The Kalman Filter is the optimal time series predictor given Gaussian errors with known variances. However, in virtually all real-world scenarios, the variances are unknown. We came up with an algorithm to estimate the covariance matrices as we make predictions.
The algorithm ends up being a hybrid of sequential Monte Carlo (SMC) and variational inference (VI). Instead of taking weighted averages of particles, we make predictions using the MAP of the covariance matrices. And when the resampling step is supposed to happen, we perform a variational update on the covariance matrices using the Expectation-Maximization algorithm.
Please read using nbviewer as the LaTeX does not render properly on GitHub.
https://nbviewer.jupyter.org/github/wicaksono-wijono/mlpp/blob/master/final-notebook.ipynb