# Calculating Optimistic Likelihoods Using (Geodesically) Convex Optimization

arXiv: https://arxiv.org/abs/1910.07817 · HTML: https://mlsanity.com/paper/1910.07817/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2019.

## Built on by (most cited first, 8 of 8)

- [Robust Bayesian Recourse](https://mlsanity.com/paper/2206.10833/) (2022)
- [Sequential Domain Adaptation by Synthesizing Distributionally Robust Experts](https://mlsanity.com/paper/2106.00322/) (2021)
- [Distributionally Robust Fair Principal Components via Geodesic Descents](https://mlsanity.com/paper/2202.03071/) (2022)
- [A Geometric Unification of Distributionally Robust Covariance Estimators: Shrinking the Spectrum by Inflating the Ambiguity Set](https://mlsanity.com/paper/2405.20124/) (2024)
- [A Greedy and Optimistic Approach to Clustering with a Specified Uncertainty of Covariates](https://mlsanity.com/paper/2204.08205/) (2022)
- [On Tractability, Complexity, and Mixed-Integer Convex Programming Representability of Distributionally Favorable Optimization](https://mlsanity.com/paper/2401.17899/) (2024)
- [Disciplined Geodesically Convex Programming](https://mlsanity.com/paper/2407.05261/) (2024)
- [Structured Regularization for Constrained Optimization on the SPD Manifold](https://mlsanity.com/paper/2410.09660/) (2024)

## Abstract (arXiv)

A fundamental problem arising in many areas of machine learning is the evaluation of the likelihood of a given observation under different nominal distributions. Frequently, these nominal distributions are themselves estimated from data, which makes them susceptible to estimation errors. We thus propose to replace each nominal distribution with an ambiguity set containing all distributions in its vicinity and to evaluate an optimistic likelihood, that is, the maximum of the likelihood over all distributions in the ambiguity set. When the proximity of distributions is quantified by the Fisher-Rao distance or the Kullback-Leibler divergence, the emerging optimistic likelihoods can be computed efficiently using either geodesic or standard convex optimization techniques. We showcase the advantages of working with optimistic likelihoods on a classification problem using synthetic as well as empirical data.