Bayesian Quantile Regression Using Stan

Exploring robust quantile estimation with Bayesian methods and Stan

bayesian regression graph on a computer screen

Key Highlights

Flexible Model Specification: Stan allows specification of custom Bayesian quantile regression models that estimate any conditional quantile function.
Use of Asymmetric Laplace Distribution: The asymmetric Laplace distribution is central to modeling error terms efficiently in quantile regression.
Practical Implementation via brms: The brms package in R provides a user-friendly interface to implement these models, despite handling one quantile at a time.

Introduction to Bayesian Quantile Regression

Bayesian quantile regression (BQR) presents a versatile framework for exploring the conditional distribution of a response variable beyond its mean. Rather than limiting analysis to the conditional mean—a characteristic of ordinary least squares regression—BQR allows for the estimation of any conditional quantile. This methodology is particularly useful when the data reveal non-normal behaviors or contain outliers that can heavily influence the mean estimate.

In the Bayesian context, prior information can be incorporated to yield robust estimates and uncertainty measures for the regression quantiles. This blend of traditional quantile regression with Bayesian inference empowers analysts to quantify the uncertainty associated with parameters and predictions, leading to more informed decision-making.

Fundamentals of Bayesian Quantile Regression with Stan

Overview of the Methodology

The Bayesian approach treats the parameters of the regression model as random variables with associated probability distributions, rather than fixed unknowns. This gives rise to a posterior distribution that reflects the uncertainty in the model’s parameters after observing the data. The central steps in implementing Bayesian quantile regression using Stan include:

Defining the Quantile Function

In quantile regression, the goal is to estimate the conditional quantile \( \text{\(\theta_p(y|x)\)} \) where \( p \) represents the quantile of interest (e.g., median when \( p = 0.5 \), or other percentiles for broader distributional insights). The regression function is typically specified as:

\( \text{\(\theta_p(y|x) = x^T \beta_p\)} \)

where \( \beta_p \) is the parameter vector corresponding to the \( p \)th quantile.

Modeling with the Asymmetric Laplace Distribution

A typical approach in Bayesian quantile regression is through the use of the asymmetric Laplace distribution (ALD). The ALD is particularly suited for quantile estimation due to its inherent properties that align with the loss function of quantile regression. In the Bayesian framework, the ALD becomes the likelihood function for the model:

\( \text{\( y_i \sim \text{ALD}(\mu_i, \sigma, p) \)} \)

Here, \( \mu_i \) represents the location parameter (often taken as a linear function of the predictors), \( \sigma \) is the scale parameter, and \( p \) specifies the quantile to be estimated.

Bayesian Framework and Prior Specification

In Bayesian quantile regression, the parameters \( \beta_p \), \( \sigma \), and possibly other auxiliary variables are assigned prior distributions. This makes it possible to incorporate previous information about these parameters. For example, a common choice is using normal priors for the regression coefficients:

\( \text{\( \beta_p \sim \text{Normal}(0, \lambda) \)} \)

where \( \lambda \) is a hyperparameter controlling the spread of the prior. Similarly, scale parameters like \( \sigma \) may be modeled using half-Cauchy or other shrinkage priors to ensure stability and prevent overfitting.

Implementing Bayesian Quantile Regression in Stan

Stan Model Specification

Implementing Bayesian quantile regression using Stan involves direct specification of the model in Stan’s modeling language. A typical Stan model includes data declarations, parameter definitions, and model blocks where the likelihood is specified and prior distributions are assigned.

Data Block and Parameters

The data block in a Stan model defines the dataset that will be used, including the number of observations, predictor variables, and the quantile \( p \) of interest. The parameter block declares the regression coefficients and scale parameters. An illustration of a simplified Stan model is shown below:


// Data declaration
data {
    int<lower=0> N;           // Number of observations
    real<lower=0, upper=1> p;  // Quantile to be estimated
    vector[N] x;             // Predictor variable (or matrix for multiple predictors)
    vector[N] y;             // Response variable
}

// Parameter declaration
parameters {
    real alpha;              // Intercept
    real beta;               // Slope coefficient
    real<lower=0> sigma;     // Scale parameter
    vector[N] w;             // Auxiliary variable for data augmentation
}

// Transformed parameters or model construction can include auxiliary measures
// depending on the method utilized (e.g., data augmentation techniques)

// Model block containing priors and likelihood specification
model {
    alpha ~ normal(0, 100);  // Non-informative prior for intercept
    beta  ~ normal(0, 100);  // Non-informative prior for slope
    sigma ~ cauchy(0, 2.5);   // Prior for scale parameter

    // Data augmentation for the Asymmetric Laplace Approximation if needed
    w ~ exponential(1.0 / sigma); // Example of data augmentation
    
    // Using data augmentation to match the likelihood required
    // y is modeled as having a distribution derived from the augmented model
    y ~ normal(alpha + beta * x + (1 - 2 * p) / (p * (1 - p)) * w, sqrt((p * (1 - p) * sigma) / (2 * w)));
}

In this example, an auxiliary variable w is introduced as part of a data augmentation strategy that can help formulate the likelihood compatible with the asymmetric Laplace formulation.

Using the brms Package for Simplified Implementation

While direct Stan coding provides complete flexibility, using the R package brms can streamline the process by interfacing with Stan directly. The brms package supports various families, including models with an asym_laplace() distribution.

