Chapter 1 Choosing the Right Bayesian Model

Introduction

Bayesian modeling has become a central approach in modern data analysis, providing a coherent framework for incorporating prior knowledge and quantifying uncertainty. With the advent of powerful tools such as Stan and user-friendly interfaces like the R package brms, practitioners can now implement a wide array of Bayesian models with relative ease. However, the flexibility of the Bayesian framework also introduces a new challenge: selecting the most appropriate model for a given dataset and research question. The landscape of Bayesian models is vast, encompassing linear and generalized linear models, robust and regularized regressions, hierarchical models, and more sophisticated approaches such as Gaussian processes and mixture models. This guide aims to offer a structured approach to model selection within the Bayesian paradigm, focusing on practical considerations, data characteristics, and modeling objectives.

Modeling Objectives and Data Characteristics

The choice of a Bayesian model should begin with a clear understanding of the research objective. In broad terms, the aim of modeling can be categorized into two primary goals: inference and prediction. Inference focuses on understanding the relationships between variables, quantifying uncertainty in parameter estimates, and testing theoretical hypotheses. Prediction, on the other hand, emphasizes the accuracy of forecasting outcomes for new observations. While the two goals are not mutually exclusive, they can lead to different modeling choices, particularly in terms of model complexity and regularization. Another factor influencing model selection is the nature of the data. Key aspects include the type of response variable (continuous, binary, count, categorical), the presence of outliers or heavy-tailed distributions, the structure of the data (e.g., hierarchical or longitudinal), and the dimensionality of the predictor space. A careful examination of these characteristics provides essential guidance for selecting an appropriate Bayesian model.

Prior Specification and Computational Considerations

An essential feature of Bayesian modeling is the specification of prior distributions. Priors can be informative, weakly informative, or non-informative, depending on the amount of domain knowledge available. Informative priors are grounded in expert knowledge or historical data, while weakly informative priors help stabilize estimates without unduly influencing the posterior. Prior predictive checks can assess the implications of the priors before seeing the data, ensuring they encode plausible assumptions. Modelers should also perform sensitivity analyses to understand how different priors affect inferences.

Computational feasibility is another practical concern. Some Bayesian models—especially nonparametric or high-dimensional ones—can be computationally intensive, requiring advanced MCMC algorithms or variational inference. Diagnostics such as the Gelman-Rubin R-hat statistic, effective sample size (ESS), and checks for divergent transitions should be used to ensure reliable inference (Gelman et al., 2013). Stan and brms provide tools to assess convergence and evaluate sampling efficiency.

Bayesian Linear Regression

Bayesian linear regression serves as the foundational model in the Bayesian framework. It assumes a linear relationship between predictors and a continuous response variable, with normally distributed residuals. This model is particularly useful for its simplicity and interpretability. When the assumptions of linearity and normality hold reasonably well, Bayesian linear regression provides reliable parameter estimates and predictive intervals. It also serves as a baseline model against which more complex models can be compared. In practice, Bayesian linear regression can be implemented in Stan with straightforward model code, specifying priors for the regression coefficients and residual variance. The flexibility of Bayesian inference allows for the incorporation of prior knowledge, which can be particularly valuable in small-sample contexts or when strong domain expertise is available.

Robust Regression for Non-Normal Residuals

Real-world data often deviate from the assumption of normally distributed residuals. Outliers or heavy-tailed distributions can exert undue influence on parameter estimates, leading to biased or unstable results. Bayesian robust regression addresses this issue by modeling the residuals using a t-distribution, which has heavier tails than the normal distribution. This approach reduces the influence of outliers, leading to more robust and reliable inferences. The implementation of robust regression in Stan involves specifying a likelihood based on the t-distribution and including an additional parameter for the degrees of freedom. This parameter controls the heaviness of the tails and can itself be estimated from the data. The robust regression model is particularly recommended when residual diagnostics from a standard linear model indicate non-normality or the presence of extreme observations.

Regularized Regression for High-Dimensional Data

When dealing with a large number of predictors or multicollinearity, regularization becomes essential to prevent overfitting and to enhance predictive performance. Bayesian regularized regression models incorporate shrinkage priors, such as the Laplace prior for Bayesian LASSO or the Gaussian prior for Bayesian ridge regression. These priors shrink the regression coefficients toward zero, effectively performing variable selection and regularization. In the Bayesian framework, regularization is naturally integrated through the prior distribution. For example, the Bayesian LASSO uses a double-exponential prior that induces sparsity by assigning higher probability mass near zero. These models are particularly useful in settings with more predictors than observations or when there is a need to identify the most influential variables.

Models for Non-Normal Data

In many applications, the response variable does not follow a normal distribution. Binary outcomes, count data, and categorical responses require specialized models. Bayesian generalized linear models (GLMs) extend the linear model framework to accommodate different types of response variables through appropriate link functions and likelihood distributions. For binary outcomes, the logistic regression model with a logit link is commonly used. For count data, Poisson and negative binomial models are appropriate, with the latter providing a flexible alternative in the presence of overdispersion. Multinomial and ordinal regression models are used for categorical outcomes, with the choice depending on whether the categories are ordered. These models are readily implemented in Stan and brms, allowing users to specify the appropriate family and link function. Model selection in this context should be guided by the distributional characteristics of the response variable and the research question at hand.

