# Probability Density Estimation through Nonparametric Adaptive Partitioning and Stitching
## 1 Introduction  
In many domains, such as climate science, epidemiology, quality assurance, finance, and market analysis, there is a need for robust high throughput probability density estimation as the amount of information processed grows. The motivation for this work is to address the computational challenges of a reliable automated method that makes fast and accurate density estimates on massive datasets. If an application is limited to similar types of input data, expert domain knowledge can be used to develop appropriate models and limit the number of relevant parameters and their accessible range. However, high throughput analysis and turnkey automation are critical for large datasets, to relieve analysts from looking at each density estimate on an individual basis to either determine a reasonable parametric model to use, or to verify if a nonparametric model presents a reasonable solution. In this context, kernel density estimation (KDE) is arguably the most common approach for nonparametric density estimation. A major contributing factor to the popularity of KDE is the intuitive interpretation of local binning and smoothing of observed data. The implementation of this conceptual idea using a simple Gaussian kernel to approximate each bin is straightforward [1; 2; 3; 4; 5; 6]. However, it becomes readily apparent that there is a trade-off between smoothness in low density regions and accurate modeling of sharp features within the ground truth. Addressing this basic trade-off has created a long history of methodological and computational improvements over many decades of active research [7; 8; 9; 10; 11; 12; 13]. Despite its popularity, however, KDE suffers from two inherent weaknesses: the user must select both an optimal bandwidth for resolution of the estimate and an appropriate kernel for modeling boundary conditions accurately. With insight and knowledge of the expected results, the two weaknesses of KDE can be largely mitigated in practice. Unfortunately, KDE does not automatically work on any type of distribution without subjective selection options set by a human during a setup process. Furthermore, there is danger that if prior conditions change in high throughput applications, the user's assumptions can become invalid, creating a risk for large errors. There is a wide range of KDE methods available for data-driven adaptive bandwidths or adaptive kernel selection. However, these methods often incur substantial computational costs, especially when applied towards large data sets and often still require guidance by domain experts. As data demands increase in various applications, parallel computing methods for density estimation have been developed to employ multiple processors and resources, aiming to expedite the calculations [14; 15; 16; 17; 18; 19; 20]. The conceptual ideas of parallel processing also extend to multivariate density estimation, and several parallel computing implementations have been developed for higher dimensions [21; 22; 23]. Given the inherent parallelizability of the KDE, this method is a natural choice for breaking up the problem. Although parallel KDE implementations offer improved computational performance, they nevertheless suffer from the same inherent weaknesses of the underlying algorithm. As an alternative to KDE, a fully automated method for univariate probability density estimation was introduced by JF and DJJ in 2018 [24]. With successive improvements since, open software packages in MATLAB, C++, Java, and R are now available [25; 26]. Our data-adaptive model-free method requires no prior knowledge of the input data characteristics, therefore the user need not set the bandwidth nor any additional parameters. Our method is based on a maximum entropy principle, while solving an inverse problem using a trial cumulative probability function by optimizing an objective function that involves sort order statistics to obtain sampled uniform random data (SURD) [24]. We will refer to this base method as NMEM for nonparametric maximum entropy method.