Copula directed acyclic graphs

Abstract

A new methodology for selecting a Bayesian network for continuous data outside the widely used class of multivariate normal distributions is developed. The ‘copula DAGs’ combine directed acyclic graphs and their associated probability models with copula C/D-vines. Bivariate copula densities introduce flexibility in the joint distributions of pairs of nodes in the network. An information criterion is studied for graph selection tailored to the joint modeling of data based on graphs and copulas. Examples and simulation studies show the flexibility and properties of the method.

Appendix: Technical details

Assumptions of Proposition 4.1 adapted from Sin and White (1996). For every node l in the graph, define let $q_{lk}(\cdot ,\varvec{\theta })=\log CV_{l,k}(\varvec{\theta }_{CV_l})-\log DV_{l,k}(\varvec{\theta }_{DV_l})$, define $\tilde{q}_{lk}(\cdot ,\varvec{\theta })=\log DV_{l,k}(\varvec{\theta }_{DV_l})$ and $\text {log-Lik}(\cdot ,\varvec{\theta };\text {node}_l)\equiv Q_{ln}(\cdot ,\varvec{\theta })=\sum _{k=1}^{n}q_{lk}(\cdot ,\varvec{\theta })$ with $\varvec{\theta }=(\varvec{\theta }_{CV_l},\varvec{\theta }_{DV_l})$ and $k=1,2,\ldots , n$. For ease of exposition we state general conditions that need to be satisfied by $q_{lk}(\cdot ,\varvec{\theta }), \ \tilde{q}_{lk}(\cdot ,\varvec{\theta }), \ Q_{ln}(\cdot ,\varvec{\theta })$ and $\varvec{\theta }$ for every model m.

Let $(\Theta ,\mathcal {F},P)$ be a complete probability space and $\Theta$ be a compact subset of $\mathbb {R}^{d}$ with $d\in \mathbb {N}$. For all $n\in \mathbb {N}$ let $Q_{ln}:\Omega \times \Theta \rightarrow \mathbb {R}$ be such that:

• i $\forall \varvec{\theta }\in \Theta , \ Q_{ln}(\cdot ,\varvec{\theta })$ is $\mathcal {F}$-measurable.

• ii $\forall \omega \in A\in \mathcal {F}$ with $P(A)=1, \ Q_{ln}(\omega ,\cdot )$ is continuously differentiable on $\Theta$.

• iii The expectation $E(Q_{ln}(\cdot ,\varvec{\theta }))$ exists and defines a function which is continuously differentiable on $\Theta$ and $\bigtriangledown E(Q_{ln}(\cdot ,\varvec{\theta }))=E(\bigtriangledown Q_{ln}(\cdot ,\varvec{\theta }))$ where $\bigtriangledown$ is the gradient operator.

• iv The least false parameter defined by $\varvec{\theta }_{0n}=\arg \sup _{\varvec{\theta }\in \Theta }\frac{1}{n}E(Q_{ln}(\cdot ,\varvec{\theta }))$ is interior to $\Theta$ uniformly (in n).

• v Given $\epsilon >0$ there exists $N_0(\epsilon )<\infty$ and $\delta (\epsilon )>0$ such that $\inf \{\min \{K_n^{*}(\varvec{\theta }):\varvec{\theta }\in \mathcal {N}_n^{*}(\epsilon )^{c}\},n>N_0(\epsilon )\} \equiv \delta (\epsilon )$, where $K_n^{*}(\varvec{\theta })\equiv n^{-1}E(Q_{ln}(\cdot ,\varvec{\theta }_{0n}))-n^{-1}E(Q_{ln}(\cdot ,\varvec{\theta })), \ \mathcal {N}_n^{*}(\epsilon )^{c}$ is the compact complement of $\mathcal {N}_n^{*}(\epsilon ) \equiv \mathcal {S}^{*}_n(\epsilon )\cap \Theta$ in $\Theta$ and $\mathcal {S}^{*}_n(\epsilon )$ is an open sphere centered at $\varvec{\theta }_{0n}$ with fixed radius $\epsilon$.

• vi For P-almost all $\omega , \ q_{lk}(\omega ,\cdot )$ is twice continuously differentiable as a function of $\varvec{\theta }$, for $k=1,2,\ldots$

• vii $q_{lk}$ and $\tilde{q}_{lk}$ satisfy a uniform weak law of large numbers (UWLLN) on $\Theta$.

