Given an undirected graph where is a finite set and is the set of undirected edges, we say that the Gaussian variable is Markov with respect to if is independent of given all the other variables whenever there no edge in . It is well-known that in that case the precision matrix belongs to the cone of positive definite matrices with whenever . One can then define the graphical Gaussian model Markov with respect to a given graph as the family of distributions
Graphical Gaussian models form nowadays one of the basic tools used to analyze high-dimensional complex continuous data. Model selection in this class of models has thus been the topic of much research both from the frequentist and Bayesian point of view. The model is defined by both the graph and the precision matrix . Model search in the frequentist framework is done by maximizing a penalized likelihood (see Friedman et al. (2008)): this yields simultaneously the best (in that sense) and
by determining which entries of the covariance matrix are zero and estimating the others. In the Bayesian framework, model search has traditionally been based on the comparison of the posterior distribution of each model, each model being represented by a graph
. The selected models are the models with the highest posterior probabilities and the correspondingis then estimated through sampling of the posterior distribution of
. The conjugate prior foris the -Wishart as defined by Roverato (2002). The density of the -Wishart can be written as
where denotes the determinant of , is the inner product of and in the space of symmetric matrices and is the normalizing constant of the -Wishart which is finite for and . Given a sample
from the Gaussian distribution in, let . The marginal density of , which is also the unnormalized posterior density of given , is then
To do model selection, one must run a Markov chain or some stochastic search to move through the space of graphs, see(Jones et al., 2005, Scott and Carvalho, 2008, Wong et al., 2003)
. This approach is not feasible for high-dimensional data for two reasons: the Markov chains are slow and the prior and posterior normalizing constants for each graphvisited are hard to evaluate numerically.
Another approach to Bayesian model selection is to run a Markov chain on the joint space of , see (Giudici and Green, 1999) or , see (Dobra and Lenkoski, 2011, Dobra et al., 2011, Wang and Li, 2012, Mohammadi and Wit, 2015). The joint posterior distribution of is
The advantage of this approach is that the posterior normalizing constant does not come into play. To compute the acceptance probabilities in the chain, the only quantity which is computationally expensive is the ratio of prior normalizing constants
where is the graph obtained from by removing one edge and . A recent paper Uhler et al. (2014) gives the exact analytic expression of these integrals. However, this expression is complicated and impossible to implement at the present time. The purpose of this paper is to give an accurate approximation to (2) which, itself, leads to accurate model selection with minimal scale-free computational burden.
. The Laplace approximation is accurate only if the density of the prior distribution is similar in shape to a multivariate normal distribution and, like for the Wishart distribution, this requires that the shape parameterbe high. Since, in order to minimize the impact of the prior distribution on inference, is traditionally chosen to be , which is not high, this approximation is not accurate. Atay-Kayis and Massam (2005) expressed as the product of a constant and an expected value. For , this expression is
where, for a given order of the vertices, is the number of neighbours of vertex which have a numbering larger than or equal to , is the incomplete upper triangular matrix with entries the free entries of the Cholesky decomposition , is a function depending on and more particularly on
. The expected value is taken with respect to a product of independent standard normal and chi-square distributions. Evaluating (2) is therefore equivalent to evaluating
where is the edge set of the graph obtained from by removing the edge . We will show that, under reasonable conditions of sparsity, we have the following approximation of (2)
where is the number of minimal paths of length 2 linking the two end points of . Under the assumption that the minimal paths (that is paths without a chord) linking the two vertices of in are disjoint, we show in equation (22) below that the accuracy of this approximation depends on the number and on the length of the paths linking the two end vertices of the edge . We illustrate the accuracy of our approximation through simulations, using numerous configurations, and we show that if the number of paths of length 3 or more is not too large, say , the approximation is of the order of . A relative error of this order does not truly affect the model search. We will follow the method of Mohammadi and Wit (2015) to do a model search using both real and simulated data. We will see that using this approximation actually yields better precision for the identification of the true model than if we actually evaluate (4). This means that graphical Gaussian model selection can be done for high-dimensional data in a fast, scale-free and accurate manner with minimal computational burden.
Though, in the numerical examples considered, the assumption that the minimal paths linking the two end vertices of are disjoints is not satisfied, the approximation seems to work and the model search gives good results. We conjecture that an approximation similar to (22) holds in general. This is the topic of future work.
2 The ratio of prior normalizing constants
Let be an undirected graph and the graph obtained by removing a given edge from . Without loss of generality, in this section, we will assume that the numbering of the vertices is such that . The aim of this section is to express ratio (4) in such a way that, in the next section, we can deduce an approximation to it. This is done in Proposition 2.1. As we shall see below, the accuracy of this approximation depends on the number of minimal paths between the two vertices and of . A path between and is said to be minimal if it has no chord.
