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Horseshoe Forests for High-Dimensional Causal Survival Analysis

This paper introduces a Bayesian tree ensemble model that employs a horseshoe prior on step heights and a reversible jump Gibbs sampler to effectively estimate heterogeneous treatment effects in high-dimensional censored survival data, demonstrating superior performance in simulations and a practical application to pancreatic cancer survival analysis.

Original authors: Tijn Jacobs, Wessel N. van Wieringen, Stéphanie L. van der Pas

Published 2026-05-08

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CC BY 4.0

Original authors: Tijn Jacobs, Wessel N. van Wieringen, Stéphanie L. van der Pas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a doctor trying to figure out which patients will benefit most from a new treatment, like radiation therapy for pancreatic cancer. You have a massive list of clues about each patient: their age, genetics, tumor size, and thousands of other biological markers. The problem is that most of these clues are just "noise"—random details that don't actually tell you anything about how the patient will respond. Only a few are the real "signals."

This paper introduces a new statistical tool called the Horseshoe Forest to solve this problem. Here is how it works, explained through simple analogies:

1. The Problem: The "Needle in a Haystack"

In high-dimensional data (where you have way more clues than patients), it's like trying to find a few needles in a giant haystack. Traditional methods often try to throw away the hay (ignore the clues) to find the needles. But in medicine, if you throw away the wrong clue, you might accidentally ignore a patient who needs help, or worse, you might confuse a symptom with the cause.

2. The Solution: The "Smart Shrinkage"

The authors created a model that doesn't just throw away clues. Instead, it uses a mathematical trick called a "Horseshoe Prior."

Think of the Horseshoe Prior as a smart, adjustable magnifying glass with a special "squeeze" feature:

The Squeeze: If a clue is just noise (like a random genetic marker that doesn't matter), the model gently squeezes its importance down to almost zero. It effectively says, "This clue is probably irrelevant, so let's ignore it."
The Stretch: If a clue is a strong signal (like a specific gene that definitely affects survival), the model lets it stretch out and stand tall. It says, "This clue is important, so we keep it."

This happens automatically for every single clue in the dataset. The model learns which clues to shrink and which to keep, without needing a human to decide in advance.

3. The Structure: A Forest of Decision Trees

The model is built using Decision Trees (which look like flowcharts). Imagine a tree that asks questions like, "Is the patient's age over 60?" or "Is this gene active?"

Old Way: In standard trees, the model tries to control complexity by limiting how deep the tree can grow or how many questions it can ask.
New Way (Horseshoe Forest): The authors changed the rules. They kept the trees flexible but put the "squeeze" (the Horseshoe Prior) directly on the answers at the bottom of the tree (the "step heights").
- If a branch of the tree leads to a dead end with no useful information, the "squeeze" makes that answer tiny.
- If a branch finds a real pattern, the answer stays big and bold.

4. The Goal: Finding the "Personalized" Effect

The main goal is to find Heterogeneous Treatment Effects.

Average Effect: "On average, this drug helps everyone by 10%." (This is easy to find but often wrong for specific people).
Personalized Effect: "This drug helps Patient A a lot, but does nothing for Patient B."

The Horseshoe Forest is designed to find these personalized patterns even when the data is messy, censored (meaning some patients are still alive at the end of the study, so we don't know their exact outcome yet), and full of thousands of confusing variables.

5. The Real-World Test: Pancreatic Cancer

The authors tested their tool on real data from 130 patients with pancreatic cancer (a very aggressive disease).

They looked at whether radiation therapy helped patients live longer.
They fed the model thousands of genetic and clinical clues.
The Result: The model found that, on average, radiation therapy did seem to help patients live longer. However, when they looked at individual patients, the model couldn't confidently say which specific patients would benefit the most. The "personalized" effects were all over the place, mostly because the data was too noisy and the patient group too small to find clear patterns for individuals.
The Takeaway: The tool worked well at handling the messy, high-dimensional data without getting confused, but it confirmed that for this specific group of patients, the treatment seems to help everyone roughly the same amount, rather than helping only a specific "lucky" subgroup.

Summary

The Horseshoe Forest is like a super-smart filter for medical data. Instead of blindly throwing away information, it gently "squeezes" out the noise while keeping the important signals. This allows researchers to look for personalized treatment effects in massive datasets without getting lost in the details, even when the data is incomplete or complex.

Technical Summary: Horseshoe Forests for High-Dimensional Causal Survival Analysis

Problem Statement
Estimating heterogeneous treatment effects (HTE) in survival data is challenging, particularly when covariates are high-dimensional ( $p \gg n$ ). Existing methods face a trade-off between flexibility and regularisation. While Bayesian Additive Regression Trees (BART) effectively model non-linear effects and interactions, their standard regularisation relies on constraints on tree structure (e.g., depth limits) or priors on splitting probabilities (e.g., Dirichlet priors in DART). In high-dimensional causal settings, these structural regularisation strategies can be insufficient or induce "regularisation-induced confounding" (RIC), where aggressive shrinkage of covariates associated with treatment assignment biases the estimated treatment effect. Furthermore, standard BART extensions to survival data (e.g., AFT-BART) often struggle to maintain accurate uncertainty quantification and variable selection properties when $p$ is large. The specific challenge addressed is estimating the Conditional Average Treatment Effect (CATE) in censored survival data while retaining all covariates to satisfy unconfoundedness assumptions, without inducing bias through over-shrinkage of weak confounders.