Multilevel and Hierarchical Models

Hierarchical data structures are common in social sciences, education, and biomedical research. In such settings, observations are nested within higher-level units, such as students within schools or patients within hospitals. Ignoring this structure can lead to biased inferences and underestimated uncertainty.

Bayesian multilevel models explicitly account for the hierarchical structure by including group-level effects. These models allow for partial pooling of information across groups, balancing between complete pooling (ignoring group differences) and no pooling (treating each group separately). The brms package offers a user-friendly interface for fitting multilevel models, handling complex random effects structures with ease. The flexibility of Bayesian multilevel modeling also facilitates the inclusion of varying slopes, cross-level interactions, and non-linear effects. When the data structure suggests hierarchical dependencies, multilevel modeling should be the default approach.

Nonlinear and Nonparametric Models

In some applications, the relationship between predictors and the response variable is inherently nonlinear or unknown. Bayesian nonparametric models, such as Gaussian process regression, offer a flexible solution by modeling the function space directly. Gaussian processes define a prior over functions and use observed data to update this prior, resulting in a posterior distribution over functions. Gaussian process regression is particularly powerful when the form of the relationship is unknown or when modeling smooth, nonlinear trends is important. However, it comes at a higher computational cost and may not scale well with large datasets. Nevertheless, for problems involving spatial data, temporal trends, or complex functional relationships, Gaussian processes provide a valuable modeling tool.

Mixture Models and Latent Structure

Data arising from heterogeneous populations may be better modeled using mixture models. Bayesian Gaussian mixture models, for instance, assume that the data are generated from a mixture of several Gaussian distributions, each representing a subpopulation. These models can uncover latent structure in the data, such as clusters or subtypes.

Mixture models introduce additional complexity due to the need to estimate both the component parameters and the mixing proportions. Bayesian inference provides a principled framework for dealing with this uncertainty, often using techniques such as latent variable augmentation and label switching adjustments.

When there is reason to believe that the data comprise distinct subgroups with different underlying characteristics, mixture models offer an effective approach to modeling such heterogeneity.

Comparative Summary Table

Model Type Use Case Key Assumptions Priors Limitations
Linear Regression Continuous outcome, low noise Linearity, normal errors Normal, Inverse-Gamma Poor with outliers
Robust Regression Heavy-tailed residuals t-distributed residuals Prior on ν Increased complexity
Regularized Regression High-dimensional predictors Sparsity Laplace, Gaussian Shrinkage may hide effects
GLMs Binary/count/categorical outcomes Appropriate link function Varied Can overfit without strong priors
Hierarchical Models Nested/grouped data Partial pooling Hierarchical priors Sensitive to group size
Gaussian Processes Unknown nonlinear function Smoothness in kernel GP prior Poor scaling (O(n³))
Mixture Models Latent structure/clustering Finite components Dirichlet, etc. Label switching, identifiability

Model Diagnostics and Comparison

Choosing the right model also involves evaluating its performance and comparing it to alternative specifications. Bayesian model diagnostics include posterior predictive checks, which assess how well the model reproduces the observed data. Graphical comparisons between observed and replicated data can reveal model misfit or systematic discrepancies.

Information criteria such as the Widely Applicable Information Criterion (WAIC) and Leave-One-Out Cross-Validation (LOO-CV) provide tools for model comparison, balancing fit and complexity. These criteria estimate the expected out-of-sample predictive performance and are particularly useful for selecting among nested or non-nested models (Vehtari, Gelman, & Gabry, 2017).

Bayes factors offer another method for model comparison, based on the ratio of marginal likelihoods. However, they are sensitive to prior specification and can be computationally intensive. In practice, WAIC and LOO-CV are often preferred for their robustness and ease of computation.

A Decision Framework for Model Selection

To aid practitioners in selecting the appropriate Bayesian model, a structured decision framework can be employed. This framework begins with identifying the type of response variable: continuous, binary, count, or categorical. Next, the data should be assessed for features such as outliers, overdispersion, hierarchical structure, and nonlinearity. Based on these characteristics, the modeler can then choose among linear models, robust regressions, generalized linear models, multilevel models, or nonparametric approaches.

This decision process is iterative and should incorporate model diagnostics and domain knowledge. Starting with a simple model and progressively introducing complexity allows for a more transparent understanding of the data and the modeling assumptions. Each modeling choice should be justified in terms of its contribution to answering the research question and improving model fit.

Conclusion

Bayesian modeling offers unparalleled flexibility and rigor in statistical inference, but this power comes with the responsibility of thoughtful model selection. This guide has outlined the key considerations for choosing among the diverse array of Bayesian models available in tools like Stan and brms. By grounding model selection in the objectives of the analysis, the characteristics of the data, and robust diagnostic procedures, practitioners can make informed choices that enhance both the interpretability and predictive performance of their models. As with all statistical modeling, the process is iterative and benefits from a combination of statistical insight, computational tools, and substantive expertise. With this guide, researchers are better equipped to navigate the Bayesian modeling landscape and apply the appropriate models to their specific challenges.