Elliptical belief propagation
Generalized least generalized squares
August 22, 2022 — August 22, 2022
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We can generalize Gaussian belief propagation to use general elliptical laws by using Mahalanobis distance without presuming the Gaussian distribution (Agarwal et al. 2013; Davison and Ortiz 2019), making it into a kind of elliptical belief propagation.
1 Robust
If we use a robust Huber loss instead of a Gaussian log-likelihood, then the resulting algorithm is usually referred to as a robust factor or as dynamic covariance scaling (Agarwal et al. 2013; Davison and Ortiz 2019). The nice thing here is that we can imagine the transition from quadratic to linear losses gives us an estimate of which observations are outliers.
2 Student-\(t\)
Surely this is around? Certainly, there is a special case in the t-process. It is mentioned, I think, in Lan et al. (2006) and possibly Proudler et al. (2007) although the latter seems to be something more ad hoc.
3 Gaussian mixture
Surely? TBD.
4 Generic
There seem to be generic update rules (Aste 2021; Bånkestad et al. 2020) which could be used to construct a generic elliptical belief propagation algorithm.