## Statistical Science

- Statist. Sci.
- Volume 32, Number 3 (2017), 405-431.

### Importance Sampling: Intrinsic Dimension and Computational Cost

S. Agapiou, O. Papaspiliopoulos, D. Sanz-Alonso, and A. M. Stuart

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#### Abstract

The basic idea of importance sampling is to use independent samples from a proposal measure in order to approximate expectations with respect to a target measure. It is key to understand how many samples are required in order to guarantee accurate approximations. Intuitively, some notion of distance between the target and the proposal should determine the computational cost of the method. A major challenge is to quantify this distance in terms of parameters or statistics that are pertinent for the practitioner. The subject has attracted substantial interest from within a variety of communities. The objective of this paper is to overview and unify the resulting literature by creating an overarching framework. A general theory is presented, with a focus on the use of importance sampling in Bayesian inverse problems and filtering.

#### Article information

**Source**

Statist. Sci. Volume 32, Number 3 (2017), 405-431.

**Dates**

First available in Project Euclid: 1 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ss/1504253124

**Digital Object Identifier**

doi:10.1214/17-STS611

**Keywords**

Importance sampling notions of dimension small noise absolute continuity inverse problems filtering

#### Citation

Agapiou, S.; Papaspiliopoulos, O.; Sanz-Alonso, D.; Stuart, A. M. Importance Sampling: Intrinsic Dimension and Computational Cost. Statist. Sci. 32 (2017), no. 3, 405--431. doi:10.1214/17-STS611. https://projecteuclid.org/euclid.ss/1504253124

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#### Supplemental materials

- Supplement to “Importance sampling: Intrinsic dimension and computational cost”. The Supplementary Material contains the proofs of all our results. It also contains some background on Gaussian measures on Hilbert spaces, and related technical aspects arising from considering measures in infinite dimensional spaces.Digital Object Identifier: doi:10.1214/17-STS611SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.

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