Human vs. machine -- 1:3. Joint analysis of classical and ML-based summary statistics of the Lyman-$\alpha$ forest

In order to compress and more easily interpret Lyman-$\alpha$ forest (Ly$\alpha$F) datasets, summary statistics, e.g. the power spectrum, are commonly used. However, such summaries unavoidably lose some information, weakening the constraining power on parameters of interest. Recently, machine learning (ML)-based summary approaches have been proposed as an alternative to human-defined statistical measures. This raises a question: can ML-based summaries contain the full information captured by traditional statistics, and vice versa? In this study, we apply three human-defined techniques and one ML-based approach to summarize mock Ly$\alpha$F data from hydrodynamical simulations and infer two thermal parameters of the intergalactic medium, assuming a power-law temperature-density relation. We introduce a metric for measuring the improvement in the figure of merit when combining two summaries. Consequently, we demonstrate that the ML-based summary approach not only contains almost all of the information from the human-defined statistics, but also that it provides significantly stronger constraints by a ratio of better than 1:3 in terms of the posterior volume on the temperature-density relation parameters.