## 5 Privacy Risk Assessment: Advancements within the PRUDEnce Framework
### Relationship between Entropy, Feature Frequency, and Re-Identification Risk
#### 5.1.2 Selecting Background Knowledge Configurations with Low-Entropy Trajectory Features Frequency  
The approach aims to enhance the identification of background knowledge instances with heightened privacy risks in the dataset. It involves mathematical definitions and procedures to systematically identify and retain instances that meet the specified criterion.  
**Definition 10** (Low-entropy Feature Set).: _Let \(x_{ui}\) represent the \(i\)-th feature of record \(x_{ui}\) of dataset \(D_{n=1}^{N}\) where \(N\) is the number of individuals, and let \(E(x_{i})\) denote the entropy associated with feature value \(x_{ui}\in D_{n=1}^{N}\). We define the set of low-entropy features \(X_{\text{low-entropy}}\) as_  
\[X_{\text{low-entropy}}=\{x_{ui}\in D_{n=1}^{N}\mid E(x_{i})\leq\text{ percentile}(E(x_{i}),y)\},\]  
_where \(\text{percentile}(E(x_{i}),y)\) is the \(y\)-th percentile of the distribution of entropy values \(E(x_{i})\)._  
**Definition 11** (Low-Entropy Selected Instances).: _If \(B_{k}\) is a set of subsets of \(D_{n=1}^{N}\), where each \(b\in B_{k}\) contains feature \(x_{ui}\in D_{n=1}^{N}\), then, for each \(b_{j}\in B_{k}\) (where \(j\) represents the \(j\)-th instance), we define the selected background knowledge instances as_  
\[b_{j}=\{x_{ui}\in b_{j}\mid x_{ui}\in X_{\text{low-entropy}}\}.\]  
_The set of all selected low-entropy subsets \(B_{\text{low-entropy}}\) is defined as_  
\[B_{\text{low-entropy}}=\{b_{j}\mid b_{j}\in B_{k}\}.\]  
In Definition 10, the low-entropy \(X_{\text{low-entropy}}\) feature set is created, selecting only the feature values with the lowest values. All possible background knowledge configurations \(B_{k}\) are generated, but only instances containing the selected feature value in the \(y\)% lowest entropy results are selected for risk evaluation. Only the background knowledge instances\(b\) that contain feature values from \(X_{low-entropy}\) are retained for risk evaluation, as defined in Definition 11. This ensures that the analysis focuses on instances featuring feature frequency with low entropy, thereby improving privacy risk assessment by considering most relevant and distinctive features from that specific dataset.  
Given their infrequency or uniqueness within the dataset, low-frequency features are prone to being distinctive or having limited occurrences. Consequently, background knowledge instances containing such features are more likely to contribute to higher risk values. We have the same or very similar maximum risk values as we compute all instances of risk. As a result, including infrequent feature values in the background knowledge raises the likelihood of identifying specific individuals.  
This formal approach provides a systematic method for selecting background knowledge instances containing low-entropy feature frequency, thereby enhancing the identification of instances with heightened privacy risks in the dataset. By incorporating the reduced background knowledge configuration into the risk assessment framework, we can formally analyze how targeting low-entropy values in the background knowledge configuration can lead to computational improvements in privacy risk assessment. This approach allows us to focus computational efforts on configurations featuring low-entropy features, thereby facilitating risk assessment and enhancing the accuracy of re-identification risk estimates.