Efficient Prior Calibration From Indirect Data
<|reference_start|>Efficient Prior Calibration From Indirect Data: Bayesian inversion is central to the quantification of uncertainty within problems arising from numerous applications in science and engineering. To formulate the approach, four ingredients are required: a forward model mapping the unknown parameter to an element of a solution space, often the solution space for a differential equation; an observation operator mapping an element of the solution space to the data space; a noise model describing how noise pollutes the observations; and a prior model describing knowledge about the unknown parameter before the data is acquired. This paper is concerned with learning the prior model from data; in particular, learning the prior from multiple realizations of indirect data obtained through the noisy observation process. The prior is represented, using a generative model, as the pushforward of a Gaussian in a latent space; the pushforward map is learned by minimizing an appropriate loss function. A metric that is well-defined under empirical approximation is used to define the loss function for the pushforward map to make an implementable methodology. Furthermore, an efficient residual-based neural operator approximation of the forward model is proposed and it is shown that this may be learned concurrently with the pushforward map, using a bilevel optimization formulation of the problem; this use of neural operator approximation has the potential to make prior learning from indirect data more computationally efficient, especially when the observation process is expensive, non-smooth or not known. The ideas are illustrated with the Darcy flow inverse problem of finding permeability from piezometric head measurements.<|reference_end|>
@article{akyildiz2024efficient,
    title={Efficient Prior Calibration From Indirect Data},
    author={O. Deniz Akyildiz, Mark Girolami, Andrew M. Stuart, Arnaud
  Vadeboncoeur},
    journal={arXiv preprint arXiv:2405.17955},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.17955},
    primaryClass={stat.ML cs.LG stat.CO}
}