Frequentist Parameter Estimation with Supervised Learning

• URL: http://arxiv.org/abs/2105.12302v1
• Date: Wed, 26 May 2021 02:24:25 GMT
• Title: Frequentist Parameter Estimation with Supervised Learning
• Authors: Samuel P. Nolan and Luca Pezz\`e and Augusto Smerzi
• Abstract summary: We use regression to infer a machine-learned point estimate of an unknown parameter. When the number of training measurements are large, this is identical to the well-known maximum-likelihood estimator (MLE) We show that the machine-learned estimator inherits the desirable properties of the MLE, up to a limit imposed by the resolution of the training grid.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recently there has been a great deal of interest surrounding the calibration of quantum sensors using machine learning techniques. In this work, we explore the use of regression to infer a machine-learned point estimate of an unknown parameter. Although the analysis is neccessarily frequentist - relying on repeated esitmates to build up statistics - we clarify that this machine-learned estimator converges to the Bayesian maximum a-posterori estimator (subject to some regularity conditions). When the number of training measurements are large, this is identical to the well-known maximum-likelihood estimator (MLE), and using this fact, we argue that the Cram{\'e}r-Rao sensitivity bound applies to the mean-square error cost function and can therefore be used to select optimal model and training parameters. We show that the machine-learned estimator inherits the desirable asymptotic properties of the MLE, up to a limit imposed by the resolution of the training grid. Furthermore, we investigate the role of quantum noise the training process, and show that this noise imposes a fundamental limit on number of grid points. This manuscript paves the way for machine-learning to assist the calibration of quantum sensors, thereby allowing maximum-likelihood inference to play a more prominent role in the design and operation of the next generation of ultra-precise sensors.

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