# Quantifying Privacy Risks for Continuous Trait Data
## 4 Model and Methodology
### Attack Methodology  
The continuous trait data and SNP data can be presented as a \(m_{t}\times n_{t}\) array \(\mathbb{T}\) and a \(m_{s}\times n_{s}\) array \(\mathbb{S}\), respectively. The \(k\)-th column in \(\mathbb{T}\), denoted by \(\mathbb{T}(\cdot,k)\), contains individual \(k\)'s all continuous traits, and the \(i\)-th row in \(\mathbb{T}\), denoted by \(\mathbb{T}(i,\cdot)\) contains \(i\)-th continuous trait for all individuals. Similarly, \(\mathbb{S}(\cdot,l)\) and \(\mathbb{S}(j,\cdot)\) represent the individual \(l\)'s genomic profile and \(j\)-th SNP for all individuals respectively. In general, the number of the genetic loci is much larger than the number of continuous traits (i.e., \(j\gg i\)). It has been shown that there are multiple \(\mathbb{S}(j,\cdot)\) corresponding to one \(\mathbb{T}(i,\cdot)\) as QTLs in the vast majority of cases [48]. Let \(\Omega(i)\) be an index set for the SNPs that are correlated with the \(i\)-th continuous trait. Then \(\mathbb{S}(\Omega(i),l)\) contains the values of SNPs that are correlated with the \(i\)-th continuous trait for individual \(l\). The deanonymization attack contains two stages. In the first stage, the adversary obtains the prior probabilities, expectations, and standard deviations as the correlations between \(\mathbb{S}(\Omega(i),\cdot)\) and \(\mathbb{T}(i,\cdot)\). In the next stage, based on this knowledge, the adversary is able to identify the best-matched pair \(\{\mathbb{S}(\cdot,l^{*}),\mathbb{T}(\cdot,k^{*})\}\) and determine whether the SNP profile \(\mathbb{S}(\cdot,l^{*})\) and continuous trait profile \(\mathbb{T}(\cdot,k^{*})\) belong to the same person. As mentioned in Section 4.1, There are two approaches for the adversary to acquire correlations between SNPs and continuous traits. For the second approach, the adversary can obtain the prior probabilities, expectations, and standard deviations directly; for the first approach, the probabilistic relationships between the continuous traits and genetic loci are derived through training data which are either publicly available or the adversary has access to, and is the focal point of this subsection. Note that the members of the continuous trait data correspond one-to-one with the members of the SNP data in the training set, for example, \(\mathbb{T}(\cdot,k)\) and \(\mathbb{S}(\cdot,k)\) are from the same individual. In addition, for the training set, each SNP in corresponding to \(\mathbb{S}(j,\cdot)\) is corresponding to at least one continuous trait in \(\mathbb{T}(i,\cdot)\), while each continuous trait in \(\mathbb{T}(i,\cdot)\) has at least one corresponding SNP in \(\mathbb{S}(j,\cdot)\). It should be noted that our approach is related to the works of [PERSON] et al. [21] and [PERSON] et al. [20]. However, these two studies limited their traits to certain types (methylation level and _cis_ RNA level), while our approach adopts continuous traits that are more general. Furthermore, different from these two studies which approximately used normal distribution to estimate the probabilities of the continuous trait values at different genotypes, since most of the continuous trait values are greater than zero, we adopt a truncated normal distribution for \(\Pr(\mathbb{S}(j,l)|\mathbb{T}(i,k))\) estimation that is more realistic. We will describe this approach in greater detail. Let \(T_{i}\) be a continuous random variable that measures the value of the \(i\)-th quantitative measure and let \(S_{\Omega(i)}\) be the region of SNPs that are mostly correlated to \(T_{i}\). The goal is to obtain the posterior probability for \(S_{\Omega(i)}\) to take value \(\mathbb{S}(\Omega(i),k)\) for an individual person \(k\) if \(T_{i}\) takes value \(\mathbb{T}(i,k)\) for the same person, that is, to obtain \(\Pr\Big{(}S_{\Omega(i)}=\mathbb{S}(\Omega(i),k)|T_{i}=\mathbb{T}(i,k)\Big{)}\) for any \(k\).