## 2 Modeling Approach  
In most studies, stable isotopic analysis is performed with a large dataset of samples to discriminate the geographical origin of food. However, in some cases, geographical discrimination could also be achieved with only a few samples. [PERSON] et al. (2015) succeed to discriminate a small number of pawns' samples ([PERSON] et al., 2015). In particular, they used only ten samples to decide which parameters are more useful in order to discriminate Australian fresh pawns from other, mainly imported, samples. Then, the method was applied to seventeen unknown samples to distinguish the geographical origin. [PERSON] et al. (2015) used stable isotopic analysis to determine honey samples from different floral sources and different areas ([PERSON] et al., 2015). They used a small number of samples (40 honey samples) to separate them in different growing areas and a smaller number of samples to separate them according to their botanical origin (13 samples of acacia, 4 of linden, 6 of sunflower, 4 of rape, 4 of honeybee, and 9 of multiforal honey). Thus, despite the number of samples, stable isotope ratio analysis led to good discrimination of honey samples in every case and is considered an effective tool for the authentication of honey.  
The modeling approach of this paper is focused on establishing a discrimination mechanism of individual bottarga samples based on a very small number of samples available for analysis. In this research, there were available five different samples (different lot numbers) from Messolongi and two different samples (different lot number) for the other three areas included in the analysis (namely Preveza--Greece, Australia & Mauritania). Each sample was used to measure stable isotopes ratios of Carbon, Nitrogen, and Sulfur. The isotopic analyses of carbon, nitrogen, and sulfur were performed by an Elementar Isoprime 100 Isotope Ratio Mass Spectrometry (IRMS) instrument (IsoPrime Ltd., Cheadle Hulme, UK) coupled to an Elemental Analyzer (Elementar Vario Isotope EL Cube, Elementar Analysensysteme GmbH, Hanau, Germany). Samples were weighed into tin capsules for measurement and were loaded on the auto-sampler of the IRMS analyzer. The results of the isotope ratio analyses were expressed as delta values \(\delta\) (%) and calculated according to the following equation:  
\[\delta\;\mathrm{X}\;(\%_{\infty})=[(\mathrm{Rsample/Rstandard})-1]\times 1000\]  
where:  
* \(\mathrm{X}\) is the isotope being studied (e.g., \({}^{13}\mathrm{C}\), \({}^{15}\mathrm{N}\), \({}^{34}\mathrm{S}\)),
* \(\mathrm{R}_{\mathrm{sample}}\) is the isotopic ratio of the measured element in its physical form (e.g., \({}^{13}\mathrm{C}\)/\({}^{12}\mathrm{C}\), \({}^{15}\mathrm{N}\)/\({}^{14}\mathrm{N}\), \({}^{34}\mathrm{S}\)/\({}^{32}\mathrm{S}\)) in the sample, and
* \(\mathrm{R}_{\mathrm{standard}}\) is the isotopic ratio of the reference material used in the analysis.  
Each sample was analyzed 5 times resulting for 25 measurements for Messolongi and 30 measurements for other areas. That is 55 measurements for each of the elements measured in the analysis.  
The whole concept of this model is based on stable isotopes ratios and quantifies the differences between the bottarga samples from Messolongi and samples from other regions (local and worldwide. Due to the probabilistic nature of the proposed machine learning model (see details in Sections 2.2-2.5), an aggregated high probability confidence is very strong statistical evidence that one sample comes from the region of interest or not (Section 2.6). The technical challenges of the problem are discussed in the next section.