  Switching dynamical systems provide a powerful, interpretable modeling
framework for inference in time-series data in, e.g., the natural sciences or
engineering applications. Since many areas, such as biology or discrete-event
systems, are naturally described in continuous time, we present a model based
on an Markov jump process modulating a subordinated diffusion process. We
provide the exact evolution equations for the prior and posterior marginal
densities, the direct solutions of which are however computationally
intractable. Therefore, we develop a new continuous-time variational inference
algorithm, combining a Gaussian process approximation on the diffusion level
with posterior inference for Markov jump processes. By minimizing the path-wise
Kullback-Leibler divergence we obtain (i) Bayesian latent state estimates for
arbitrary points on the real axis and (ii) point estimates of unknown system
parameters, utilizing variational expectation maximization. We extensively
evaluate our algorithm under the model assumption and for real-world examples.
