Simulation models of complex dynamics in the natural and social sciences commonly lack a tractable likelihood function, rendering traditional likelihood-based statistical inference impossible. Recent advances in machine learning have introduced novel algorithms for estimating otherwise intractable likelihood functions using a likelihood ratio trick based on binary classifiers. Consequently, efficient likelihood approximations can be obtained whenever good probabilistic classifiers can be constructed. We propose a kernel classifier for sequential data using path signatures based on the recently introduced signature kernel. We demonstrate that the representative power of signatures yields a highly performant classifier, even in the crucially important case where sample numbers are low. In such scenarios, our approach can outperform sophisticated neural networks for common posterior inference tasks.