# Quantifying Privacy Risks for Continuous Trait Data
## 4 Model and Methodology
### Attack Methodology
By the Bayes rule, we have \[\Pr\Big{(}S_{\Omega(i)}=\mathbb{S}(\Omega(i),k)|T_{i}=\mathbb{T}(i,k) \Big{)} \tag{1}\] \[=\frac{\Pr\Big{(}T_{i}=\mathbb{T}(i,k)|S_{\Omega(i)}=\mathbb{S}( \Omega(i),k)\Big{)}\Pr\Big{(}S_{\Omega(i)}=\mathbb{S}(\Omega(i),k)\Big{)}}{\sum _{i=1}^{m}\Pr\Big{(}T_{i}=\mathbb{T}(i,k)|S_{\Omega(i)}=\mathbb{S}(\Omega(i),k) \Big{)}\Pr\Big{(}S_{\Omega(i)}=\mathbb{S}(\Omega(i),k)\Big{)}}\]  
To make \(\mathbb{S}(\Omega(i),\cdot)\) easier to handle, let \(\mathbb{S}(\Omega(i),\cdot)\approx\mathbb{S}(i,\cdot)\), where the SNP represented by \(\mathbb{S}(i,\cdot)\) is the SNP that most relevant to \(\mathbb{T}(i,\cdot)\). Note that \(i_{1}\
eq i_{2}\Leftrightarrow\mathbb{T}(i_{1},\cdot)\
eq\mathbb{T}(i_{2}, \cdot),\mathbb{S}(i_{1},\cdot)\
eq\mathbb{S}(i_{2},\cdot)\), then we have \(m_{s}=m_{t}\) and \(n_{s}=n_{t}\) for the training set. In Equation (1), \(\Pr\Big{(}S_{\Omega(i)}=\mathbb{S}(\Omega(i),k)\Big{)}\) can be calculated using MAF, which is mentioned in Section 2.3, and may also be available from genetic database such as _dbSNP_[51]. \(\Pr\Big{(}T_{i}=\mathbb{T}(i,k)|S_{\Omega(i)}=\mathbb{S}(\Omega(i),k)\Big{)}\) is characterized by a continuous conditional probability density \(f_{\pi_{1}|S_{\Omega(i)}}\). Figure 1: Privacy-breaching model for trait-gene profile reidentification. The adversary first infers the correlations between continuous trait data and gene data (central box) with training datasets (upper box left) or existing research results (upper box right) and then maps the anonymous continuous trait profile (coloured with black in the left box) to the genotype (coloured with orange in the right box) with correlation obtained for reidentification. Then, the adversary verifies whether an individual with the continuous trait data is the member of the genotype dataset. If the matching between the continuous trait profile and genotype is sufficiently significant, it is considered correct (lower box, reidentified individuals are indicated as human shapes with the same colour of that in the right box). Each human shape represent an individual; each trait data or gene data in a box represent a continuous trait profile or a genotype profile from an individual. Both [PERSON] et al. [21] and [PERSON] et al. [20] verified that most of the continuous traits related to SNPs satisfied the normal distribution. However, unlike their solutions that used the normal distribution directly, since most continuous traits have non-negative values, the distribution with range \((0,+\infty)\) is more appropriate to model the continuous distribution functions. Specifically, let the mean and variance parameters of this truncated normal distribution be \(\mu_{i}\) and \(\sigma_{i}\), then its density is  
\[f_{T_{i}|S_{\Omega(i)}}=\frac{1}{\sigma_{i}}\frac{\phi\Big{(}\frac{T_{i}-\mu_{ i}}{\sigma_{i}}\Big{)}}{1-\Phi\Big{(}\frac{-\mu_{i}}{\sigma_{i}}\Big{)}}=\frac{ \phi\Big{(}\frac{T_{i}-\mu_{i}}{\sigma_{i}}\Big{)}}{\sigma_{i}\Phi\Big{(}\frac {\mu_{i}}{\sigma_{i}}\Big{)}}\]  
where \(\phi(\cdot)\) and \(\Phi(\cdot)\) are the probability density function (p.d.f.) and cumulative distribution function (c.d.f.) respectively.