Convergence theory for large n

harrispopgen / mushi

[mu]tation [s]pectrum [h]istory [i]nference

https://harrispopgen.github.io/mushi/

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Convergence theory for large n #7

Closed wsdewitt closed 4 years ago

wsdewitt commented 5 years ago

Some scribbling with @kharris today thinking about convergence of the SFS as n (number of sampled haplotypes) goes to infinity. We probably need to model the continuous normalized SFS, instead of the raw sampled SFS of Rosen et al (whose normalization diverges in n). See the \phi(x) in the forward time diffusion models.

[ ] TeX this up
[ ] Can we promote the An matrix of Polanski and Kimmel to a continuous kernel operator? Seems like this will get funky since the matrix rows are defined recursively in terms of n.

kamdh commented 5 years ago

What happened to the whiteboard photo?

kamdh commented 5 years ago

Ah it reappeared

wsdewitt commented 5 years ago

Yun has looked here:

Terhorst, Jonathan, and Yun S. Song. 2015. Fundamental Limits on the Accuracy of Demographic Inference Based on the Sample Frequency Spectrum. Proceedings of the National Academy of Sciences of the United States of America 112 (25): 7677–82.

kamdh commented 5 years ago

Interesting, from quick skim their bound O(1/log(s)) is by the number of segregating sites. But via our scribblings above, this is related to the number of individuals n.

wsdewitt commented 4 years ago

Also see: https://www.sciencedirect.com/science/article/pii/S0040580908000038