model selection to identify the most likely causal model, with the probability density functions (PDF) for the models below:

-   $p(E,A,B)=p(E)p(A\mid E)p(B\mid A)$,

-   $p(E,A,B)=p(E)p(B\mid E)p(A\mid B)$,

-   $p(E,A,B)=p(E)p(A\mid E) p(B\mid E,A)$,

where the dependence on $A$ in the last term of the last model indicates that there may be a residual correlation between $B$ and $A$ not explained by $E$. The minimal additive model assumes the distributions are [@schadt2005integrative] [$$\begin{aligned}
  E&\sim&\mathrm{Bernoulli}(q),\nonumber\\
  A\mid E&\sim&\mathrm{N}(\mu_{A\mid E},\sigma_A^2),\nonumber\\
  B\mid A&\sim&\mathrm{N}\left(\mu_B+\rho\frac{\sigma_B}{\sigma_A}(A-\mu_A),(1-\rho^2)\sigma_B^2\right),\nonumber\\
  B\mid E,A&\sim&\mathrm{N}\left(\mu_{B\mid E}+\rho\frac{\sigma_B}{\sigma_A}(A-\mu_{A\mid E}),(1-\rho^2)\sigma_B^2\right),\nonumber\end{aligned}$$]{} so that $E$ fulfills a Bernoulli distribution, $A\mid E$ undergoes a normal distribution whose mean depends on $E$, and that $B\mid A$ has a conditional normal distribution whose mean and variance are contributed in part by $A$. For $(B\mid E,A)$, the mean of $B$ also depends on $E$. The parameters of all distributions can be estimated by maximum likelihood, and the model with the highest likelihood is selected as the most likely causal model. The number of free parameters can be accounted using penalties like the Akaike information criterion (AIC) [@schadt2005integrative].

The approach has been extended in various ways. In [@chen2007harnessing], likelihood ratio tests, comparison to randomly permuted data, and false discovery rate estimation techniques are used to convert 