Modular Bayesian Hierarchical Shrinkage Models

Implements a two-stage Bayesian hierarchical modeling framework for applying shrinkage to subgroup-specific effects. The package separates model fitting (Stage 1) from hierarchical shrinkage (Stage 2), enabling modular sensitivity analyses without refitting expensive Markov chain Monte Carlo (MCMC) chains. Supports flexible prior specifications through the 'distributional' package, mixture approximations via 'mclust', and efficient 'Stan'-based inference.

shrinkr

Modular Bayesian Hierarchical Shrinkage Models

shrinkr provides a flexible framework for two-stage Bayesian hierarchical modeling (BHM). By separating model fitting from hierarchical shrinkage, it allows subgroup-specific estimates from virtually any Bayesian model to be incorporated into a common borrowing framework.

Table of Contents

• Key Idea
• Installation
• Quick Start
• Key Features
• Why Use shrinkr?
• Performance Considerations
• Documentation
• Mathematical Foundation
• Citation
• Getting Help

Key Idea

1. Stage 1: Fit any Bayesian model (e.g., in Stan, NIMBLE, JAGS) without shrinkage using flat/uninformative priors on the parameters of interest.
   Extract posterior draws of subgroup-specific parameters ($\theta_g$, the treatment effect or estimand of interest for each subgroup $g = 1, \ldots G$).

2. Stage 2: Approximate those subgroup posteriors with a mixture of multivariate normals,
   and then fit a hierarchical model across groups to borrow strength ("shrink").

This modular approach lets you separate shrinkage from the first model—ideal for simulation pipelines, sensitivity checks, and Bayesian data borrowing.

When is this especially useful? The mixture approximation makes shrinkr particularly valuable in situations where the typical two-stage approach assuming a normal likelihood for Stage 1 posteriors is unlikely to hold—for example, small subgroups, rare events, skewed or multimodal posteriors, or complex survival models. By capturing non-normality through the Gaussian mixture, the Stage 2 shrinkage better reflects the actual uncertainty from Stage 1.

This approach is mathematically justified through marginalization—see Mathematical Foundation for the full derivation.

Installation

# On CRAN
install.packages("shrinkr") # installs current release


# Install from GSK-Biostatistics repository
# install.packages("devtools")
devtools::install_github("GSK-Biostatistics/shrinkr") # installs developmental version

# Install from internal repository or local source
# install.packages("remotes")
remotes::install_local("path/to/shrinkr")

# Or install dependencies first
install.packages(c("rstan", "distributional", "mclust", "posterior"))

Note: This package requires a working installation of rstan. See the RStan Getting Started guide for platform-specific installation instructions.

Troubleshooting: If you encounter C++ compilation issues with rstan:

• Windows: Install Rtools
• Mac: Ensure Xcode command line tools are installed (xcode-select --install)
• Linux: Install r-base-dev package

Quick Start

library(shrinkr)
library(distributional)

# Step 1: Prepare posterior samples from your Stage 1 model
# Here we simulate posterior draws for 4 subgroups as an example.
# theta_g represents the subgroup-specific treatment effect for group g.
# In practice, these come from Stan, NIMBLE, JAGS, brms, etc.
# IMPORTANT: Use flat/uninformative priors on theta in Stage 1
set.seed(1104)
samples <- list(
  group1 = matrix(rnorm(2000, mean = 0.0, sd = 0.5), ncol = 1),
  group2 = matrix(rnorm(2000, mean = 0.5, sd = 0.5), ncol = 1),
  group3 = matrix(rnorm(2000, mean = 1.0, sd = 0.5), ncol = 1),
  group4 = matrix(rnorm(2000, mean = 0.3, sd = 0.6), ncol = 1)
)

# Step 2: Fit mixture approximation (up to K_max = 3 Gaussian components)
# K_max controls the maximum number of mixture components. mclust selects
# the best K (<= K_max) via BIC. Default is 5; 3 is often sufficient.
mix <- fit_mixture(samples, K_max = 3, verbose = TRUE)

# Step 3: Apply hierarchical shrinkage with custom priors
priors <- list(
  mu = dist_normal(0, 5),
  tau = dist_truncated(dist_student_t(3, 0, 1), lower = 0)
)

# Check what the priors imply about pairwise differences
prior_pred <- sample_prior_predictive(priors, n_groups = 4, n_draws = 2000)
pw <- prior_pairwise_differences(prior_pred)
plot(pw)  # density of |theta_i - theta_j|

fit <- shrink(
  mixture = mix,
  hierarchical_priors = priors,
  chains = 4,
  iter = 2000,
  warmup = 1000
)

# Step 4: Examine results
summary(fit)
plot(fit)

Key Features

Flexible Prior Specification

shrinkr supports a wide range of priors through the distributional package:

