bias-variance trade-off, anomalies

In another day out of commission, I spoke to Foreman-Mackey at length about various matters statistical, and wrote some text for a Tinker, Blanton, & Hogg NSF proposal. The statistics discussion ranged all around, but perhaps the most important outcome is that Foreman-Mackey clarified for me some things about the cryo-EM and galaxy deprojection projects I have been thinking about. The question is: Can averaging (apparently) similar projected images help with inferring angles and reconstruction? Foreman-Mackey noted that if we condition on the three-dimensional model, the projections are independent. Therefore there can be no help from neighboring images in image space. They might decrease the variance of any estimator, but they would do so at the cost of bias. You can't decrease variance without increasing bias if you aren't bringing in new information. At first I objected to this line of argument and then all of a sudden I had that “duh” moment. Now I have to read the literature to see if mistakes have been made along these lines.

The text I wrote for the proposal was about CMB anomalies and the corresponding large-scale-structure searches that might find anomalies in the three-space. Statistical homogeneity and isotropy are such fundamental predictions of the fundamental model, it is very worth testing them. Any anomalies found would be very productive.

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