# Neural Likelihoods via Cumulative Distribution Functions

arXiv: https://arxiv.org/abs/1811.00974 · HTML: https://mlsanity.com/paper/1811.00974/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2018.

## Built on by (most cited first, 25 of 25)

- [Survival Regression with Proper Scoring Rules and Monotonic Neural Networks](https://mlsanity.com/paper/2103.14755/) (2021)
- [Neural Fine-Gray: Monotonic neural networks for competing risks](https://mlsanity.com/paper/2305.06703/) (2023)
- [Generative Archimedean Copulas](https://mlsanity.com/paper/2102.11351/) (2021)
- [Dynamic Hawkes Processes for Discovering Time-evolving Communities' States behind Diffusion Processes](https://mlsanity.com/paper/2105.11152/) (2021)
- [Deep Copula-Based Survival Analysis for Dependent Censoring with Identifiability Guarantees](https://mlsanity.com/paper/2312.15566/) (2023)
- [Marginal Causal Flows for Validation and Inference](https://mlsanity.com/paper/2411.01295/) (2024)
- [Advancing Constrained Monotonic Neural Networks: Achieving Universal Approximation Beyond Bounded Activations](https://mlsanity.com/paper/2505.02537/) (2025)
- [Neural Copula: A unified framework for estimating generic high-dimensional Copula functions](https://mlsanity.com/paper/2205.15031/) (2022)
- [HACSurv: A Hierarchical Copula-Based Approach for Survival Analysis with Dependent Competing Risks](https://mlsanity.com/paper/2410.15180/) (2024)
- [Diffusion and Flow-based Copulas: Forgetting and Remembering Dependencies](https://mlsanity.com/paper/2509.19707/) (2025)
- [Inference and Sampling for Archimax Copulas](https://mlsanity.com/paper/2205.14025/) (2022)
- [Marginalizable Density Models](https://mlsanity.com/paper/2106.04741/) (2021)
- [Bayesian Neural Hawkes Process for Event Uncertainty Prediction](https://mlsanity.com/paper/2112.14474/) (2021)
- [Cumulative Hazard Function Based Efficient Multivariate Temporal Point Process Learning](https://mlsanity.com/paper/2404.13663/) (2024)
- [Prediction of Survival Outcomes under Clinical Presence Shift: A Joint Neural Network Architecture](https://mlsanity.com/paper/2508.05472/) (2025)
- [Regularization Strategies for Quantile Regression](https://mlsanity.com/paper/2102.05135/) (2021)
- [HyperHawkes: Hypernetwork based Neural Temporal Point Process](https://mlsanity.com/paper/2210.00213/) (2022)
- [Forecasting the 2016-2017 Central Apennines Earthquake Sequence with a Neural Point Process](https://mlsanity.com/paper/2301.09948/) (2023)
- [Cumulative Distribution Function based General Temporal Point Processes](https://mlsanity.com/paper/2402.00388/) (2024)
- [Efficient CDF Approximations for Normalizing Flows](https://mlsanity.com/paper/2202.11322/) (2022)
- [Learning Multivariate CDFs and Copulas using Tensor Factorization](https://mlsanity.com/paper/2210.07132/) (2022)
- [Reinterpreting survival analysis in the universal approximator age](https://mlsanity.com/paper/2307.13579/) (2023)
- [2-Cats: 2D Copula Approximating Transforms](https://mlsanity.com/paper/2309.16391/) (2023)
- [Spectral Survival Analysis](https://mlsanity.com/paper/2505.22641/) (2025)
- [Smooth Neural Point Processes via B-Splines](https://mlsanity.com/paper/2607.21098/) (2026)

## Abstract (arXiv)

We leverage neural networks as universal approximators of monotonic functions to build a parameterization of conditional cumulative distribution functions (CDFs). By the application of automatic differentiation with respect to response variables and then to parameters of this CDF representation, we are able to build black box CDF and density estimators. A suite of families is introduced as alternative constructions for the multivariate case. At one extreme, the simplest construction is a competitive density estimator against state-of-the-art deep learning methods, although it does not provide an easily computable representation of multivariate CDFs. At the other extreme, we have a flexible construction from which multivariate CDF evaluations and marginalizations can be obtained by a simple forward pass in a deep neural net, but where the computation of the likelihood scales exponentially with dimensionality. Alternatives in between the extremes are discussed. We evaluate the different representations empirically on a variety of tasks involving tail area probabilities, tail dependence and (partial) density estimation.