## 1 Introduction
Part of the issue derives from the use of only limited human variability, for example, during the training phase of the ML applications: as mentioned above, studies on individual populations, or groups of populations, have been prioritised [29; 32; 33], probably because of the difficulty of accessing worldwide data of known sex [34]. Furthermore, testing is often performed on samples too small to provide statistical reliance [35; 36; 37], and those samples are often part of the same body of data (e.g., collected by the same observer, belonging to the same population/group) used during the training phase [38], which potentially limits the population-inclusive application of the estimation achieved. The shortcomings of previous approaches are not only limited to the source and size of the sample. The number of variables considered can be problematic for several reasons. Cranial variables can be highly correlated with each other because of morphological integration [24], and this can make the dataset redundant. The more the number of variables, the higher the redundancy, which increases the risk of overfitting (i.e., when the prediction performs well on the training data but cannot generalise on unseen data [39]). Furthermore, the use of several variables can become problematic when dealing with incomplete cranial remains, a common occurrence with archaeological and forensic material. An additional limitation is the over-reliance on \"accuracy\" as a measure of performance. The accuracy of an estimation reflects the proportion of cases correctly assigned to their class: in a binary situation, such as the estimation of sex, the accuracy is computed by counting how many males and females are identified as males or females, respectively. Quantitative algorithms, though, provide the estimation in terms of probability, and, in a binary case, observation is attributed to a class if it has a probability higher than 0.5 for that class, although the threshold can be different. Therefore, accuracy does not account for the probability of the estimation (its \"strength\", in other words) because the same result is obtained regardless of whether the probability of an observation being male or female is 0.51 or 0.99. The usefulness of accuracy is dependent on where we set the cut-off probability [40]. The result of the abovementioned limitations is that, as of today, we lack a clear understanding of the potential of quantitative applications for the estimation of sex from the human cranium. In fact, although previous applications have fuelled research on the subject, the training and testing frameworks they used were prone to potential bias to an extent difficult to pinpoint, thus limiting their use as standardised methods for estimating sex. Here, we try to overcome those limitations using a population-inclusive neural network approach based on large training and testing datasets from different sources, requiring a limited number of traits and maximising the probability of estimation rather than accuracy. The advances produced by this work are three-fold: (I) it provides an estimation easy to transfer to other datasets, regardless of ancestry; (II) it clarifies the potential of quantitative, algorithm-based estimation of sex from human crania, maximising probability over accuracy; (III) it presents a step-by-step protocol for the application of ML-based predictions to solve basic problems in the field of physical anthropology.