Retinoic acid treated human dendritic cells induce T regulatory cells via the expression of CD141 and GARP which is impaired with age.

It is common in nature to see aggregation of objects in space. Exploring the mechanism associated with the locations of such clustered observations can be essential to understanding the phenomenon, such as the source of spatial heterogeneity, or comparison to other event generating processes in the same domain. Log-Gaussian Cox processes (LGCPs) represent an important class of models for quantifying aggregation in a spatial point pattern. However, implementing likelihood-based Bayesian inference for such models presents many computational challenges, particularly in high dimensions. In this paper, we propose a novel likelihood-free inference approach for LGCPs using the recently developed BayesFlow approach, in which invertible neural networks are employed to approximate the posterior distribution of parameters of interest. BayesFlow is a neural simulation-based method based on "amortized" posterior estimation. That is, after an initial training procedure, fast feed-forward operations allow rapid posterior inference for any data within the same model family. Comprehensive numerical studies validate the reliability of the framework and show that BayesFlow achieves substantial computational gain in repeated application, especially for two-dimensional LGCPs. We demonstrate the utility and robustness of the method by applying it to two distinct oral microbial biofilm images.