# Bayesian Joint Chance Constrained Optimization: Approximations and Statistical Consistency

arXiv: https://arxiv.org/abs/2106.12199 · HTML: https://mlsanity.com/paper/2106.12199/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 54 references that are in the map.
- First posted on arXiv 23 Jun 2021.

## Builds on (most cited first, 11 of 11)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Frequentist Consistency of Variational Bayes](https://mlsanity.com/paper/1705.03439/) (2017)
- [Convergence Rates of Variational Posterior Distributions](https://mlsanity.com/paper/1712.02519/) (2017)
- [On Distributionally Robust Chance Constrained Programs with Wasserstein Distance](https://mlsanity.com/paper/1806.07418/) (2018)
- [Solving Chance-Constrained Problems via a Smooth Sample-Based Nonlinear Approximation](https://mlsanity.com/paper/1905.07377/) (2019)
- [Asymptotic Consistency of α-R'enyi-Approximate Posteriors](https://mlsanity.com/paper/1902.01902/) (2019)
- [Data-Driven Chance Constrained Optimization under Wasserstein Ambiguity Sets](https://mlsanity.com/paper/1805.06729/) (2018)
- [Asymptotic Consistency of Loss-Calibrated Variational Bayes](https://mlsanity.com/paper/1911.01288/) (2019)
- [Consistency of Distributionally Robust Risk- and Chance-Constrained Optimization under Wasserstein Ambiguity Sets](https://mlsanity.com/paper/2012.08850/) (2020)
- [Variational Bayesian Methods for Stochastically Constrained System Design Problems](https://mlsanity.com/paper/2001.01404/) (2020)
- [Solving Bayesian Risk Optimization via Nested Stochastic Gradient Estimation](https://mlsanity.com/paper/2007.05860/) (2020)

## Built on by (most cited first, 1 of 1)

- [Differentiability and Approximation of Probability Functions under Gaussian Mixture Models](https://mlsanity.com/paper/2411.02721/) (2024)

## Abstract (arXiv)

This paper considers data-driven chance-constrained stochastic optimization problems in a Bayesian framework. Bayesian posteriors afford a principled mechanism to incorporate data and prior knowledge into stochastic optimization problems. However, the computation of Bayesian posteriors is typically an intractable problem, and has spawned a large literature on approximate Bayesian computation. Here, in the context of chance-constrained optimization, we focus on the question of statistical consistency (in an appropriate sense) of the optimal value, computed using an approximate posterior distribution. To this end, we rigorously prove a frequentist consistency result demonstrating the convergence of the optimal value to the optimal value of a fixed, parameterized constrained optimization problem. We augment this by also establishing a probabilistic rate of convergence of the optimal value. We also prove the convex feasibility of the approximate Bayesian stochastic optimization problem. Finally, we demonstrate the utility of our approach on an optimal staffing problem for an M/M/c queueing model.