Methodology
The authors propose the Causal Horseshoe Forest, a Bayesian regression tree ensemble model adapted for high-dimensional survival analysis. The core innovation lies in shifting the focus of regularisation from the tree structure to the leaf node parameters (step heights).

Model Framework: The method employs an Accelerated Failure Time (AFT) model for potential survival times:
$\log T(a) = f(x, \hat{e}(x)) + a \cdot \tau(x) + \varepsilon$
where $f(x, \hat{e}(x))$ models the baseline prognosis (incorporating the estimated propensity score $\hat{e}(x)$ to mitigate confounding) and $\tau(x)$ models the heterogeneous treatment effect. Both functions are modeled as sums of regression trees.
Horseshoe Prior on Step Heights: Instead of standard Gaussian priors on step heights, the authors place a global-local horseshoe prior directly on the step heights ( $h_\ell$ ) of the trees.
$h_\ell | \lambda_\ell^2, \tau^2, \omega \sim N(0, \omega \lambda_\ell^2 \tau^2)$
Here, $\tau$ is a global shrinkage parameter shared across a tree, and $\lambda_\ell$ are local shrinkage parameters for each leaf. The global and local parameters follow half-Cauchy distributions. This allows for adaptive shrinkage: weak signals (noise) are shrunk toward zero, while strong signals are preserved, avoiding the "all-or-nothing" sparsity of methods like LASSO or DART. This continuous shrinkage is crucial for causal inference, as it retains covariates that may be weak confounders, thereby preserving the unconfoundedness assumption.
Computational Inference: The non-conjugate nature of the horseshoe prior breaks the analytical tractability of standard BART's Gibbs sampler. To address this, the authors develop a reversible jump Metropolis-Hastings (RJMH) algorithm within a Gibbs sampling framework.
- Outer Loop: Updates the prognostic and treatment effect forests separately using a Bayesian backfitting strategy. Censored times are handled via data augmentation (sampling from truncated normal distributions).
- Inner Loop (RJMH): Jointly updates the tree structure (via GROW, PRUNE, CHANGE moves) and the associated step heights/local shrinkage parameters. The proposal mechanism uses "pseudo-Gibbs" updates informed by parent node parameters to improve mixing.
Implementation: The method is implemented in the R package ShrinkageTrees. It supports binary outcomes (for propensity score estimation) and censored survival outcomes.

Key Contributions

Novel Regularisation Strategy: The paper introduces a step-height prior based on the horseshoe prior for Bayesian tree ensembles. This shifts regularisation from structural constraints to parameter shrinkage, enabling adaptive control of model complexity in high dimensions.
Computational Framework: The authors develop a reversible jump Gibbs sampler to accommodate the non-conjugate horseshoe prior within the tree ensemble framework, allowing for joint updates of tree topology and step heights.
Application to Survival CATE: The framework is specifically tailored for high-dimensional causal survival analysis, addressing the complexities of censoring and the need for robust confounding adjustment without variable exclusion.

Results
The authors evaluate the method through extensive simulations and a real-world case study:

Simulation Studies:
- General Performance: In scenarios with linear and non-linear treatment effects across dimensions $p \in \{100, 1000, 5000\}$ , the Causal Horseshoe Forest achieves Root Mean Squared Error (RMSE) comparable to or better than AFT-BCF (Bayesian Causal Forest) and significantly outperforms AFT-BART and AFT-DART in high dimensions.
- Uncertainty Quantification: Unlike AFT-BCF, which produces overly wide credible intervals to maintain coverage, the Horseshoe Forest maintains stable coverage (near 90%) with significantly shorter, more precise credible intervals.
- Regularisation-Induced Confounding (RIC): In simulations designed to isolate weak confounders (covariates predictive of treatment but weakly associated with outcome), the Horseshoe Forest maintains stable RMSE and coverage. In contrast, methods relying on sparsity-inducing priors (like AFT-Shrinkage BCF) show increased bias and reduced coverage as weak confounders are added, demonstrating the advantage of continuous shrinkage over exact sparsity.
Case Study (Pancreatic Cancer):
- The method was applied to The Cancer Genome Atlas (TCGA) data for pancreatic ductal adenocarcinoma (PDAC) to estimate the effect of adjuvant radiotherapy on survival.
- The model identified a positive average treatment effect (ATE) with a posterior mean of approximately 0.75 (95% CI: 0.40 to 1.11).
- Analysis of individual CATEs revealed substantial uncertainty for most patients, with wide credible intervals including zero. However, the posterior means showed a consistent rightward shift, suggesting a generally beneficial but homogeneous effect rather than strong heterogeneity.
- A preliminary landmark analysis suggested that immortal time bias may have inflated the effect size in the main analysis, though the positive effect persisted.

Significance and Claims
The paper claims that the proposed method offers a robust alternative to existing high-dimensional causal inference tools for survival data. By retaining all covariates and applying adaptive continuous shrinkage, the Causal Horseshoe Forest mitigates the risk of omitted variable bias and regularisation-induced confounding, which are critical concerns in observational survival studies. The authors position their work as filling a gap in extending high-dimensional causal methods to survival outcomes, providing a tool that balances flexible non-linear modeling with the rigorous statistical requirements of causal inference. The availability of the ShrinkageTrees package facilitates the application of these methods to real-world high-dimensional biomedical data.