Uncertainty of a Brownian BridgePosted by Yuling Yao on Oct 30, 2019.
As a Bayesian statistician, I almost love the use of the word “uncertainty” as if that manifests my expertise in inference–until recently someone talks to me that they will stop talking to me as there is too much “uncertainty” in my life, which makes total sense as after integrating out the location where I will settle down in a few years as an international phd student which is indeed nearly uniform on every pixel on this planet, my life path will portrait a Cauchy distribution or maybe even improper, and that is, evidently, plenty of uncertainty.
Technically that is also not totally true as in the long run I will die so it cannot be a Cauchy distribution: in best it is a Brownian bridge or a Cauchy bridge, if it is a name, in a way that has a to end in 0. Nevertheless it is still a lot of uncertainty in the bulk.
I cannot deny that a heavy tail, or myself, is hatful if we have a convex utility function, or if we are a statistician and we want to prove some Bestensein von mises rate. But hey, I just realize it is misleading to think about how/why should I minimize the uncertainty too– for a brownian bridge, the optimal path that minimizes the point-wise variance is probably degenerated. It is a trivial path that stays in 0. Ahhh, we solve the optimal transport problem!