SB-ETAS: using simulation based inference for scalable, likelihood-free inference for the ETAS model of earthquake occurrences

Performing Bayesian inference for the Epidemic-Type Aftershock Sequence (ETAS) model of earthquakes typically requires MCMC sampling using the likelihood function or estimating the latent branching structure. These tasks have computational complexity $O(n^2)$ with the number of earthquakes and therefore do not scale well with new enhanced catalogs, which can now contain an order of $10^6$ events. On the other hand, simulation from the ETAS model can be done more quickly $O(n \log n )$. We present SB-ETAS: simulation-based inference for the ETAS model. This is an approximate Bayesian method which uses Sequential Neural Posterior Estimation (SNPE), a machine learning based algorithm for learning posterior distributions from simulations. SB-ETAS can successfully approximate ETAS posterior distributions on shorter catalogues where it is computationally feasible to compare with MCMC sampling. Furthermore, the scaling of SB-ETAS makes it feasible to fit to very large earthquake catalogs, such as one for Southern California dating back to 1932. SB-ETAS can find Bayesian estimates of ETAS parameters for this catalog in less than 10 hours on a standard laptop, which would have taken over 2 weeks using MCMC. Looking beyond the standard ETAS model, this simulation based framework would allow earthquake modellers to define and infer parameters for much more complex models that have intractable likelihood functions.