  We evaluate the robustness of a probabilistic formulation of system
identification (ID) to sparse, noisy, and indirect data. Specifically, we
compare estimators of future system behavior derived from the Bayesian
posterior of a learning problem to several commonly used least squares-based
optimization objectives used in system ID. Our comparisons indicate that the
log posterior has improved geometric properties compared with the objective
function surfaces of traditional methods that include differentially
constrained least squares and least squares reconstructions of discrete time
steppers like dynamic mode decomposition (DMD). These properties allow it to be
both more sensitive to new data and less affected by multiple minima ---
overall yielding a more robust approach. Our theoretical results indicate that
least squares and regularized least squares methods like dynamic mode
decomposition and sparse identification of nonlinear dynamics (SINDy) can be
derived from the probabilistic formulation by assuming noiseless measurements.
We also analyze the computational complexity of a Gaussian filter-based
approximate marginal Markov Chain Monte Carlo scheme that we use to obtain the
Bayesian posterior for both linear and nonlinear problems. We then empirically
demonstrate that obtaining the marginal posterior of the parameter dynamics and
making predictions by extracting optimal estimators (e.g., mean, median, mode)
yields orders of magnitude improvement over the aforementioned approaches. We
attribute this performance to the fact that the Bayesian approach captures
parameter, model, and measurement uncertainties, whereas the other methods
typically neglect at least one type of uncertainty.
