Parametric reduced-order models built by least-squares projection, including POD, reduced basis methods, and Gaussian process emulation, can be viewed as conditional expectations in a Bayesian updating framework.
Bobrowski, Functional analysis for probability and stochastic processes, Cambridge University Press, Cambridge, 2005
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Reduced Order Models and Conditional Expectation -- Analysing Parametric Low-Order Approximations
Parametric reduced-order models built by least-squares projection, including POD, reduced basis methods, and Gaussian process emulation, can be viewed as conditional expectations in a Bayesian updating framework.