Unbinned likelihood fits aim at maximizing the information one can extract from experimental data, yet their application in realistic statistical analyses is often hindered by the computational cost of profiling systematic uncertainties. Additionally, current machine learning-based inference methods are typically limited to estimating scalar parameters in a multidimensional space rather than full differential distributions. We propose a general framework for Simulation-Based Inference (SBI) that efficiently profiles nuisance parameters while measuring multivariate Distributions of Interest (DoI), defined as learnable invertible transformations of the feature space. We introduce Factorizable Normalizing Flows to model systematic variations as parametric deformations of a nominal density, preserving tractability without combinatorial explosion. Crucially, we develop an amortized training strategy that learns the conditional dependence of the DoI on nuisance parameters in a single optimization process, bypassing the need for repetitive training during the likelihood scan. This allows for the simultaneous extraction of the underlying distribution and the robust profiling of nuisances. The method is validated on a synthetic dataset emulating a high-energy physics measurement with multiple systematic sources, demonstrating its potential for unbinned, functional measurements in complex analyses.

翻译：无分箱似然拟合旨在最大化从实验数据中提取的信息，然而其在现实统计分析中的应用常受限于剖析系统不确定性的计算成本。此外，当前基于机器学习的推断方法通常仅限于估计多维空间中的标量参数，而非完整的微分分布。我们提出了一个模拟推断通用框架，该框架能在测量多元兴趣分布的同时高效地剖析冗余参数；兴趣分布被定义为特征空间的可学习可逆变换。我们引入可分解归一化流，将系统变异建模为标称密度的参数化形变，从而在避免组合爆炸的同时保持可处理性。关键的是，我们开发了一种摊销训练策略，可在单次优化过程中学习兴趣分布对冗余参数的条件依赖关系，绕过了似然扫描期间重复训练的需求。这使得能够同时提取基础分布并对冗余参数进行稳健剖析。该方法在一个模拟高能物理测量、包含多重系统源的合成数据集上得到验证，展示了其在复杂分析中实现无分箱函数式测量的潜力。