In a first step, we recall how (3) was obtained in Atay-Kayis and Massam (2005). Let be an upper triangular matrix with positive diagonal elements such is the Cholesky decomposition of . Let be the complement of in the set of all possible edges in a graph with vertex set , i.e. indexes the missing edges of . Similarly denotes the missing edges of . It was shown that
are free variables in the sense that there is a 1-1 correspondence between and and that can be expressed in terms of and are thus non-free variables. Then, with a change of variables from to , we have
This expression immediately yields (3) with
where the expected value is taken with respect to a product of independent for and distribution for . It remains to give the expression of the non-free variables in terms of the free ones. Equation (31) in Atay-Kayis and Massam (2005), in the particular case where , yields
This equation shows that each non-free variables is equal to multiplied by the sum, over the rows which are numbered to of the products of the entries in the column and the column .
It is well-known (see formula (22) in Atay-Kayis and Massam (2005)) that, if is a perfect sequence of prime components of with corresponding separators and induced graphs , then can be decomposed as
where and are the submatrices of indexed by the vertices of and respectively. In a second step, we show that the ratio in (2) can be reduced to the ratio
where is the graph induced by and all the vertices contained in the minimal path between and and not containing the edge while is the graph obtained from by adding the edge . We say that a path between and is minimal if this path has no chord. Indeed, if a vertex is not on a path linking and , it is either not linked to either or or it is linked to one of them but not both. In the first case, the graph is disconnected and we clearly only have to consider the connected part of containing and . In the second case, assuming, without loss of generality, that is not linked to , means that there is a separator between the prime components containing and the prime components containing . Since in both and , the prime component are the same, their corresponding terms will cancel out in (8) above and so will the terms for the corresponding separators. Finally if belongs to a path between and but not on a minimal path, then such that is a chord. Then the graph induced by the path and the chord is a prime component which will appear in both the factorization of and . This proves that in our study of (2), it is sufficient to consider a graph which contains only the minimal paths between and . To keep notation simple, from now in this paper, we will write and for and respectively.
In a third step, we will set up the numbering of the vertices in a convenient way. From now on, we assume that is the graph induced by the minimal paths between and . Let be the set of these minimal paths. A path between and with vertices between and will be written
We let , and be, respectively, the set of edges, the set of interior vertices of and the set of interior points deprived of , i.e.
If is the total number of minimum paths, we set an arbitrary order of the path where, for convenience, we list the paths of length 2, i.e. last. Within each path , we order the vertices in following the path. The vertices and are ranked last so that the order of the vertices is
We are now ready to state the following.
Since is equal to the cardinality of , since the only change between and is the edge and since the neighbours of are the same in and , given the order (9), only is different in and . Indeed while . This yields immediately equation (10). Let us now prove (11). From (7) and given (9), we have
We want to compute the entries in terms of
We now note three important facts. First, the elements of the first row of the matrix are all zero except for those corresponding to the edges of the path , i.e.
Third, due to the first entry of column being free, none of the entries of column are necessarily zero. However, for each , using iteratively (7), we see that the entries of column are zero except for the last one which is a free variable,i.e.
which is identical to (11). ∎
Consider the graph of Figure 1 where for simplicity, we write for
The matrix is as follows.
where the entries marked with a are the non-free entries and are given as follows:
3 An approximation to the ratio of normalizing constants
3.1 The results
From (5), we see that is the sum of ratios of product of independent normal by the square root of a chi-square distribution with degrees of freedom and similarly is the sum of ratios of products of standard normals by a product of square roots of chi-square distributions. To obtain an approximation to (21), we proceed in two steps. We will first show that is a good approximation to We will then see that is independent of and thus
which can easily be obtained explicitly. The following theorem is the main result of this paper and yields an accurate approximation to ratio (2).
The approximation (24) gives an analytically explicit expression for ratio (2) and thus removes the need to do the simulations that were previously necessary to the evaluation of (4) in model search. This makes the search algorithm scale free. We will illustrate this fact in Section 4. Before doing so and before giving the proof of Theorem 3.1 above, we give, in Table 1 the value of under different scenarios for graphs having five different paths between and .
|0.0136 (0)||0.0036 (0)||0.0014 (0)|
|0.0333 (0)||0.0213 (0)||0.0186 (0)|
|0.0591 ()||0.0452 (0)||0.0411 (0)|
|0.0944 ()||0.0797 (0)||0.076 ()|
|0.1541 ()||0.1369 (0)||0.1323 ()|
3.2 The proof
We express here and in terms of