• Standard families: Normal, Student-t, Cauchy, Lognormal
• Heavy-tailed: Inverse-Gamma, Half-Cauchy, Half-t
• Uniform priors for bounded heterogeneity
• Truncated distributions for constraining parameter ranges on both mu and tau

# Examples of different tau priors
priors <- list(
  mu = dist_normal(0, 5),
  tau = dist_truncated(dist_normal(0, 2.5), lower = 0)  # Half-normal
)

# Or with half-t for heavier tails
priors <- list(
  mu = dist_normal(0, 5),
  tau = dist_truncated(dist_student_t(3, 0, 1), lower = 0)  # Half-t
)

# Truncated mu (e.g., constrain global mean to be positive)
priors <- list(
  mu = dist_truncated(dist_normal(0, 5), lower = 0),
  tau = dist_truncated(dist_normal(0, 2.5), lower = 0)
)

# Spike-and-slab prior on tau for testing homogeneity
priors <- list(
  mu = dist_normal(0, 5),
  tau = dist_truncated(prior_spike_slab(spike_prob = 0.5, spike_scale = 0.01, slab_scale = 1), lower = 0)
)

Two Input Methods

1. Full posterior samples (recommended): Provides mixture approximation that captures posterior uncertainty

fit <- shrink(mixture = mix, hierarchical_priors = priors)

1. Point estimates + variance: When full posteriors are unavailable

fit <- shrink(
  mle = c(0.5, 1.2, -0.3, 0.8),
  var_matrix = c(0.1, 0.15, 0.12, 0.09),  # Vector assumes independent components
  #can also be variance-covariance matrix
  hierarchical_priors = priors
)

Prior Predictive Checks

Before fitting, check the implications of your prior choices:

# Check prior implications
prior_pred <- sample_prior_predictive(
  hierarchical_priors = priors,
  n_groups = 4,
  n_draws = 1000
)

# Visualize hyperparameter and theta distributions
plot(prior_pred)

# Check implied spread
summary(prior_pred)

# implied_range: for each draw, this is max(theta) - min(theta) across groups.
# It tells you how spread out the subgroup effects are under your prior.
median(prior_pred$implied_range)  # Typical range across groups
quantile(prior_pred$implied_range, c(0.025, 0.975))  # 95% CI

# Prior predictive pairwise differences: |theta_i - theta_j|
# This is recommended for calibrating priors — see whether your prior
# implies plausible subgroup differences on the clinical scale.
pw <- prior_pairwise_differences(prior_pred)
print(pw)
plot(pw)                      # Pooled density
plot(pw, by_pair = TRUE)      # Violin per pair

Diagnostics and Visualization

# Check model fit
summary(fit)
fit$diagnostics

# Visualize shrinkage
plot(fit)

# Check mixture approximation quality
plot(mix, draws=samples, type="density")
plot(mix, draws=samples, type="qq")

# Extract specific parameters
extract_mu_tau(fit)
summarize_mu_tau(fit)

# Standard rstan diagnostics work directly on fit$fit:
# rstan::traceplot(fit$fit, pars = c("mu", "tau"))
# rstan::stan_rhat(fit$fit)
# bayesplot::mcmc_trace(fit$fit, pars = c("mu", "tau"))

Why Use shrinkr?

Modularity

• Fit complex Stage 1 models once, then explore different shrinkage approaches
• No need to refit expensive MCMC chains for sensitivity analyses
• Separates scientific model specification from statistical regularization

Flexibility

• Works with any Bayesian software (Stan, NIMBLE, JAGS, etc.)
• Supports diverse prior families and custom priors
• Handles both univariate and multivariate parameters

Efficiency

• Reuses posterior samples for multiple shrinkage scenarios
• Precompiled Stan models for fast fitting
• Mixture approximation enables efficient Stage 2 sampling

Use Cases

• Meta-analysis: Shrink study-specific effects with flexible heterogeneity priors
• Clinical trials: Borrow information across subgroups or historical controls
• Simulation studies: Compare shrinkage methods systematically
• Federated learning: Combine insights from distributed data sources

Performance Considerations

• Stage 1: Can be computationally expensive (full Bayesian model fitting)
• Stage 2: Fast (typically < 1 second for 4-5 groups with 2000 iterations)
• Mixture fitting: Time scales with number of groups and posterior samples
• Memory: Storing full posterior samples requires more memory than point estimates

Documentation

• vignette("getting_started"): Basic workflow and examples
• vignette("tidy_bayesian_workflow"): Tidy workflows with tidybayes, posterior, and bayesplot
• vignette("brms_integration"): Working with brms models (survival analysis example)
• vignette("federated_learning"): Using shrinkr in the context of federated learning
• vignette("map_prior_with_beastt"): Building meta-analytic-predictive (MAP) priors with shrinkr and beastt

Mathematical Foundation

The two-stage approach is mathematically justified through marginalization. The key insight is that the Stage 1 prior on θ must be uninformative (flat/improper) for the equivalence to hold.

