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Due to the ease of modern data collection, applied statisticians often have access to a large set of covariates that they wish to relate to some observed outcome. Generalized linear models (GLMs) offer a particularly interpretable framework for such an analysis. In these high-dimensional problems, the number of covariates is often large relative to the number of observations, so we face non-trivial inferential uncertainty; a Bayesian approach allows coherent quantification of this uncertainty. Unfortunately, existing methods for Bayesian inference in GLMs require running times roughly cubic in parameter dimension, and so are limited to settings with at most tens of thousand parameters. We propose to reduce time and memory costs with a low-rank approximation of the data in an approach we call LR-GLM. When used with the Laplace approximation or Markov chain Monte Carlo, LR-GLM provides a full Bayesian posterior approximation and admits running times reduced by a full factor of the parameter dimension. We rigorously establish the quality of our approximation and show how the choice of rank allows a tunable computational–statistical trade-off. Experiments support our theory and demonstrate the efficacy of LR-GLM on real large-scale datasets.
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A statistical framework for domain shape estimation in Stokes flows
Abstract We develop and implement a Bayesian approach for the estimation of the shape of a two dimensional annular domain enclosing a Stokes flow from sparse and noisy observations of the enclosed fluid. Our setup includes the case of direct observations of the flow field as well as the measurement of concentrations of a solute passively advected by and diffusing within the flow. Adopting a statistical approach provides estimates of uncertainty in the shape due both to the non-invertibility of the forward map and to error in the measurements. When the shape represents a design problem of attempting to match desired target outcomes, this ‘uncertainty’ can be interpreted as identifying remaining degrees of freedom available to the designer. We demonstrate the viability of our framework on three concrete test problems. These problems illustrate the promise of our framework for applications while providing a collection of test cases for recently developed Markov chain Monte Carlo algorithms designed to resolve infinite-dimensional statistical quantities.
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- PAR ID:
- 10427696
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 39
- Issue:
- 8
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- Article No. 085009
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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