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Bayesian structured additive quantile regression for inflated bounded data

This paper proposes a Bayesian structured additive quantile regression framework for inflated bounded data that simultaneously models the conditional quantiles of the continuous component and the probabilities of boundary values using flexible predictors, with inference performed via the Liesel software and validated through simulations and real-world applications.

Original authors: Francisco F. Queiroz, Johannes Brachem, Paul F. V. Wiemann, Thomas Kneib

Published 2026-03-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francisco F. Queiroz, Johannes Brachem, Paul F. V. Wiemann, Thomas Kneib

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 trying to predict how much rain will fall in a city. Most days, it rains a little bit or a lot, and you can measure that on a scale. But sometimes, it doesn't rain at all (0 inches), and on rare occasions, the rain gauge breaks because it's overflowing (1 inch, or the maximum).

Standard weather models often struggle with these "perfect zeros" and "perfect ones." They try to force everything into a smooth curve, which doesn't make sense when you have days with absolutely no rain or days that are just "too wet to measure."

This paper introduces a new, super-flexible tool called Bayesian Structured Additive Quantile Regression to solve exactly this kind of problem. Here is how it works, broken down into simple concepts:

1. The "Two-Part" Strategy: The Gatekeeper and the Meter

Instead of trying to fit one giant, messy equation to the whole picture, the authors split the problem into two separate jobs, like a bouncer at a club and a bartender inside.

The Gatekeeper (The Discrete Part): This part answers the question: "Will it rain at all, or will it be a total flood?" It calculates the probability of getting a 0 (no rain) or a 1 (max rain). It decides if the day is a "boundary" day.
The Meter (The Continuous Part): If the day isn't a 0 or a 1, this part measures the actual amount of rain between 0 and 1. It focuses on the "in-between" days.

By separating these two, the model doesn't get confused. It knows that "no rain" is a different kind of event than "a little bit of rain."

2. The "Quantile" Lens: Not Just the Average

Most models only tell you the average (the mean). But averages can be misleading.

Example: If you have 9 days with 1 inch of rain and 1 day with 100 inches, the average is 11 inches. But that doesn't tell you that 90% of the time, it's just a drizzle.

This paper uses Quantile Regression. Instead of asking "What is the average?", it asks:

"What is the rain level for the driest 10% of days?" (The 10th percentile)
"What is the rain level for the wettest 10% of days?" (The 90th percentile)
"What is the median (middle) day?"

This gives you a full picture of the weather, not just a single number. It helps you understand the extremes, not just the middle.

3. The "Structured Additive" Magic: The Swiss Army Knife

The "Structured Additive" part is the tool's superpower. It allows the model to handle complex real-world messiness that simple math can't catch. Think of it as a Swiss Army knife that can switch blades depending on the job:

Non-linear effects: Maybe rain doesn't increase steadily with temperature. Maybe it's low at 20°C, spikes at 30°C, and drops at 40°C. This model can draw that curved line automatically.
Spatial effects: It knows that rain in São Paulo might be different from rain in Rio, even if the temperature is the same. It can map out geographic patterns.
Random effects: If you are testing the same person multiple times (like in the hearing test example), it knows that Person A is naturally different from Person B, and it accounts for that without messing up the overall trend.

4. Real-World Examples from the Paper

The authors tested their new tool on two very different problems:

A. Traffic Fatalities in Brazil (The "Zero" Problem)
They looked at traffic death rates in thousands of Brazilian towns.

The Problem: Many towns had zero deaths in a year. A standard model would get confused by all these zeros.
The Insight: The model found that while bigger cities have more total accidents, the proportion of deaths relative to the population is actually lower in big cities (maybe because they have better roads and ambulances).
The Twist: The model also showed that in some northern states, if a death did happen, it was more likely to be part of a larger tragedy (higher conditional quantiles), whereas in the south, deaths were more evenly spread out. The "average" model would have missed these regional nuances.

B. Hearing Tests for Cochlear Implants (The "One" Problem)
They analyzed how well people with hearing implants understood speech.

