# Graph-dependent shrinkage priors for Bayesian trend filtering

arXiv: https://arxiv.org/abs/2608.23802 · HTML: https://mlsanity.com/paper/2608.23802/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [A Survey on Graph Representation Learning Methods](https://mlsanity.com/paper/2204.01855/) (2022)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Trend Filtering on Graphs](https://mlsanity.com/paper/1410.7690/) (2014)
- [Vector-Valued Graph Trend Filtering with Non-Convex Penalties](https://mlsanity.com/paper/1905.12692/) (2019)
- [The Graphical Horseshoe Estimator for Inverse Covariance Matrices](https://mlsanity.com/paper/1707.06661/) (2017)
- [Ancillarity-Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC Estimation of Stochastic Volatility Models](https://mlsanity.com/paper/1706.05280/) (2017)
- [Dynamic Shrinkage Processes](https://mlsanity.com/paper/1707.00763/) (2017)
- [Approximate l0-penalized estimation of piecewise-constant signals on graphs](https://mlsanity.com/paper/1703.01421/) (2017)
- [Precision Matrix Estimation under the Horseshoe-like Prior-Penalty Dual](https://mlsanity.com/paper/2104.10750/) (2021)
- [Multivariate Trend Filtering for Lattice Data](https://mlsanity.com/paper/2112.14758/) (2021)
- [Locally adaptive smoothing with Markov random fields and shrinkage priors](https://mlsanity.com/paper/1512.06505/) (2015)
- [Structured Shrinkage Priors](https://mlsanity.com/paper/1902.05106/) (2019)
- [Variance estimation in graphs with the fused lasso](https://mlsanity.com/paper/2207.12638/) (2022)
- [Scalable Multiple Network Inference with the Joint Graphical Horseshoe](https://mlsanity.com/paper/2206.11820/) (2022)

## Abstract (arXiv)

Many common data dependencies can be characterized by graphs: time series data are sequential (chain graph), images appear as pixels (lattice graph), areal data are defined by neighboring units (spatial adjacency graph), etc. Graph trend filtering seeks to smooth and predict such data. However, classical trend filtering only incorporates the graph for estimation of the trend, which limits its adaptivity, and is brittle in the presence of missing data. Further, it lacks uncertainty quantification and faces certain computing challenges. We address these limitations with a comprehensive Bayesian framework for (graph-) dependent data. Our approach leverages the graph at three critical junctures: 1) the trend, to enable smoothing, imputation, and prediction; 2) the local shrinkage, to enhance adaptivity and precision; and 3) the MCMC sampling algorithm, to deliver scalable posterior (predictive) inference via sparse and banded operations. For the proposed graph-dependent shrinkage priors, we study the local concentration and adaptivity properties and establish conditions for posterior propriety. Simulation studies demonstrate that, relative to state-of-the-art frequentist and Bayesian alternatives, this framework provides more accurate point estimates, more precise interval estimates, and highly competitive computing. We apply our methods for spatio-temporal modeling and forecasting of local area unemployment data for every county in the continental U.S. during the 2020 COVID-19 unemployment shock.