Before revision:  Finally, we validate our theoretical results empirically  , demonstrating that the approximate algorithm exhibits faster overall runtime with low error in a simulation study .
After revision:  Finally, we validate our theoretical results empirically , demonstrating that the approximate algorithm exhibits faster overall runtime with low error . Finally, we extend the setting and application of our methods to hidden Markov models, and illustrate the potential use of the proposed algorithms in practice with an application to computer-generated music .