The Problem: People either understood everything (100%) or nothing (0%). Very few got it "half right."
The Insight: The model helped researchers see that under difficult noise conditions, one specific algorithm helped people avoid total failure (reducing the "zeros"). Under good conditions, a different algorithm helped them achieve perfect scores (increasing the "ones").
The Benefit: By looking at the "middle" (the people who got some words right), they could see that the algorithms behaved differently for different types of sentences, something a simple average would have smoothed over.

Summary: Why This Matters

Think of this new model as a high-definition camera for data.

Old models were like black-and-white sketches: they gave you a rough idea of the "average" shape.
This new model is a 4K video: it captures the zeros, the ones, the curves, the geography, and the extremes.

It allows scientists to ask deeper questions: "How does this factor affect the worst-case scenario?" or "How does this factor affect the best-case scenario?" rather than just "What is the average effect?" This is crucial for making better decisions in medicine, public policy, and engineering.

1. Problem Statement

The paper addresses the statistical challenge of modeling bounded continuous data on the unit interval $[0, 1]$ that frequently exhibit inflation at the boundaries (i.e., a non-negligible proportion of exact zeros and/or ones).

Context: Such data are common in fields like medicine, ecology, and sociology (e.g., mortality rates, proportion of land cover, speech intelligibility scores).
Limitations of Existing Methods:
- Standard Regression: Traditional mean-based models (like Beta regression) often fail to capture heterogeneity, asymmetry, and tail behavior.
- Two-Part Models: While existing "two-part" models separate the discrete (boundary) and continuous components, they typically rely on simple linear predictors and parametric distributional assumptions (e.g., Beta, Unit-Weibull).
- Quantile Regression Gaps: Previous quantile regression approaches for inflated data were limited to simple linear specifications and often could not handle simultaneous zero- and one-inflation. They lacked the flexibility to model complex effects like spatial dependencies, random effects, or non-linear covariate relationships.

2. Methodology

The authors propose a Bayesian Structured Additive Quantile Regression framework for zero- and/or one-inflated bounded data.

A. Model Formulation (Two-Part Framework)

The response variable $y_i \in [0, 1]$ is modeled as a mixture of a discrete component (boundary values) and a continuous component (values in $(0, 1)$ ).

Discrete Component: Modeled via a multinomial (or Bernoulli for single inflation) regression to estimate probabilities $p_{0i} = P(y_i=0)$ $p_{0 i} = P (y_{i} = 0)$ and $p_{1i} = P(y_i=1)$ $p_{1 i} = P (y_{i} = 1)$ .
- Predictors: Structured additive predictors $\eta_{0i}$ and $\eta_{1i}$ using link functions (e.g., logit) to model the log-odds of inflation.
Continuous Component: Models the conditional quantiles of the transformed response $h(y_i)$ $h (y_{i})$ for $y_i \in (0, 1)$ $y_{i} \in (0, 1)$ .
- Link Function: A strictly increasing link $h(\cdot)$ (e.g., logit) maps $(0, 1)$ to $\mathbb{R}$ .
- Quantile Definition: The $\tau$ -th conditional quantile is modeled as $Q_{h(y_i)|y_i \in (0,1)}(\tau | x_i) = \eta_{2i, \tau}$ .

B. Structured Additive Predictors

The core innovation is the use of structured additive predictors for both parts of the model:
$\eta = \beta_0 + \sum f_k(x_k)$
These predictors can include:

Linear effects: Standard parametric coefficients.
Non-linear smooth effects: Modeled via penalized splines (e.g., cubic splines).
Spatial effects: Modeled via Gaussian Markov Random Fields (GMRF) to capture geographic adjacency.
Random effects: To account for hierarchical data structures (e.g., subject-specific variability).
Varying-coefficient terms: Where the effect of one covariate changes based on another.

