Orthogonal calibration via posterior projections with applications to the Schwarzschild model
arxiv(2024)
Abstract
The orbital superposition method originally developed by Schwarzschild (1979)
is used to study the dynamics of growth of a black hole and its host galaxy,
and has uncovered new relationships between the galaxy's global
characteristics. Scientists are specifically interested in finding optimal
parameter choices for this model that best match physical measurements along
with quantifying the uncertainty of such procedures. This renders a statistical
calibration problem with multivariate outcomes. In this article, we develop a
Bayesian method for calibration with multivariate outcomes using orthogonal
bias functions thus ensuring parameter identifiability. Our approach is based
on projecting the posterior to an appropriate space which allows the user to
choose any nonparametric prior on the bias function(s) instead of having to
model it (them) with Gaussian processes. We develop a functional projection
approach using the theory of Hilbert spaces. A finite-dimensional analogue of
the projection problem is also considered. We illustrate the proposed approach
using a BART prior and apply it to calibrate the Schwarzschild model
illustrating how a multivariate approach may resolve discrepancies resulting
from a univariate calibration.
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