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Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball

Hamze Bajgiran, Hamed and Batlle Franch, Pau and Owhadi, Houman and Scovel, Clint and Shirdel, Mahdy and Stanley, Michael and Tavallali, Peyman (2021) Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20220524-180308552

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Abstract

There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of risk is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample x, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties (a) an optimal prior is identified after measuring the data, and the notion of risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (which is fast and not subject to the curse of dimensionality). The method is characterized by a parameter in [0,1] acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near 1, the method produces a posterior distribution concentrated around a maximum likelihood estimate with tight but low confidence UQ estimates. When that parameter is near 0, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/2108.10517arXivDiscussion Paper
ORCID:
AuthorORCID
Owhadi, Houman0000-0002-5677-1600
Scovel, Clint0000-0001-7757-3411
Tavallali, Peyman0000-0001-7166-5489
Additional Information:© 2021. California Institute of Technology. Government sponsorship acknowledged. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The authors gratefully acknowledge support from Beyond Limits (Learning Optimal Models) through CAST (The Caltech Center for Autonomous Systems and Technologies) and partial support from the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning).
Group:Center for Autonomous Systems and Technologies (CAST)
Funders:
Funding AgencyGrant Number
NASA/JPL/CaltechUNSPECIFIED
Beyond LimitsUNSPECIFIED
Air Force Office of Scientific Research (AFOSR)FA9550-18-1-0271
Record Number:CaltechAUTHORS:20220524-180308552
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20220524-180308552
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:114897
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:24 May 2022 20:26
Last Modified:24 May 2022 20:26

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