• viii Each element of $\bigtriangledown q_{lk}(\cdot ,\varvec{\theta }_{0n})$ satisfies a central limit theorem.

• ix $\exists \epsilon , \alpha >0$ such that for P-almost all $\omega$ and for all n sufficiently large and for all $\varvec{\theta } \in \mathcal {N}_n^{*}(\epsilon ), \det (n^{-1}\bigtriangledown ^{2}Q_{ln}(\omega ,\varvec{\theta }))\ge \alpha$, with $\mathcal {N}^{*}_n(\epsilon )$ as in Asumption v.

• x For all n sufficiently large and for all $\varvec{\theta } \in \mathcal {N}_n^{*}(\epsilon ), E[n^{-1}\bigtriangledown ^2Q_{ln}(\cdot ,\varvec{\theta })]$ is $\varvec{O}(1)$.

• xi Each element of $\bigtriangledown ^2 q_{ln}$ satisfies a UWLLN on $\mathcal {N}^{*}_n(\epsilon )$.

We assume that the copula densities are such that the above conditions are satisfied. These are basic assumptions that guarantee that $\hat{\varvec{\theta }}_n-\varvec{\theta }_{0n}=\varvec{O}_p(n^{-1/2})$ and $Q_n(\cdot ,\hat{\varvec{\theta }}_n)-Q_n(\cdot ,\varvec{\theta }_{0n})=\varvec{O}_p(1)$. The asymptotic normality of $\sqrt{n}(\hat{\varvec{\theta }}_n-\varvec{\theta }_{0n})$ for the models we consider has been shown in Hobæk Haff (2013).

1.1 Penalty conditions in Lemma 4 for the penalty in cDAG-IC

Proof

Define ${\varDelta }\widehat{\hbox {pen}}_{\mathrm{cDAG}} = \widehat{\hbox {pen}}_{\mathrm{cDAG}}^1(n,\hat{\varvec{\theta }}^1)- \widehat{\hbox {pen}}_{\mathrm{cDAG}}^2(n,\hat{\varvec{\theta }}^2)$. For (i) it holds that

$$\begin{aligned} \begin{aligned} {\varDelta }\widehat{\hbox {pen}}_{\mathrm{cDAG}}/n&=\left( \frac{E\log DV^{1}_{l}}{|pa^{1}(l)|}-\frac{E\log DV^{2}_{l}}{|pa^{2}(l)|}\right) \frac{1}{\log n}\\&\quad +o_{P}(1) = o_P(1). \end{aligned} \end{aligned}$$

The first equality holds due to Assumption vii.

For (ii) and (iii) it follows that

$$\begin{aligned} \begin{aligned}&\frac{{\varDelta }\widehat{\hbox {pen}}_{\mathrm{cDAG}}}{\sqrt{n}}=\Big (\frac{E\log DV^{1}_{l}}{|pa^{1}(l)|}-\frac{E\log DV^{2}_{l}}{|pa^{2}(l)|}\Big )\frac{\sqrt{n}}{\log n}+ o_P(\sqrt{n}),\\&{\varDelta }\widehat{\hbox {pen}}_{\mathrm{cDAG}}=\Big (\frac{E\log DV^{1}_{l}}{|pa^{1}(l)|}-\frac{E\log DV^{2}_{l}}{|pa^{2}(l)|}\Big )\frac{n}{\log n}+ o_P(n). \end{aligned} \end{aligned}$$

By the assumed positiveness of the penalty difference, the conditions hold. $\square$

Definition of ‘d-separation’ between $\mathcal {X}$ and $\mathcal {Y}$ by $\mathcal {Z}$ (Barber 2012). For every node $x \in \mathcal {X}$ and $y \in \mathcal {Y}$, check every path $\mathcal {U}$ between x and y (that is, a sequence of nodes that starts in x and by following the directionality of the arrows leads to y). A path $\mathcal {U}$ is blocked if there is a node w in $\mathcal {U}$ such that either: (i) w is a collider (a collider node has two incoming arrows to it) and neither w nor any of its descendants is in $\mathcal {Z}$, or (ii) w is not a collider on $\mathcal {U}$ and w is in $\mathcal {Z}$. If all such paths are blocked then the sets of nodes $\mathcal {X}$ and $\mathcal {Y}$ are d-separated by $\mathcal {Z}$. If the sets of nodes $\mathcal {X}$ and $\mathcal {Y}$ are d-separated by $\mathcal {Z}$, they are independent conditional on $\mathcal {Z}$.