Conditional Power Prior

2026-10-03

Posterior distribution

With pooling, the prior distribution of $\theta_T$ is given by:

$$p(\theta_T) = \mathcal{N}(\theta_T | \hat{\theta}_S, \sigma^2_S/N_S)$$

We observe $N_T$ data points with sample mean $\hat{\theta}_T$ and known variance $\sigma^2_T$. This implies the likelihood is given by:

$$p(\hat{\theta}_T | \theta_T) = \mathcal{N}(\hat{\theta}_T | \theta_T, \sigma^2_T/N_T)$$

Since both the prior and the likelihood are Gaussian distributions, the posterior will also be a Gaussian distribution.

The posterior distribution $p(\theta_T | \hat{\theta}_T)$ is :

$$\theta_T | \hat{\theta}_T \sim \mathcal{N}(\mu_{\text{post}}, \sigma^2_{\text{post}})$$

Posterior Mean:

$$\mu_{\text{post}} = \sigma_{\text{post}}^2 \left( \frac{\hat{\theta}_S N_S}{\sigma^2_S} + \frac{\hat{\theta}_T N_T}{\sigma^2_T} \right)$$

Posterior Variance:

$$\sigma_{\text{post}}^2 = \left( \frac{N_S}{\sigma^2_S} + \frac{N_T}{\sigma^2_T} \right)^{-1}$$

Thus, the posterior distribution $p(\theta_T | \hat{\theta}_T)$ is:

$$\theta_T | \hat{\theta}_T \sim N\left( \left( \frac{N_S}{\sigma^2_S} + \frac{N_T}{\sigma^2_T} \right)^{-1} \left( \frac{\hat{\theta}_S N_S}{\sigma^2_S} + \frac{\hat{\theta}_T N_T}{\sigma^2_T} \right), \left( \frac{N_S}{\sigma^2_S} + \frac{N_T}{\sigma^2_T} \right)^{-1} \right)$$

In the limit where $\sigma_T$ goes to infinity:

$$\sigma_{\text{post}}^2 \rightarrow \frac{\sigma^2_S}{N_S}$$

$$\mu_{\text{post}} \rightarrow \hat{\theta}_S$$

That is, the target study data are discarded because the likelihood becomes flat, and the posterior is the same as the prior.

Code example

Pooling the data is equivalent to defining a Gaussian prior with mean and standard deviation corresponding to the treatment effect estimate and standard error on the treatment effect in the source study.

Set working directory and load packages

Load the case study configuration

Load the simulation configuration and the Belimumab case study configuration from YAML files.

set.seed(42)
case_study_config <- yaml::yaml.load_file(system.file("conf/case_studies/belimumab.yml", package = "BExTE"))

Create data objects

Create a source_data instance, where information about the source data is stored

source_data <- ObservedSourceData$new(case_study_config)

Set the observed target data (in the paediatrics population)

target_data <- ObservedTargetData$new(treatment_effect_estimate = case_study_config$target$treatment_effect, treatment_effect_standard_error = case_study_config$target$standard_error, target_sample_size_per_arm = as.integer(case_study_config$target$total / 2), summary_measure_likelihood = case_study_config$summary_measure_likelihood)

Create model

method <- "pooling"
method_parameters <- list(
  initial_prior = "noninformative" # This corresponds to the fact that the posterior for the adults data is derived from an uninformative prior (for consistency with other methods).
)

Now, we define the model we want to use for inferring the treatment effect in the target study.

model <- Model$new()

model <- model$create(
  case_study_config = case_study_config,
  method = method,
  method_parameters = method_parameters,
  source_data = source_data
)

This model calls RBesT in the backend, leveraging the fact that it is a special case of a Robust Mixture Prior.

Inference

# Perform Bayesian inference based on observed target data.
model$inference(target_data = target_data)

## [1] "Success"

model$plot_pdfs(xmin = 0, xmax = 1, resolution = 100)

Remarks

Note that pooling and separate analysis are two extreme cases of a Conditional Power Prior, with $\gamma = 1$ and $\gamma = 0$ respectively.