C. Bayesian Inference & Computation

Likelihood Approximation: To enable Bayesian inference for quantile regression, the authors utilize the Asymmetric Laplace Distribution (ALD) as a working likelihood for the continuous component. The ALD's location parameter corresponds to the desired quantile.
Location-Scale Mixture: The ALD is represented as a scale mixture of normals involving an exponential latent variable ( $w_i$ ). This transforms the problem into a conditionally Gaussian regression, facilitating efficient sampling.
Priors:
- Coefficients for smooth/spatial/random effects follow multivariate normal priors with rank-deficient penalty matrices (inducing smoothing).
- Smoothing variances follow Inverse Gamma hyperpriors.
Sampling Algorithm: Posterior inference is performed using Markov Chain Monte Carlo (MCMC) implemented in Liesel, a probabilistic programming framework.
- Metropolis-Hastings: Used for regression coefficients with iteratively weighted least squares proposals.
- Gibbs Sampling: Used for smoothing variances and latent variables ( $w_i$ ).
- NUTS (No-U-Turn Sampler): Used for other parameters on the log-level.

3. Key Contributions

Flexibility in Inflation: The model handles simultaneous zero- and one-inflation, a capability missing in previous quantile regression literature for bounded data.
Structured Additivity: It extends quantile regression beyond simple linear models, allowing for non-linear, spatial, and random effects in both the discrete (inflation probability) and continuous (quantile) parts.
Comprehensive Distributional Analysis: Unlike mean-based models, this approach captures the full conditional distribution, revealing how covariates affect different parts of the distribution (e.g., tails vs. median) differently.
Software Implementation: The methodology is implemented in Liesel, providing a robust, open-source tool for semiparametric regression with complex structures.

4. Results

Simulation Studies

Setup: Data were generated from zero- and one-inflated models with non-linear covariate effects. Scenarios included correct specification (ALD) and misspecification (GJS-t distribution).
Findings:
- The model accurately estimated parameters for both discrete and continuous components.
- Robustness: Even under likelihood misspecification (using GJS-t data), the model provided low Root Mean Squared Error (RMSE) for quantile predictions.
- Coverage: 95% credible intervals achieved nominal coverage rates, particularly for larger sample sizes and moderate inflation probabilities.

Real-Data Applications

Traffic Fatalities in Brazil (Zero-Inflated):
- Data: Proportion of traffic deaths across 5,499 municipal districts.
- Findings: The model revealed non-linear and spatial patterns that linear models missed.
  - Population Size: Larger municipalities had lower traffic death proportions, but this effect was stronger at higher quantiles (tail behavior).
  - Youth Population: Positive association with traffic deaths, particularly in the upper quantiles.
  - Spatial Effects: Significant regional disparities (e.g., higher mortality in Northern states like Acre and Roraima).
- Comparison: The structured additive quantile model outperformed zero-inflated Beta regression in capturing heterogeneity, especially in the tails of the distribution.
Speech Intelligibility (Zero- and One-Inflated):
- Data: Proportion of correctly identified morphemes by cochlear implant recipients under different sound-processing algorithms.
- Findings:
  - Random Effects: Successfully captured subject-specific variability, which was significant for the probability of extreme outcomes (0 or 1) but less so for intermediate performance.
  - Algorithm Performance: The effect of sound-processing algorithms varied by Signal-to-Noise Ratio (SNR) and quantile level. For instance, Algorithm B reduced complete failures (zeros) at low SNR, while Algorithm A performed better for perfect recognition (ones) at high SNR.
  - Quantile Insights: The impact of SNR was stronger at lower to median quantiles than at the upper quantile, a nuance missed by mean-based models.

5. Significance

This paper provides a unified, flexible framework for analyzing bounded data with boundary inflation. Its significance lies in:

Methodological Advancement: Bridging the gap between quantile regression (distributional flexibility) and structured additive models (complex covariate effects).
Practical Utility: Offering a tool that can handle real-world complexities like spatial correlation and hierarchical data structures without relying on restrictive parametric assumptions.
Insight Generation: Demonstrating that covariate effects often vary significantly across the distribution (e.g., a variable might affect the median but not the tails, or vice versa), providing deeper insights for policy and scientific understanding than traditional mean-based approaches.
Extensibility: The framework is designed to be extendable to other classes of inflated models and unbounded positive data (e.g., zero-inflated count or continuous data on $[0, \infty)$ ).