## 2 Modeling Approach
### Model Details  
Modern machine learning offers a plethora of models, algorithms, and software frameworks [30,31]. They are based on a wide range of mathematical/statistical ideas and methods, while they vary significantly in their complexity and performance. From this multitude of available models, only a tiny subset fulfills and addresses the technical challenges of Section 2.1. The very small size of the dataset (11 different samples) significantly restricts the model complexity. In addition, the small size implies a significant uncertainty of the model parameters, which should be visible during the decision process; thus, the proposed model should explicitly report it. The measurement error, which is not negligible, is considered by a hierarchical structure, as well. Such a machine learning model does not explicitly exist in a predefined form, but as a generic part of a probabilistic machine learning approach [32]. Before presenting the model structure and parameters, the features and target variable should be clarified. In machine learning, features are the data used as input to the model, while the target is the desired output. In this case, primary features are the ratios of stable isotopes while their combinations up to second-order are the derived ones. The list of primary features is summarized in Table 1. It is noted that d will be used instead of 8 notation, on the caption of plots due to software font support limitations. The target variable is the sample origin which follows the following encoding:  
\[target= \begin{cases}1.&\quad\text{if origin of sample is Messolongi, \ Greece}\\ 0.&\quad\text{if origin of sample is not Messolongi, \ Greece}\end{cases} \]  
By this encoding, the problem is transformed into an equivalent binary classification. The proposed probabilistic model is a generalized linear model with parameters that will be estimated by their full distribution rather than point estimates (full distribution estimation will reveal and quantify the model uncertainty). By construction, it has the desired simplicity due to the small data set, it is hierarchical and able to cope with external sources of variations (the measurement error in our case). \begin{table}
\begin{tabular}{c c} \hline \hline
**Feature** & **Is Influenced By** \\ \hline \(\delta^{15}\text{N}_{\text{AIR}}\) (\%) & Nutrient uptake, digestion, metabolism, Cultivation Conditions (i.e., fertilizers)—Techniques \\ \hline \(\delta^{13}\text{C}_{\text{V-PDB}}\)(\%) & Nutrient uptake (e.g., C3 versus C4), digestion and metabolism, adulteration of foods \\ \hline \(\delta^{34}\text{S}_{\text{V-CDT}}\)(\%) & Bacterial action, Distance to Sea (e.g., inland versus coastal geography) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of primary features. The standard notation of probabilistic modeling is used [33; 34], where \(p(\cdot)\) which denotes a marginal distribution, and \(p(\cdot|\cdot)\) a conditional probability density for both continues and discrete random variables. The \"\(\sim\)\" symbol is used to denote that a random variable follows known distributions. In this paper \(\theta\sim N\big{(}\mu,\sigma^{2}\big{)}\) denotes that \(\theta\) has a normal distribution with mean \(\mu\) and variance \(\sigma^{2}\), which is equivalent to \(p(\theta)\sim N\big{(}\theta|\mu,\sigma^{2}\big{)}\) or even more explicitly \(p\big{(}\theta|\mu,\sigma^{2}\big{)}\sim N\big{(}\theta|\mu,\sigma^{2}\big{)}\). \(\theta\sim B(p)\) denotes a Bernoulli distribution with success probability \(p\), or equivalently a binomial distribution when \(n=1\), \(\theta\sim Bin(p,n)\). The basic idea of the proposed probabilistic classification model is that we consider each sample of bottarga as a Bernoulli trial, where success is outcome 1 (i.e., origin of sample is Messolongi--Greece according to our encoding) with success probability p that is defined for each trial by a non-linear function which depends on the linear combinations of our features (i.e., linear combinations of ratios of stable isotopes up to second-order).