# Neural Spline Search for Quantile Probabilistic Modeling

arXiv: https://arxiv.org/abs/2301.04857 · HTML: https://mlsanity.com/paper/2301.04857/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 12 Jan 2023.

## Builds on (most cited first, 14 of 14)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [Sample-Efficient Reinforcement Learning with Stochastic Ensemble Value Expansion](https://mlsanity.com/paper/1807.01675/) (2018)
- [High-Quality Prediction Intervals for Deep Learning: A Distribution-Free, Ensembled Approach](https://mlsanity.com/paper/1802.07167/) (2018)
- [Deep Factors for Forecasting](https://mlsanity.com/paper/1905.12417/) (2019)
- [Neuro-Symbolic Program Synthesis](https://mlsanity.com/paper/1611.01855/) (2016)
- [Learning Quantile Functions without Quantile Crossing for Distribution-free Time Series Forecasting](https://mlsanity.com/paper/2111.06581/) (2021)
- [Prediction Intervals: Split Normal Mixture from Quality-Driven Deep Ensembles](https://mlsanity.com/paper/2007.09670/) (2020)
- [A deep learning approach to diabetic blood glucose prediction](https://mlsanity.com/paper/1707.05828/) (2017)
- [On the benefits of maximum likelihood estimation for Regression and Forecasting](https://mlsanity.com/paper/2106.10370/) (2021)

## Built on by (most cited first, 2 of 2)

- [Deep Evidential Learning for Bayesian Quantile Regression](https://mlsanity.com/paper/2308.10650/) (2023)
- [Simulation-Based Inference with Quantile Regression](https://mlsanity.com/paper/2401.02413/) (2024)

## Abstract (arXiv)

Accurate estimation of output quantiles is crucial in many use cases, where it is desired to model the range of possibility. Modeling target distribution at arbitrary quantile levels and at arbitrary input attribute levels are important to offer a comprehensive picture of the data, and requires the quantile function to be expressive enough. The quantile function describing the target distribution using quantile levels is critical for quantile regression. Although various parametric forms for the distributions (that the quantile function specifies) can be adopted, an everlasting problem is selecting the most appropriate one that can properly approximate the data distributions. In this paper, we propose a non-parametric and data-driven approach, Neural Spline Search (NSS), to represent the observed data distribution without parametric assumptions. NSS is flexible and expressive for modeling data distributions by transforming the inputs with a series of monotonic spline regressions guided by symbolic operators. We demonstrate that NSS outperforms previous methods on synthetic, real-world regression and time-series forecasting tasks.