A tutorial on Bayesian Normal linear regression

Abstract

Regression is a common task in metrology and often applied to calibrate instruments, evaluate inter-laboratory comparisons or determine fundamental constants, for example. Yet, a regression model cannot be uniquely formulated as a measurement function, and consequently the Guide to the Expression of Uncertainty in Measurement (GUM) and its supplements are not applicable directly. Bayesian inference, however, is well suited to regression tasks, and has the advantage of accounting for additional a priori information, which typically robustifies analyses. Furthermore, it is anticipated that future revisions of the GUM shall also embrace the Bayesian view.Guidance on Bayesian inference for regression tasks is largely lacking in metrology. For linear regression models with Gaussian measurement errors this tutorial gives explicit guidance. Divided into three steps, the tutorial first illustrates how a priori knowledge, which is available from previous experiments, can be translated into prior distributions from a specific class. These prior distributions have the advantage of yielding analytical, closed form results, thus avoiding the need to apply numerical methods such as Markov Chain Monte Carlo. Secondly, formulas for the posterior results are given, explained and illustrated, and software implementations are provided. In the third step, Bayesian tools are used to assess the assumptions behind the suggested approach.These three steps (prior elicitation, posterior calculation, and robustness to prior uncertainty and model adequacy) are critical to Bayesian inference. The general guidance given here for Normal linear regression tasks is accompanied by a simple, but real-world, metrological example. The calibration of a flow device serves as a running example and illustrates the three steps. It is shown that prior knowledge from previous calibrations of the same sonic nozzle enables robust predictions even for extrapolations.

Quantifying parameter uncertainties in optical scatterometry using Bayesian inversion

“…This can be used to also quantify uncertainties of the reconstructed parameter values. [2][3][4] Here we demonstrate an efficent method for parameter reconstruction and uncertainty quantification using a Newton method to solve the inverse problem, an efficient finite-element based solver for the forward-problem, and a Bayesian approach for relating measurement uncertainties and prior knowledge to the reconstruction results. The paper is structured as follows: An optical scatterometry setup which serves as application example is presented in Section 2.…”

Traceably calibrated scanning Hall probe microscopy at room temperature

“…The graphene samples were grown on silicon carbide (SiC) (0001) substrates with a size of 5 mm × 10 mm using a so-called polymer-assisted sublimation growth technique (Kruskopf et al, 2016;Momeni Pakdehi et al, 2018, 2019. The high morphological and electronic homogeneity of the graphene samples utilizes scalable realization of Hall sensors on true two-dimensional carbon sheets without bilayer inclusions.…”

“…out electronics, including the stability of the current source and the voltmeter noise as well as thermoelectric voltages; (iv) the positioning accuracy of the scanning system; and (v) the influence of the sensor on the sample, for example in terms of the magnetic stray field generated by the supply current. The multiplicity of uncertainty sources and the fact that standard uncertainty analysis is not sufficient for linear regression tasks (Klauenberg et al, 2015), as used in the Hall sensor calibration, rule out a conventional uncertainty propagation calculation. Therefore, the uncertainties of the main contributions were analyzed separately to evaluate their impact on the measurement.…”

A Novel Active Learning Regression Framework for Balancing the Exploration-Exploitation Trade-Off

“…According to Reference [ 40 ], the conjugate posterior distribution is a Normal Inverse Gamma (NIG) distribution as follows: …”

“…In this section, we have provided the final formulations for Bayesian linear regression model. The interested readers can find more details in References [ 40 , 42 ].…”