We present a data-driven approach to characterizing nonidentifiability of a
model's parameters and illustrate it through dynamic as well as steady kinetic
models. By employing Diffusion Maps and their extensions, we discover the
minimal combinations of parameters required to characterize the output behavior
of a chemical system: a set of effective parameters for the model. Furthermore,
we introduce and use a Conformal Autoencoder Neural Network technique, as well
as a kernel-based Jointly Smooth Function technique, to disentangle the
redundant parameter combinations that do not affect the output behavior from
the ones that do. We discuss the interpretability of our data-driven effective
parameters, and demonstrate the utility of the approach both for behavior
prediction and parameter estimation. In the latter task, it becomes important
to describe level sets in parameter space that are consistent with a particular
output behavior. We validate our approach on a model of multisite
phosphorylation, where a reduced set of effective parameters (nonlinear
combinations of the physical ones) has previously been established
analytically.

该研究使用数据驱动方法表征模型参数的不可识别性，并通过动态和稳定动力学模型进行说明。通过 Diffusion Maps 及其扩展，确定了化学系统输出行为所需的最小参数组合：模型的有效参数。使用自动编码神经网络技术和基于核的平滑函数技术，分离对输出行为无影响的冗余参数组合和对其有影响的参数组合。在参数估计中，描述与特定输出行为一致的参数空间中的级集非常重要。在多位点磷酸化模型上验证了该方法的可行性。

关于重要和不重要参数组合的讨论

On the Parameter Combinations That Matter and on Those That do Not

We develop a penalized likelihood estimation framework to estimate the
structure of Gaussian Bayesian networks from observational data. In contrast to
recent methods which accelerate the learning problem by restricting the search
space, our main contribution is a fast algorithm for score-based structure
learning which does not restrict the search space in any way and works on
high-dimensional datasets with thousands of variables. Our use of concave
regularization, as opposed to the more popular $\ell_0$ (e.g. BIC) penalty, is
new. Moreover, we provide theoretical guarantees which generalize existing
asymptotic results when the underlying distribution is Gaussian. Most notably,
our framework does not require the existence of a so-called faithful DAG
representation, and as a result the theory must handle the inherent
nonidentifiability of the estimation problem in a novel way. Finally, as a
matter of independent interest, we provide a comprehensive comparison of our
approach to several standard structure learning methods using open-source
packages developed for the R language. Based on these experiments, we show that
our algorithm is significantly faster than other competing methods while
obtaining higher sensitivity with comparable false discovery rates for
high-dimensional data. In particular, the total runtime for our method to
generate a solution path of 20 estimates for DAGs with 8000 nodes is around one
hour.

本文提出了一种基于得分的结构学习快速算法的罚似然估计框架，该快速算法不受限制地处理高维数据集，并使用凹规则化以解决估计问题的固有不可识别性。其中，高斯贝叶斯网络是重点研究领域之一。