Importance Sampling: Computational Complexity and Intrinsic Dimension
The basic idea of importance sampling is to use independent samples from one measure in order to approximate expectations with respect to another measure. Understanding how many samples are needed is key to understanding the computational complexity of the method, and hence to understanding when it will be effective and when it will not. It is intuitive that the size of the difference between the measure which is sampled, and the measure against which expectations are to be computed, is key to the computational complexity. An implicit challenge in many of the published works in this area is to find useful quantities which measure this difference in terms of parameters which are pertinent for the practitioner. The subject has attracted substantial interest recently from within a variety of communities. The objective of this paper is to overview and unify the resulting literature in the area by creating an overarching framework. The general setting is studied in some detail, followed by deeper development in the context of Bayesian inverse problems and filtering.
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Mixture models with symmetric errors
A semiparametric mixture of two populations with the same probability density and different locations can be identified and estimated under the assumption that the common probability density is symmetric. We use indentifiability results in Bordes et al. (2006) and propose a new estimation algorithm based on techniques from the theory of inverse problems.
We consider next a semiparametric mixture of regression models and study its identifiability. We propose an estimation procedure of the mixing proportion and of the location functions locally at a fixed point. Our estimation procedure is based on the symmetry of the errors' distribution and does not require finite moments on the errors. We establish under mild conditions minimax properties and asymptotic normality of our estimators. We study the finite sample performance on synthetic data and on the positron emission tomography imaging data in a cancer study in Bowen et al. (2012).
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