Combining summary statistics with simulation-based inference for the 21 cm signal from the Epoch of Reionisation

The 21 cm signal from the Epoch of Reionisation will be observed with the upcoming Square Kilometer Array (SKA). We expect it to yield a full tomography of the signal, which opens up the possibility to explore its non-Gaussian properties. This raises the question of how can we extract the maximum information from tomography and derive the tightest constraint on the signal. In this work, instead of looking for the most informative summary statistics, we investigate how to combine the information from two sets of summary statistics using simulation-based inference. To this end, we trained neural density estimators (NDE) to fit the implicit likelihood of our model, the LICORICE code, using the Loreli II database. We trained three different NDEs: one to perform Bayesian inference on the power spectrum, one to perform it on the linear moments of the pixel distribution function (PDF), and one to work with the combination of the two. We performed ∼ 900 inferences at different points in our parameter space and used them to assess both the validity of our posteriors with a simulation-based calibration (SBC) and the typical gain obtained by combining summary statistics. We find that our posteriors are biased by no more than ∼ 20 % of their standard deviation and under-confident by no more than ∼ 15 %. Then, we established that combining summary statistics produces a contraction of the 4D volume of the posterior (derived from the generalised variance) in $91.5 %$ of our cases, and in $70$ to $80 %$ of the cases for the marginalised 1D posteriors. The median volume variation is a contraction of a factor of a few for the 4D posteriors and a contraction of 20 to 30 $%$ in the case of the marginalised 1D posteriors. This shows that our approach is a possible alternative to looking for so-called sufficient statistics in the theoretical sense.