Compare the optimal worst case probability pwc with

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Unformatted text preview: ere ai ∈ Rn are known, vi are IID N (0, σ 2 ) random noises, and φ : R → R is an unknown monotonic increasing function, known to satisfy α ≤ φ′ (u) ≤ β, for all u. (Here α and β are known positive constants, with α < β .) We want to find a maximum likelihood estimate of x and φ, given yi . (We also know ai , σ , α, and β .) This sounds like an infinite-dimensional problem, since one of the parameters we are estimating is a function. In fact, we only need to know the m numbers zi = φ−1 (yi ), i = 1, . . . , m. So by estimating φ we really mean estimating the m numbers z1 , . . . , zm . (These numbers are not arbitrary; they must be consistent with the prior information α ≤ φ′ (u) ≤ β for all u.) 54 (a) Explain how to find a maximum likelihood estimate of x and φ (i.e., z1 , . . . , zm ) using convex optimization. (b) Carry out your method on the data given in nonlin_meas_data.m, which includes a matrix ˆ A ∈ Rm×n , with rows aT , . . . , aT . Give xml , the maximum likelihood estimate of x. Plot your m 1 ˆml . (You can do this by p...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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