# Comparative Analysis of Local Differential Privacy Schemes in Healthcare Datasets
## 1 Introduction
This concept asserts that as the data's dimensionality grows, introducing sufficient noise to protect privacy becomes increasingly challenging without excessively distorting the data. For instance, consider a dataset with 10 features. Achieving local differential privacy would require perturbing each feature with a specific amount of random noise. However, if we increase the number of features to 100, the noise to maintain the same level of privacy also increases. To tackle these challenges, several studies have been conducted. [PERSON] et al. proposed LOPUB [6], a privacy-preserving technique where users first encode their data using Bloom filters and then perturb the encoded data using the randomized response (RR) algorithm [7]. Finally, the perturbed-encoded data are sent to a central server, which estimates the multi-dimensional joint probability distribution by applying the LASSO [8] and expectation-maximization (EM) algorithms [9]. This method was tested on open datasets and demonstrated good performance for high-dimensional data with low multi-dimensional joint probability distributions. Meanwhile, LOCOP [10] represents a novel approach for preserving privacy in high-dimensional data through synthesis. A univariate marginal distribution is computed using LASSO regression from the perturbed-encoded data, and a multivariate Gaussian copula is used to model the attribute dependence. This method aims to provide a privacy-preserving solution for sharing high-dimensional data with a central server. Regrettably, approaches like LOPUB encounter diminished data utility as \(k\)-way increases, where \(k\) is the dimensionality of joint probability distribution estimation. For certain datasets, the error between \(k=2\) and \(k=3\) evaluations can be two or three times higher, contingent upon the cardinality of the attributes. Another crucial factor to take into account is the number of elements in each attribute domain. Consequently, these methods only offer a partial resolution to the challenge of handling extensive datasets with numerous attributes and high cardinality. Unfortunately, both LOPUB and LOCOP rely on LASSO regression, initially introduced to enhance prediction accuracy in regression models. However, the LASSO regularization technique, employed in linear regression, grapples with high cardinality--a prevalent issue for large datasets. The encoding and perturbation method introduced by [6] has also been utilized and built upon in subsequent works such as [PERSON] [10] and [PERSON] [11]. We propose the application of the Bayesian Ridge Regression (BRR) of [PERSON] [3] on the central server to address the challenge of evaluating highly correlated attributes. The differences between our approach [6], and [10] are outlined in Table 1.  
This work is supported by the research of [PERSON] et al. [12], [PERSON] et al. [13], and [PERSON] et al. [14]. [PERSON]'s investigation focused on computing the posterior probability from the prior probability, providing several benefits, including the construction of robust prior distributions. Additionally, [PERSON] [13] applied a Bayesian approach in sparse modeling and machine learning. [PERSON] et al. [14] demonstrated the effectiveness of this approach in constructing reliable prior probability distributions. This collective evidence underscores the value of investigating BRR, suggesting practical applications across diverse fields. Our previous work (a preliminary version of this work was published in [15]) laid the groundwork for integrating BRR within the LDP mechanism, and the extended version of this work delves deeply into exploring attribute correlations using the average absolute Pearson correlation. This comprehensive expansion goes beyond the mere application of BRR; it involves nuanced analyses of privacy guarantees and complexity trade-offs, employing various metrics such as R-squared. Additionally, it includes a comparison between regression techniques applied to estimate joint probability distributions in LDP.