We introduced \textit{Surjective Sequential Neural Likelihood} estimation, a new neural likelihood method for simulation-based inference for high-dimensional data. SSNL uses dimensionality-reducing surjections to embed the data in a lower-dimensional space while simultaneously learning the likelihood function.

SSNL performs particularly well when applied to high-dimensional time series data outperforming state-of-the-art methods and is on par with them in other experiments, making it an excellent off-the-shelf estimator for high-dimensional data sets. As a limitation, we identified that due to introducing an additional conditional density, SSNL in some cases requires a higher sample size than other flow-based models like SNL or SNPE-C.

We anticipate that our method will complement other approaches in SBI well, but do not argue that it should be generally preferred over others. For instance, when posteriors have a simple geometry and the likelihood function is complicated to approximate, posterior estimators like SNPE-C often provide superior performance. Similarly, neural ratio methods, such as SNRE-C, are typically easier to fit as flow-based methods, since they only require training a classifier.

Future research could elucidate how deeper (stacked) surjective architectures impact posterior inference and if more suitable surjective architectures can be developed for SBI. The application of information-theoretic criteria to more rigorously determine the dimensionality of the latent space, for instance, as done in \citet{chen2023learning}, is another possible research direction.
