In this paper we consider standard convex relaxations for such problems.

If it is clear from the context, we will mostly use the favorable uncluttered notation. Using Leibniz' theorem, we have ˆ A.2.1 Calculation of the Filtering Distribution The filtering distribution is defined as α ( y,z,t):= p (y,z,t | x Consider the case where there is no observation in the interval [t,t + h], h > 0. We compute α ( y,z,t + h) = p (y,z,t + h | x A.2.2 Calculation of the Backward Distribution The backward distribution is defined as β (y,z,t):= p( x We find the dynamics of the smoothing distribution by calculating its time derivative. Using the terms in Eq. (28) we have The second term of Eq. (31) does not depend on the Accordingly, both sides of Eq. (35) have Eq. Appendix A.3.4) and provide the gradient with respect to the dispersion A comprehensive overview over the ground-truth and learned parameters is given in Table 2. Note that we utilize this procedure for all experiments.

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.