Full One-Stage Model

In a standard hierarchical model, the joint posterior is:

$$\pi(\theta, \mu, \tau, \psi | D) \propto L(\theta, \psi | D) \cdot \pi(\theta | \mu, \tau) \cdot \pi(\mu) \cdot \pi(\tau) \cdot \pi(\psi)$$

where:

• $\theta$ = subgroup-specific parameters of interest
• $\mu, \tau$ = hierarchical mean and variance parameters
• $\psi$ = nuisance parameters from the likelihood
• $D$ = observed data

Marginalizing out $\psi$:

$$\pi(\theta, \mu, \tau | D) \propto \left[\int L(\theta, \psi | D) \pi(\psi) d\psi \right] \cdot \pi(\theta | \mu, \tau) \cdot \pi(\mu) \cdot \pi(\tau)$$

The Two-Stage Equivalence

Stage 1 uses a flat/uninformative prior on θ:

$$\pi_1(\theta | D) \propto \int L(\theta, \psi | D) \pi(\psi) d\psi$$

This is the posterior when $\pi(\theta) \propto 1$ (flat prior). Notice this matches the bracketed term above.

Stage 2 treats Stage 1 posterior as "data":

$$\pi(\theta, \mu, \tau | D) \propto \pi_1(\theta | D) \cdot \pi(\theta | \mu, \tau) \cdot \pi(\mu) \cdot \pi(\tau)$$

Why This Works

The equivalence holds because:

1. Stage 1 prior is flat: $\pi(\theta) \propto 1$, so it doesn't appear in the Stage 1 posterior
2. All shrinkage comes from Stage 2: The hierarchical prior $\pi(\theta | \mu, \tau)$ is applied in Stage 2 only
3. No "double priors": We don't have both $\pi_1(\theta)$ and $\pi(\theta | \mu, \tau)$ competing

Important Implementation Details

Always use flat/weakly informative priors in Stage 1:

# Stan: Don't specify a prior (defaults to improper uniform)
parameters {
  vector[n_groups] theta;  // No prior = flat prior
}

# JAGS: Use very wide prior
theta[g] ~ dnorm(0, 1e-6)  # Precision = 1e-6 means very wide

# NIMBLE: Similar to JAGS
theta[g] ~ dnorm(0, sd = 1000)

Apply hierarchical structure in Stage 2:

priors <- list(
  mu = dist_normal(0, 5),      # This is the ONLY prior on location
  tau = dist_truncated(dist_student_t(3, 0, 1), lower = 0)
)

fit <- shrink(mixture = mix, hierarchical_priors = priors)

The shrinkr package implements this by approximating $\pi(\theta | D)$ with a mixture of normals in Stage 2, enabling efficient exploration of different hierarchical priors without refitting the Stage 1 model.

Why a Mixture of Normals?

The Stage 1 posterior $\pi_1(\theta | D)$ can be arbitrarily complex — skewed, multimodal, heavy-tailed — depending on the model and data. We need a tractable density that Stan can evaluate efficiently in its log-likelihood. A mixture of multivariate normals is the natural choice because:

1. Universal approximation: Any continuous density can be approximated to arbitrary precision by a finite mixture of normals (given enough components). This is a classical result in density estimation.
2. Closed-form log-density: Each component contributes a multi_normal_cholesky_lpdf term that Stan evaluates analytically — no numerical integration needed.
3. Automatic model selection: mclust selects the number of components $K$ and covariance structure via BIC, so the user does not need to specify this manually. The K_max argument sets an upper bound (default 5); in practice, 2-5 components suffice for most posterior shapes.
4. Captures non-normality: Unlike the point-estimate-plus-variance approach (which implicitly assumes a normal posterior), the mixture path faithfully represents skewness, heavy tails, and multimodality from Stage 1.

Should I assess sensitivity to K_max? In general, the BIC-selected $K$ is stable once $K_{\text{max}}$ is large enough. You can check this by comparing fit_mixture(..., K_max = 3) vs K_max = 5 vs K_max = 8 and inspecting the QQ plots. If the selected $K$ and the QQ diagnostics are unchanged, the result is robust. The function fit_mixture() returns the selected $K$ and BIC for comparison.

Citation

If you use shrinkr in your work, please cite:

Maronge, J. M. (2026). shrinkr: Modular Bayesian Hierarchical Shrinkage Models. 
R package version 0.4.5.

Getting Help

For questions and issues:

• Open an issue on the internal repository.
• Contact the package maintainer: jacob.m.maronge@gsk.com.

License

This package is licensed under the GNU General Public License v3.0 or later (GPL-3.0-or-later). You are free to use, modify, and redistribute it under the terms of the GPL. For the full text visit https://www.gnu.org/licenses/gpl-3.0.html.

Portions of this package are derived from templates provided by the rstantools project (Copyright © Trustees of Columbia University), also licensed under GPL-3.