $D_$KL$(q \Vert p)$ between $q$ and $p$ (Ghahramani and Beal, 2001).