  Bayesian analysis of state-space models includes computing the posterior
distribution of the system's parameters as well as filtering, smoothing, and
predicting the system's latent states. When the latent states wander around
$\mathbb{R}^n$ there are several well-known modeling components and
computational tools that may be profitably combined to achieve these tasks.
However, there are scenarios, like tracking an object in a video or tracking a
covariance matrix of financial assets returns, when the latent states are
restricted to a curve within $\mathbb{R}^n$ and these models and tools do not
immediately apply. Within this constrained setting, most work has focused on
filtering and less attention has been paid to the other aspects of Bayesian
state-space inference, which tend to be more challenging. To that end, we
present a state-space model whose latent states take values on the manifold of
symmetric positive-definite matrices and for which one may easily compute the
posterior distribution of the latent states and the system's parameters, in
addition to filtered distributions and one-step ahead predictions. Deploying
the model within the context of finance, we show how one can use realized
covariance matrices as data to predict latent time-varying covariance matrices.
This approach out-performs factor stochastic volatility.
