It is well known in astronomy that propagating non-Gaussian prediction uncertainty in photometric redshift estimates is key to reducing bias in downstream cosmological analyses. Similarly, likelihood-free inference approaches, which are beginning to emerge as a tool for cosmological analysis, require a characterization of the full uncertainty landscape of the parameters of interest given observed data. However, most machine learning (ML) or training-based methods with open-source software target point prediction or classification, and hence fall short in quantifying uncertainty in complex regression and parameter inference settings such as the applications mentioned above. As an alternative to methods that focus on predicting the response (or parameters) $\mathbf{y}$ from features $\mathbf{x}$, we provide nonparametric conditional density estimation (CDE) tools for approximating and validating the entire probability density function (PDF) $\mathrm{p}\left(\mathbf{y}\right|\mathbf{x})$ of $\mathbf{y}$ given (i.e., conditional on) $\mathbf{x}$. This density approach offers a more nuanced accounting of uncertainty in situations with, e.g., nonstandard error distributions and multimodal or heteroskedastic response variables that are often present in astronomical data sets. As there is no one-size-fits-all CDE method, and the ultimate choice of model depends on the application and the training sample size, the goal of this work is to provide a comprehensive range of statistical tools and open-source software for nonparametric CDE and method assessment which can accommodate different types of settings – involving, e.g., mixed-type input from multiple sources, functional data, and images – and which in addition can easily be fit to the problem at hand. Specifically, we introduce four CDE software packages in Python and R based on ML prediction methods adapted and optimized for CDE: NNKCDE, RFCDE, FlexCode, and DeepCDE. Furthermore, we present the cdetools package with evaluation metrics. This package includes functions for computing a CDE loss function for tuning and assessing the quality of individual PDFs, together with diagnostic functions that probe the population-level performance of the PDFs. We provide sample code in Python and R as well as examples of applications to photometric redshift estimation and likelihood-free cosmological inference via CDE.

在天文学中众所周知，在光度红移估计中传播非高斯预测不确定性是减少下游宇宙学分析中偏差的关键。同样，无可能性推断方法（已开始作为宇宙学分析的工具出现）要求在给定观测数据的情况下对目标参数的全部不确定性特征进行表征。但是，大多数带有开源软件的基于机器学习（ML）或基于训练的方法都将目标点预测或分类为目标，因此，在量化复杂回归和参数推断设置（例如上述应用程序）中的不确定性方面不足。作为专注于预测响应（或参数）的方法的替代方法$\mathbf{ÿ}$ 从功能 $\mathbf{X}$，我们提供了非参数条件密度估计（CDE）工具，用于近似和验证整个概率密度函数（PDF） $\mathrm{p}（\mathbf{ÿ}|\mathbf{X}）$ 的 $\mathbf{ÿ}$ 给定的（即有条件的） $\mathbf{X}$。这种密度方法可以更精确地解决不确定性问题，例如在天文学数据集中经常出现的非标准误差分布和多峰或异方差响应变量。由于没有一种千篇一律的CDE方法，并且模型的最终选择取决于应用程序和训练样本的大小，因此，这项工作的目标是为以下方面提供全面的统计工具和开源软件：非参数CDE和方法评估可以适应不同类型的设置-涉及例如来自多个源，功能数据和图像的混合类型输入-并且还可以轻松地解决当前的问题。具体来说，我们在Python和R中引入了四个CDE软件包基于为CDE调整和优化的ML预测方法：NNKCDE，RFCDE，FlexCode和DeepCDE。此外，我们为cdetools软件包提供了评估指标。该软件包包括用于计算CDE损失函数以调整和评估单个PDF的质量的功能，以及探测PDF总体水平性能的诊断功能。我们提供Python和R中的示例代码，以及通过CDE进行光度红移估计和无可能性宇宙学推断的应用示例。