  The problem of system identification for the Kalman filter, relying on the
expectation-maximization (EM) procedure to learn the underlying parameters of a
dynamical system, has largely been studied assuming that observations are
sampled at equally-spaced time points. However, in many applications this is a
restrictive and unrealistic assumption. This paper addresses system
identification for the continuous-discrete filter, with the aim of generalizing
learning for the Kalman filter by relying on a solution to a continuous-time
It\^o stochastic differential equation (SDE) for the latent state and
covariance dynamics. We introduce a novel two-filter, analytical form for the
posterior with a Bayesian derivation, which yields analytical updates which do
not require the forward-pass to be pre-computed. Using this analytical and
efficient computation of the posterior, we provide an EM procedure which
estimates the parameters of the SDE, naturally incorporating irregularly
sampled measurements. Generalizing the learning of latent linear dynamical
systems (LDS) to continuous-time may extend the use of the hybrid Kalman filter
to data which is not regularly sampled or has intermittent missing values, and
can extend the power of non-linear system identification methods such as
switching LDS (SLDS), which rely on EM for the linear discrete-time Kalman
filter as a sub-unit for learning locally linearized behavior of a non-linear
system. We apply the method by learning the parameters of a latent,
multivariate Fokker-Planck SDE representing a toggle-switch genetic circuit
using biologically realistic parameters, and compare the efficacy of learning
relative to the discrete-time Kalman filter as the step-size irregularity and
spectral-radius of the dynamics-matrix increases.
