# Implicit Maximum a Posteriori Filtering via Adaptive Optimization

arXiv: https://arxiv.org/abs/2311.10580 · HTML: https://mlsanity.com/paper/2311.10580/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 17 Nov 2023.

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

- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [A Century of Portraits: A Visual Historical Record of American High School Yearbooks](https://mlsanity.com/paper/1511.02575/) (2015)
- [Online Natural Gradient as a Kalman Filter](https://mlsanity.com/paper/1703.00209/) (2017)
- [Early Stopping is Nonparametric Variational Inference](https://mlsanity.com/paper/1504.01344/) (2015)
- [Low-rank extended Kalman filtering for online learning of neural networks from streaming data](https://mlsanity.com/paper/2305.19535/) (2023)
- [Weak in the NEES?: Auto-tuning Kalman Filters with Bayesian Optimization](https://mlsanity.com/paper/1807.08855/) (2018)
- [The Extended Kalman Filter is a Natural Gradient Descent in Trajectory Space](https://mlsanity.com/paper/1901.00696/) (2019)
- [Bayesian filtering unifies adaptive and non-adaptive neural network optimization methods](https://mlsanity.com/paper/1807.07540/) (2018)
- [On Computational Complexity Reduction Methods for Kalman Filter Extensions](https://mlsanity.com/paper/1512.03077/) (2015)

## Built on by (most cited first, 4 of 4)

- [Outlier-robust Kalman Filtering through Generalised Bayes](https://mlsanity.com/paper/2405.05646/) (2024)
- [Bayesian Online Natural Gradient (BONG)](https://mlsanity.com/paper/2405.19681/) (2024)
- [Martingale Posterior Neural Networks for Fast Sequential Decision Making](https://mlsanity.com/paper/2506.11898/) (2025)
- [A Metalearned Neural Circuit for Nonparametric Bayesian Inference](https://mlsanity.com/paper/2311.14601/) (2023)

## Abstract (arXiv)

Bayesian filtering approximates the true underlying behavior of a time-varying system by inverting an explicit generative model to convert noisy measurements into state estimates. This process typically requires either storage, inversion, and multiplication of large matrices or Monte Carlo estimation, neither of which are practical in high-dimensional state spaces such as the weight spaces of artificial neural networks. Here, we frame the standard Bayesian filtering problem as optimization over a time-varying objective. Instead of maintaining matrices for the filtering equations or simulating particles, we specify an optimizer that defines the Bayesian filter implicitly. In the linear-Gaussian setting, we show that every Kalman filter has an equivalent formulation using K steps of gradient descent. In the nonlinear setting, our experiments demonstrate that our framework results in filters that are effective, robust, and scalable to high-dimensional systems, comparing well against the standard toolbox of Bayesian filtering solutions. We suggest that it is easier to fine-tune an optimizer than it is to specify the correct filtering equations, making our framework an attractive option for high-dimensional filtering problems.