# Uncertainty quantification and out-of-distribution detection using surjective normalizing flows

arXiv: https://arxiv.org/abs/2311.00377 · HTML: https://mlsanity.com/paper/2311.00377/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 42 references that are in the map.
- First posted on arXiv 1 Nov 2023.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Anchor regression: heterogeneous data meets causality](https://mlsanity.com/paper/1801.06229/) (2018)
- [SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates](https://mlsanity.com/paper/2008.10546/) (2020)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020)
- [How do you go where? Improving next location prediction by learning travel mode information using transformers](https://mlsanity.com/paper/2210.04095/) (2022)
- [Simulation-based Inference for High-dimensional Data using Surjective Sequential Neural Likelihood Estimation](https://mlsanity.com/paper/2308.01054/) (2023)
- [Vision Paper: Causal Inference for Interpretable and Robust Machine Learning in Mobility Analysis](https://mlsanity.com/paper/2210.10010/) (2022)
- [Characteristics of human mobility patterns revealed by high-frequency cell-phone position data](https://mlsanity.com/paper/1907.03604/) (2019)
- [Funnels: Exact maximum likelihood with dimensionality reduction](https://mlsanity.com/paper/2112.08069/) (2021)

## Abstract (arXiv)

Reliable quantification of epistemic and aleatoric uncertainty is of crucial importance in applications where models are trained in one environment but applied to multiple different environments, often seen in real-world applications for example, in climate science or mobility analysis. We propose a simple approach using surjective normalizing flows to identify out-of-distribution data sets in deep neural network models that can be computed in a single forward pass. The method builds on recent developments in deep uncertainty quantification and generative modeling with normalizing flows. We apply our method to a synthetic data set that has been simulated using a mechanistic model from the mobility literature and several data sets simulated from interventional distributions induced by soft and atomic interventions on that model, and demonstrate that our method can reliably discern out-of-distribution data from in-distribution data. We compare the surjective flow model to a Dirichlet process mixture model and a bijective flow and find that the surjections are a crucial component to reliably distinguish in-distribution from out-of-distribution data.