To break this equivalence and uncover a more specific causal gene regulation network, genotype data is incorporated in the model inference process. The most straightforward approach is to use any of the methods in the previous section to calculate the probability $P(X_i\to X_j)$ of a causal interaction from $X_i$ to $X_j$ [@zhu2004; @zhu2008; @zhu2012stitching; @zhang2013integrated], for example by defining the prior as $p({\mathcal{G}})=\prod_{X_i}\left(\prod_{X_j\in{\mathbf{Pa}}^{({\mathcal{G}})}(X_i)}P(X_j\to X_i)
\prod_{X_j\not\in{\mathbf{Pa}}^{({\mathcal{G}})}(X_i)}(1-P(X_j\to X_i))\right)$. A more ambitious approach is to jointly learn the eQTL associations and causal trait (i.e. gene or phenotype) networks. In [@neto2010causal], EM is used to alternatingly map eQTLs given the current DAG structure, and update the DAG structure and model parameters given the current eQTL mapping. In [@scutari2014multiple], Bayesian networks are learned where SNPs and traits both enter as variables in the model, with the constraint that traits can depend on SNPs, but not vice versa. However, the additional complexity of both methods means that they are computationally expensive and have only been applied to problems with a handful of traits [@neto2010causal; @scutari2014multiple].

A few additional “tips and tricks” are worth mentioning:

-   First, when the number of vertices is much larger than the sample count, we may break the problem into independent sub-problems 