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Implementing marginal likelihood

Working on generalizing marginal likelihood for all three foreground models.

Considered briefly how to use the matrix inversion lemma to improve numerical stability of all models. Decided it could be difficult to build in a general way; will proceed with the direct method for now, until there is evidence of numerical instability.

Generalized how covariance is generated. Wrote several new functions for generating differenct covariance matrices. Will need to rework some due to the developments I describe later.

Theoretical Developments

Thought extensively about perturbation models I described yesterday. I realized there is a serious problem with modeling motion using brownian motion -- the prior variance grows without bound as time approaches infinity. Thus, the marginal prior for the curve in view 36 has much greater prior variance than the curve one in view 1. This doesn't make sense -- ideally, they should all have the same marginal prior.

This led to a reading session in Williams and Rasmussen, which led me to develop two new motion models, which I describe extensively in today's accompanying post.

Tomorrow

Finish implementing new covariance functions.

Implement generalized marginal likelihood.

implement fast generalized marginal likelihood (approximation)

Test against existing ML for old models.

Test with new models. Are the ML's higher? (will probably need training).

Build training data set from hand-traced ground-truth.

Write training code. Train all three candidate models.