Sequential experimental design and response optimisation |
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Authors: | Luc Pronzato Éric Thierry |
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Institution: | (1) Laboratoire I3S, CNRS/Université de Nice-Sophia Antipolis, bat. Euclide, Les Algorithmes, 2000 route des Lucioles, BP 121, 06903 Sophia-Antipolis Cedex, France |
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Abstract: | We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy
k
. This may correspond to the case of a regression model, where one observesy
k
=f(θ,x
k
)+ε
k
, with ε
k
some random error, or to the Bernoulli case wherey
k
∈{0, 1}, with Pry
k
=1|θ,x
k
|=f(θ,x
k
). Special attention is given to sequences given by
, with
an estimated value of θ obtained from (x1, y1),...,(x
k
,y
k
) andd
k
(x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon,
where one wants to maximize ∑
i=1
N
w
i
f(θ, x
i
) with {w
i
} a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method
for a binary response problem. |
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Keywords: | Adaptive control Bernoulli trials binary response dose-response optimum design parameter estimation response optimisation sequential design |
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