Vaniver comments on A Fervent Defense of Frequentist Statistics - Less Wrong
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I do wonder if it would have been better to include something along the lines of "with probability 1" to the claim that non-Bayesian methods can solve it easily. Compressed sensing isn't magic, even though it's very close.
Humans get tripped up by context changes very frequently. It's not obvious to me where you think this robustness would come from.
Compressed sensing isn't even magic, if you're halfway versed in signal processing. I understood compressed sensing within 30 seconds of hearing a general overview of it, and there are many related analogs in many fields.
The convex optimization guys I know are all rather impressed by compressed sensing- but that may be because they specialize in doing L1 and L2 problems, and so compressed sensing makes the things they're good at even more important.