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.

翻译：最初由Schwarzschild(1979)提出的轨道叠加方法用于研究黑洞及其宿主星系的增长动力学，并揭示了星系整体特征之间的新关联。科学家特别关注如何为该模型寻找与物理测量值最佳匹配的最优参数选择，同时量化此类过程的不确定性。这构成了一个多变量输出的统计校准问题。本文提出了一种采用正交偏差函数实现多变量输出校准的贝叶斯方法，从而确保参数可识别性。该方法通过将后验投影至恰当空间，允许用户选择偏差函数的任意非参数先验，而无需像高斯过程那样对其建模。我们利用希尔伯特空间理论构建了函数投影方法，并考虑了投影问题的有限维等价形式。通过BART先验验证所提方法，并将其应用于史瓦西模型的校准，展示了多变量方法如何解决单变量校准产生的偏差问题。