Implementing a brms Model

With brms, fitting a Bayesian quantile regression model is as straightforward as specifying the model formula, data, and the family function. For instance, the following R code fits a median regression model (with quantile \( p = 0.5 \)):


# R code using the brms package
library(brms)

# Define your model formula, e.g., y ~ x
formula <- y ~ x

# Fit the Bayesian quantile regression model for the median (p = 0.5)
fit <- brm(
  formula = formula,
  data = df, 
  family = asym_laplace(),  # Use the asymmetric laplace distribution family
  quantile = 0.5            # Set the desired quantile
)

# Examine the model summary
print(summary(fit))

Note that the asym_laplace() family in brms is designed to handle a single quantile at a time. If your analysis requires multiple quantiles, separate models must be executed for each quantile of interest.

Challenges and Considerations

Limitations When Using the Asymmetric Laplace Distribution

Although the asymmetric Laplace distribution simplifies the specification of the likelihood in quantile regression, it does come with certain limitations:

Single Quantile Focus

One of the notable drawbacks is that the approach inherently focuses on one quantile at a time. Consequently, to analyze multiple quantiles (e.g., 10th, 50th, 90th percentiles), it requires separate model fittings, which can be computationally taxing and less efficient compared to frequentist approaches that can estimate several quantiles simultaneously.

Model Complexity and Data Augmentation

Implementing Bayesian quantile regression with data augmentation strategies to emulate the asymmetric Laplace likelihood may introduce additional complexity in model specification. The selection of appropriate priors and ensuring convergence of the model via Markov Chain Monte Carlo (MCMC) sampling are critical. Sensitivity to outliers can be reduced using quantile-based methods, but successful execution requires careful consideration of the model’s structure.

Practical Advice for Researchers and Practitioners

Practical implementation of Bayesian quantile regression involves several steps which include:

Thoroughly assessing the data quality and distribution to justify the choice of quantile regression over mean regression.
Deciding on the appropriate quantile(s) for analysis dependent on the research question.
Starting with simpler models, such as those using the brms package, to avoid modeling mis-specification before progressing to more advanced custom Stan models.
Running separate models for different quantiles if your study demands insights across various parts of the distribution. Although slightly repetitive, this procedure ensures clarity in estimation and uncertainty quantification.
Utilizing the flexibility of Stan to incorporate informative priors when there is adequate prior knowledge about the regression coefficients or scale parameters.

These considerations play a pivotal role in ensuring that the Bayesian quantile regression model is both robust and interpretable.

Comparative Analysis: Bayesian vs. Frequentist Quantile Regression

Why Choose Bayesian Methods?

Compared to the frequentist approach to quantile regression, the Bayesian framework offers several distinct advantages:

Incorporation of Prior Knowledge

One of the main benefits is the ability to incorporate prior beliefs about the parameters of interest. This is particularly beneficial when data is limited or when there is strong, domain-specific knowledge about the expected relationships. Through the use of informative priors, the Bayesian model can yield more stabilized estimates.

Uncertainty Quantification

Bayesian methods produce full posterior distributions for model parameters rather than point estimates alone. This comprehensive uncertainty quantification is instrumental in making probabilistic statements about the parameters and in assessing the reliability of predictions.

Model Flexibility

Stan’s modeling language allows for complex customization and the incorporation of hierarchical structures and random effects. This flexibility can be particularly beneficial when dealing with multi-level datasets or when a more nuanced understanding of the variability in the data is desired.

Comprehensive Summary Table

Aspect	Description
Objective	Estimate conditional quantiles rather than the conditional mean.
Key Distribution	Asymmetric Laplace Distribution (ALD) used to model the error distribution.
Implementation Tools	Stan for model specification via MCMC and brms for an easy R interface.
Bayesian Advantages	Incorporation of prior information, full uncertainty quantification, and model flexibility.
Challenges	Handling only one quantile per model and increased model complexity, often requiring data augmentation.
Application Areas	Robust regression analysis particularly when data is not normally distributed or contains outliers.

Additional Considerations and Future Directions

Advanced Modeling Techniques

As research in Bayesian quantile regression continues to evolve, there are innovative approaches that aim to overcome the limitations associated with the single-quantile framework. Ongoing academic work is exploring methods to handle multiple quantiles within a single model by extending the traditional asymmetric Laplace methodology. Some of these methods incorporate hierarchical modeling techniques and more complex data augmentation schemes, offering the promise of simultaneous multi-quantile estimation.

Multi-Quantile Estimation

Researchers are developing models that attempt to estimate several quantiles concurrently, thereby addressing computational inefficiencies. Such approaches are designed to achieve coherent estimates across various points in the distribution while preserving the inherent uncertainty of each estimate. These advancements could prove highly valuable in fields such as economics, environmental science, and biomedical statistics where understanding the full distribution is critical.

Integration with Modern Computational Tools

The integration of Stan with modern computational libraries and packages like brms represents a significant step forward. The ease of interfacing Stan with R and Python has democratized access to sophisticated Bayesian techniques. Continued improvements in these tools are making it simpler to incorporate complex model structures and perform rigorous diagnostic checks on model convergence and posterior estimates.

Future directions also include the adoption of methods that blend machine learning techniques with Bayesian statistics, providing a pathway to even more flexible and robust predictive modeling. These combined approaches are expected to yield greater insights while handling large datasets efficiently.

References

An rstan implementation of Bayesian Quantile Regression - GitHub
Ruler - Quantile Regressions in Stan: Part I - Spinkney
Bayesian Quantile Regression - The Stan Forums
Quantile Regression in Stan - Kamran Afzali's Blog
brms: Bayesian Regression Models using Stan - Paul Bürkner