In all panels, each marker represents a family in 5. For each statistical or information measure (rows), different scores (columns) comparing generated and natural sequences are the coordinates of these points. All scores are such that being close to 0 is better, and their value for multiple sequence alignments (MSAs) generated by Boltzmann machine DCA (bmDCA) with default parameters is shown versus that for MSAs generated by MSA Transformer. The statistical or information measures considered in each row are defined in ‘Analyzing the statistics of MSAs’ – from the top: one-body frequency, two- and three-body connected correlations, entropy, mutual information, and co-information. For each of them, we consider its values over all MSA columns (or pairs or triplets of columns), and all amino acids if appropriate, for both natural and synthetic MSAs. To obtain the vertical and horizontal coordinates (respectively) of the markers in each panel, we compare these values for each natural MSA with the values from the corresponding synthetic MSAs generated by bmDCA with default parameters or by our method based on MSA Transformer (respectively). We use four different scores for this comparison, and devote each column of the figure to one of these scores – from the left: where denotes the Pearson correlation; where ‘Slope’ means the slope of best linear fit (see Figure 4—figure supplement 2 and Figure 4—figure supplement 3 for illustrations of these first two quantities in the case of two- and three-body connected correlations for families PF00072 and PF00153); the Jensen–Shannon divergence between the distributions of values; the Wasserstein distance between these distributions. For each statistical or information measure (row) and each score (column), and for each family in 5, we have one value of the score comparing the natural and bmDCA-generated MSAs and another one comparing the natural and MSA-Transformer–generated MSAs. We plot the former value versus the latter, yielding one marker per protein family in each plot. Thus, each plot compares the ability of bmDCA and MSA Transformer to reproduce the statistics of the natural data. Blue markers (above the diagonal) mean that the scores for MSA-Transformer–generated MSAs are better, while green markers (below the diagonal) mean the opposite.