We are writing to promote a novel Bayesian dynamic borrowing approach, the empirical profile Bayesian estimation, in the published article "Yaoshi Wu, Jianan Hui, Qiqi Deng. Empirical profile Bayesian estimation for extrapolation of historical adult data to pediatric drug development. Pharmaceutical statistics 2020; 19; 787-802. http://dx.doi.org/10.1002/pst.2031."

Bayesian borrowing has been a long-standing concept, but the difficulty has been that if the current data deviates from the historical information, often referred to as the prior-data conflict, then the simple traditional Bayesian approach will introduce bias and lead to a false positive decision. The Bayesian dynamic borrowing, which considers the prior-data conflict, refers to that the efficacy information borrowed informatively from external sources using the Bayesian approach is discounted due to the potential discrepancy between the historical and current data. For example, a study medication under investigation for the pediatric drug development may bear a different treatment effect compared to that for the adult population, which may be the consequence of the difference in mode of action or baseline characteristics such as age, weight, BMI, etc. Some Bayesian dynamic borrowing approaches in the current literature are listed as follows.

• Hierarchical Bayesian model [1]: discount by multiple layers of priors.
• Power prior [2], [3]: discount by raising the historical data to some power \a".
• Robust mixture prior [4]: discount by different weights of informative and weakly informative priors.
• Empirical profile Bayesian estimation [5]: discount for the clinical expectation of pediatric drug effect size compared to adult and only borrow the information about the variability.

The selection of a proper Bayesian dynamic borrowing approach should be based on its interpretability, simplicity, practicality, and communicability. Note that there is no regulatory authority (U.S. Food and Drug Administration (FDA), European Medicines Agency (EMA), etc.) recommended Bayesian dynamic borrowing approach, and mathematically, these approaches may be equivalent. In fact, FDA encourages sponsors to explore a variety of design options in the planning phase and to discuss their considerations with the appropriate FDA review division at regulatory meetings such as End-of-Phase-2 (EOP2) or Type C meetings.

The empirical profile Bayesian estimation is a restricted informative borrowing when observed pediatric mean in the current pediatric trial exceeds a predefined threshold r based on clinical expectation of the pediatric drug effect compared to adult drug effect under the transfer-ability assumption. Most of the existing Bayesian approaches assume exchange-ablity meaning the pediatric and adult drug effects are exchangeable. The transfer-ability assumption, unlike the exchangeability assumption in other Bayesian approaches, allows the treatment effect difference in the pediatric and adult populations. The empirical profile Bayesian has the following benefits:

• The idea is easy to interpret, implement, and understand.

The implementation of the empirical profile Bayesian is illustrated in the following example. If the observed pediatric treatment effect mean is at least 50%, or any reasonable clinical expectation provided by the trial physician, of the observed adult treatment effect mean, the efficacy information from the adult phase III trial will be borrowed to the pediatric trial. The false positive rate and power gain will be studied to evaluate the trade-off of the borrowing and further determine whether the 50% borrowing threshold is appropriate and beneficial.

• Improve communications with clinical trial physicians about the informative borrowing and achieving sensible decision making.

When the authors developed this approach, one of the motivations was to largely incorporate the clinical judgment on decision boundary rather than completely relying on statistical inputs and mathematical derivation. Such a unique benefit is achieved by pre-specifying a threshold r for informative borrowing to consolidate clinical knowledge for a clinically meaningful treatment effect size on the pediatric population compared to the adult.

• Achieve a good trade-off between the false positive rate and power gain.

A good decision is made to seek a sensible balance between the chance of making a false positive decision and a true positive decision. The substantial numerical studies presented in the article show that the empirical profile Bayesian approach provides a very nice trade-off between power gain (making a true positive decision) and type I error inflation (making a false positive decision), compared to other Bayesian approaches and frequentist approach, especially for a small pediatric sample size.

• Consistent decision-making boundary with clinical practice.

The decision making boundaries using different approaches shown in the rejection  figure (see Figure 4 in the article) demonstrates that the decision making boundary using profile Bayesian is consistent with clinical practice for pediatric drug development as its critical region for rejecting the null hypothesis for no effect is the most sensible for decision making compared to all other approaches (see Section 4 in the article). For example, if the pediatric treatment effect is much higher than the adult, it's expected the informative borrowing should strengthen the signal of pediatric treatment benefit. However, the other Bayesian dynamic approaches discount the informative borrowing due to the pediatric and adult difference and result in weakening the pediatric efficacy signal. On the contrary, the empirical profile Bayesian enables the borrowing using the one-sided threshold r rather than two-sided, the signal will be strengthened as long as the pediatric effect is large enough.

• Unbiasedness of the posterior mean and mathematically simple.

As mentioned earlier, since only the information about variability is borrowed under the transfer-ability assumption, the posterior mean is indeed an unbiased estimate of pediatric treatment effect size, on the other hand, the other Bayesian borrowing approaches will result in a biased estimate if pediatric and adult treatment effect distribution are not exchangeable. It's also worth noting that the empirical profile Bayesian approach is mathematically simple and can be combined with other Bayesian dynamic approaches.

A potential limitation of the approach is that the posterior mean using empirical profile Bayesian is an unbiased estimator of the pediatric treatment effect under transfer-ability, which may be a trade-off that the approach may underestimate the variability when pediatric sample size is small, however, the restrictive borrowing from the prior variability mitigates such a risk to some extent.

With the benefits of unbiased Bayes estimate under scenario when the treatment effect is different between pediatric and adult, and dynamic informative borrowing depending on a clinically meaningful threshold r, the profile Bayesian not only reaches a more attractive balance of type I error inflation and power gain than other Bayesian approaches but also has the unique benefit to proactively incorporate clinical judgment on the decision boundary to make a more sensible decision and improve the communication with clinical trial physicians or non-statisticians. The authors believe that the empirical profile Bayesian estimation is a competitive and attractive dynamic borrowing approach for pediatric drug development because of its interpret-ability, simplicity, practicality, and communicability.