A MCMC Approach for Learning the Structure of Gaussian Acyclic Directed Mixed Graphs

Abstract

Graphical models are widely used to encode conditional independence constraints and causal assumptions, the directed acyclic graph (DAG) being one of the most common families of models. However, DAGs are not closed under marginalization: that is, if a distribution is Markov with respect to a DAG, several of its marginals might not be representable with another DAG unless one discards some of the structural independencies. Acyclic directed mixed graphs (ADMGs) generalize DAGs so that closure under marginalization is possible. In a previous work, we showed how to perform Bayesian inference to infer the posterior distribution of the parameters of a given Gaussian ADMG model, where the graph is fixed. In this paper, we extend this procedure to allow for priors over graph structures.

Appendix

We describe the parameters referred to in the sampler of Sect. 3. The full derivation is based on previous results described by Silva and Ghahramani (2009). Let HH be the statistic $\sum _{n=1}^{d}\mathbf{H}_{\mathit{sp}(i)}^{(d)}{\mathbf{H}_{\mathit{sp}(i)}^{(d)}}^{T}$. Likewise, let $\mathbf{YH} \equiv \sum _{n=1}^{d}Y _{i}^{(d)}\mathbf{H}_{\mathit{sp}(i)}^{(d)}$ and $\mathbf{YY} \equiv \sum _{n=1}^{d}{Y _{i}^{(d)}}^{2}$. Recall that the hyperparameters for the $\mathcal{G}$-inverse Wishart are δ and U, as given by Eq. (1) and as such we are computing a “conditional normalizing constant” for the posterior of V integrating over only one of the row/columns of V.

First, let

$$\displaystyle{ \begin{array}{rcl} \mathbf{A}_{i}& \equiv &\mathbf{V}_{\mathit{sp}(i),\mathit{nsp}(i)}\mathbf{V}_{\mathit{nsp}(i),\mathit{nsp}(i)}^{-1} \\ \mathbf{M}_{i}& \equiv &{(\mathbf{U}_{\setminus i,\setminus i})}^{-1}\mathbf{U}_{\setminus \mathit{i,i}} \\ \mathbf{m}_{i}& \equiv &(\mathbf{U}_{\mathit{ss}} -\mathbf{A}_{i}\mathbf{U}_{\mathit{ns}})\mathbf{M}_{\mathit{sp}(i)} + (\mathbf{U}_{\mathit{sn}} -\mathbf{A}_{i}\mathbf{U}_{\mathit{nn}})\mathbf{M}_{\mathit{nsp}(i)}\\ & & \\ \mathbf{K}_{\mathcal{B}}^{-1} & \equiv &\mathbf{U}_{\mathit{ss}} -\mathbf{A}_{i}\mathbf{U}_{\mathit{ns}} -\mathbf{U}_{\mathit{sn}}\mathbf{A}_{i}^{T} + \mathbf{A}_{i}\mathbf{U}_{\mathit{nn}}\mathbf{A}_{i}^{T} \\ \mu _{\mathcal{B}}&\equiv &\mathbf{K}_{\mathcal{B}}\mathbf{m}_{i}\\ \end{array} }$$

(11)

where

$$\displaystyle\begin{array}{rcl} & & \left [\begin{array}{cc} \mathbf{U}_{\mathit{ss}} & \mathbf{U}_{\mathit{sn}} \\ \mathbf{U}_{\mathit{ns}} & \mathbf{U}_{\mathit{nn}} \end{array} \right ] \equiv \left [\begin{array}{cc} \mathbf{U}_{\mathit{sp}(i),\mathit{sp}(i)} & \mathbf{U}_{\mathit{sp}(i),\mathit{nsp}(i)} \\ \mathbf{U}_{\mathit{nsp}(i),\mathit{sp}(i)} & \mathbf{U}_{\mathit{nsp}(i),\mathit{nsp}(i)} \end{array} \right ]{}\end{array}$$

(12)

Moreover, let

$$\displaystyle\begin{array}{rcl} & & \begin{array}{rcl} \mathcal{U}_{i}& \equiv &\mathbf{M}_{i}^{T}\mathbf{U}_{\setminus i,\setminus i}\mathbf{M}_{i} -\mathbf{m}_{i}^{T}\mathbf{K}_{i}\mathbf{m}_{i} \\ u_{\mathit{ii}.\setminus i}& \equiv &\mathbf{U}_{\mathit{ii}} -\mathbf{U}_{i,\setminus i}{(\mathbf{U}_{\setminus i,\setminus i})}^{-1}\mathbf{U}_{\setminus \mathit{i,i}} \\ & & \\ \alpha _{i}& \equiv &\left (\delta +p - 1 + \#\mathit{nsp}(i)\right )/2 \\ \beta _{i}& \equiv &\left(u_{\mathit{ii}.\setminus i} + \mathcal{U}_{i}\right )/2 \\ \mathbf{T}& \equiv &\mathbf{K}_{\mathcal{B}}^{-1} + \mathbf{HH} \\ \mathbf{q}& \equiv &\mathbf{YH} + \mathbf{K}_{\mathcal{B}}^{-1}\mu _{\mathcal{B}} \end{array} {}\end{array}$$

(13)

where #nsp(i) is the number of non-spouses of Y _i (i.e., $(p - 1) -\sum _{j=1}^{p}z_{\mathit{ij}}$).

Finally,

$$\displaystyle\begin{array}{rcl} & & \begin{array}{rcl} \alpha _{i}^{\prime}& \equiv &\frac{N} {2} +\alpha _{i}, \\ \beta _{i}^{\prime}& \equiv &\frac{\mathbf{YY} +\mu _{ \mathcal{B}}^{T}\mathbf{K}_{\mathcal{B}}^{-1}\mu _{\mathcal{B}}-{\mathbf{q}}^{T}{\mathbf{T}}^{-1}\mathbf{q}} {2} +\beta _{i} \end{array} {}\end{array}$$

(14)

Notice that each calculation of A _i (and related products) takes $\mathcal{O}({p}^{3})$ steps (assuming the number of non-spouses is $\mathcal{O}(p)$ and the number of spouses is $\mathcal{O}(1)$, which will be the case in sparse graphs). For each vertex Y _i , an iteration could take $\mathcal{O}({p}^{4})$ steps, and a full sweep would take prohibitive $\mathcal{O}({p}^{5})$ steps. In order to scale this procedure up, some tricks can be employed. For instance, when iterating over each candidate spouse for a fixed Y _i , the number of spouses increases or decreases by 1: this means fast matrix update schemes can be implemented to obtain a new A _i from its current value. However, even in this case the cost would still be $\mathcal{O}({p}^{4})$. More speed-ups follow from solving for $\mathbf{V}_{\mathit{sp}(i),\mathit{nsp}(i)}\mathbf{V}_{\mathit{nsp}(i),\mathit{nsp}(i)}^{-1}$ using sparse matrix representations, which should cost less than $\mathcal{O}({p}^{3})$ (but for small to moderate p, sparse matrix inversion might be slower than dense matrix inversion). Moreover, one might not try to evaluate all pairs $Y _{i} \leftrightarrow Y _{j}$ if some pre-screening is done by looking only at pairs where the magnitude of corresponding correlation sampled in the last step lies within some interval.