taken in [@Rasmussen2013]; derive a Gibbs sampler using a data augmentation scheme which associates a latent variable to each interaction. However, such an algorithm is unlikely to scale well with the number of interactions. Additionally, we can make use of existing code for posterior inference with Hawkes processes and graphs based on compound CRMs, and therefore propose a two-step approximate inference procedure, motivated by modular Bayesian inference [@Jacob2017].

Let $\mathcal Z=(z_{ij}(T))_{1\leq i,j\leq V}$ be the adjacency matrix defined by $z_{ij}(T)=1$ if there is at least one interaction between $i$ and $j$ in the interval $[0,T]$, and 0 otherwise. We have $$\begin{aligned}
\pi(\phi,\xi|\mathcal D)&=\pi(\phi,\xi|\mathcal D,\mathcal Z)=\pi(\xi|\mathcal D,\mathcal Z)\pi(\phi|\xi ,\mathcal D).\end{aligned}$$ The idea of the two-step procedure is to (i) Approximate $\pi(\xi|\mathcal D,\mathcal Z)$ by $\pi(\xi|\mathcal Z)$ and obtain a Bayesian point estimate $\widehat\xi$ then (ii) Approximate $\pi(\phi|\xi ,\mathcal D)$ by $\pi(\phi|\widehat \xi ,\mathcal D)$.

The full posterior is thus approximated by $\widetilde\pi(\phi,\xi)=\pi(\xi|\mathcal Z)\pi(\phi|\widehat \xi ,\mathcal D).$ As mentioned in Section \[sec:properties2\], the statistical model for the binary adjacency matrix $\mathcal Z$ is the same as in [@Todeschini2016]. We use the MCMC scheme of [@Todeschini2016] and the accompanying software package SNetOC[^2] to perform inference. The MCMC sampler is a Gibbs sampler which 