## 2 Background
### Association Rule Mining  
The goal of association rule mining is to find correlations between variables [6]. When applied to decisional processes, association rule mining aims to derive rules describing the process from observational data. A _decisional process_\(D\) can be represented as a set of _observations_, each describing a process instance for a given subject. Given a set of _variables_\(\mathcal{V}=\left\{\mathit{Var}_{1},\ldots,\mathit{Var}_{n}\right\}\), each representing a characteristic of the subject, an observation \(t\) is a tuple \(\left(\mathit{Var}_{1}=v_{1},\ldots,\mathit{Var}_{n}=v_{n}\right)\), where \(v_{i}\in\mathit{dom}(\mathit{Var}_{i})\) is the value of variable \(\mathit{Var}_{i}\) in \(t\). Every pair \(\mathit{Var}_{i}=v_{i}\) is called _item_, and a set of items _itemset_. An _association rule_\(r\) is an implication of the form \(r:X\to Y\), where \(X\) and \(Y\) are two itemsets, respectively, called _antecedent_ and _consequent_ of the rule. Intuitively, an association rule indicates that if \(X\) occurs in a record, then \(Y\) will also likely occur in that record. In this work, we consider _class association rules_[11], i.e., association rules whose consequent consists of a single class item. For the sake of simplicity, hereafter we use the term 'rule' to refer to a class association rule. To assess the _relevance_ of the mined rules, in this work, we consider two well-known and largely adopted metrics [12]: _support_, which represents the percentage of records in the dataset covered by a rule, and _confidence_, which represents the percentage of records covered by the rule among those covered by its antecedent. Formally, the support and confidence of an association rule \(r:X\to Y\) with respect to a dataset \(D\) are defined as  
\[supp(r)=\frac{|\{t\in D\mid X\cup Y\subseteq t\}|}{|D|} \tag{1}\] \begin{table}
\begin{tabular}{p{85.4 pt} p{341.4 pt}} \hline \hline
**Desideratum** & **Description and Motivations** \\ \hline _Data agnostic_ & The hypothesis generation should be agnostic of the _data-generation_ process, i.e., the (often unknown) composition of decisional processes leading to a potentially biased outcome. Ideally, an approach to uncover these latent biases should be able to determine a set of hypotheses without requiring a priori knowledge of the composition of the underlying _data-generation_ processes; indeed, in most contexts, little or no knowledge on how the decisions are taken is available, or the decisional environment is so complex that it is impossible to know, a priori, whether the decisions in outcome will be affected by biases (latent or explicitly manifest) in the process. \\ \hline _No parameter tuning_ & The solution should not require the tuning of parameters to avoid guess-work during the setup phase, and improve the stability and interpretability of the evaluation it generates, and operate. \\ \hline _Feature level and Feature value level_ & Biases can be analyzed at different levels of granularity. Capturing biases at a feature level would allow determining which (set of) variable(s) has an impact on the outcome, i.e., whether there is a significant correlation between the type of characteristics of an individual and the outcome variable. In certain domains such as in discrimination discovery, an analyst might require a more fine-grained analysis able to identify whether individuals with certain characteristics have been treated differently. To this end, it is desirable that the outcome of the analysis highlights possible correlations between feature _values_ and the outcome variable. \\ \hline _Change impact_ & The solution should enable the analysis of the impact of changing the value of one (or more) feature(s) on the value of the outcome, allowing the user to assess both the direction (i.e., positive or negative) and the magnitude of this impact. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Desiderata for bias detection and investigation.