Kunal Anilkumar 1,2, Niklas Hartung 1, Charlotte Kloft 2,3, Wilhelm Huisinga 1,2
1 Institute of Mathematics, University of Potsdam (, Germany), 2 Graduate Research Training Program PharMetrX (, ), 3 Dept. of Clinical Pharmacy and Biochemistry, Institute of Pharmacy,Freie Universitaet (, Germany)
Objectives: In Model-Informed Precision Dosing (MIPD), Bayesian hierarchical models utilize prior knowledge from clinical trials and real world data to update posteriors (population parameter uncertainty) [4]. However, since patient-specific latent parameters (e.g., CL, Vd) are unobserved, the likelihood needs to be marginalised. But marginal likelihoods become intractable, except in artificial settings. While pseudo-marginal frameworks like Grouped Independence Metropolis Hastings (GIMH) address this by using likelihood estimators, standard batch MCMC methods face a paradox: estimation becomes increasingly difficult as more data becomes available because the likelihood “concentrates,” hindering posterior exploration.
Although Sequential Monte Carlo methods can mitigate this, their validation still requires “ground truth” distributions from batch MCMC. This study examines how the choice and variance of likelihood estimators affect the construction of such ground truths. We outline the associated methodological challenges and consider how the conditional independence structure of hierarchical data may be leveraged in the design of estimation strategies within this context.
Methods: A Bayesian hierarchical model was considered in which individual-specific latent parameters were independently distributed conditional on a population-level parameter of interest. The intractable marginal likelihood was approximated using Monte Carlo integration. Two likelihood estimators were implemented within an otherwise identical GIMH framework.
1. The Product-Form estimator (PFGIMH), constructing the likelihood as a product of independent individual-level Monte Carlo estimates, thereby explicitly leveraging the conditional independence inherent in the patient data[1].
2. The Joint estimator (JGIMH), which approximates the total population likelihood based on a single Monte Carlo approximation of the full hierarchical likelihood[3].
Both estimators were unbiased[2] and used the same computational budget, proposal distributions, and tuning parameters. To validate the implementations, we tested them on simpler hierarchical models (i.e. conjugate hierarchical normal models) where the analytical posterior distribution was available. The input data consisted of multiple individuals, each contributing a number of noisy observations, depending on rich/sparse sampling. Individual-specific parameters were unobserved and varied around a shared population-level parameter, while observations were subject to measurement error.
Results: Across all scenarios, the Product-Form estimator consistently exhibited more stable performance than the Joint estimator. Using analytically tractable models, convergence was quantified via posterior mean bias (normalised by the analytical posterior standard deviation). In a simple setting with three individuals, the Product-Form estimator required 5-fold fewer Monte Carlo samples per iteration to achieve target acceptance rates (15% – 45%) and posterior mean bias comparable to the Joint estimator. As the number of individuals increased, the Product-Form estimator continued to perform better than its Joint counterpart since it had lower variance, resulting in improved mixing of the population-level Markov chain, achieving more stable acceptance behaviour, and lower autocorrelation.
However, after increasing data richness and number of individuals, while other tuning parameters were held constant, both estimators exhibited performance degradation consistent with the known batch MCMC limitation under concentrated posteriors. We observed sharply reduced acceptance rates (<1%) and large posterior mean bias for both estimators. The Product-Form estimator overcame these issues by a 12-fold increase in the number of Monte Carlo samples, whereas the Joint estimator continued to show unstable performance. Conclusions: The choice of the unbiased likelihood estimator is a critical determinant of the performance in the GIMH framework. By explicitly leveraging the independence of individual subjects through the Product-Form estimator, the variance of the likelihood estimate is drastically reduced. This allows for valid Bayesian inference with significantly lower computational overhead (fewer Monte Carlo samples per iteration). These findings suggest that for hierarchical models, the PFGIMH provides a robust alternative to standard MCMC methods when individual likelihoods are complex but independent. References: [1] Kuntz, Juan, et al. ‘Product-Form Estimators: Exploiting Independence to Scale up Monte Carlo’. Statistics and Computing, vol. 32, no. 1, Dec. 2021, p. 12. [2] Andrieu, Christophe, and Gareth O. Roberts. ‘The Pseudo-Marginal Approach for Efficient Monte Carlo Computations’. The Annals of Statistics, vol. 37, no. 2, 2009, pp. 697–725. [3] Chopin, Nicolas, and Omiros Papas Iliopoulos. An Introduction to Sequential Monte Carlo. Springer International Publishing, 2020. Springer Series in Statistics. [4] Maier, Corinna, et al. ‘A Continued Learning Approach for Model-Informed Precision Dosing: Updating Models in Clinical Practice’. CPT: Pharmacometrics & Systems Pharmacology, vol. 11, no. 2, Feb. 2022, pp. 185–98.
Reference: PAGE 34 (2026) Abstr 11905 [www.page-meeting.org/?abstract=11905]
Poster: Methodology - Estimation Methods