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We consider two connected aspects of maximum likelihood estimation of theparameter for high-dimensional discrete graphical models: the existence of themaximum likelihood estimate (mle) and its computation.  When the data is sparse, there are many zeros in the contingency table andthe maximum likelihood estimate of the parameter may not exist. Fienberg andRinaldo (2012) have shown that the mle does not exists iff the data vectorbelongs to a face of the so-called marginal cone spanned by the rows of thedesign matrix of the model. Identifying these faces in high-dimension ischallenging. In this paper, we take a local approach : we show that one suchface, albeit possibly not the smallest one, can be identified by looking at acollection of marginal graphical models generated by induced subgraphs$G_i,i=1,\ldots,k$ of $G$. This is our first contribution.  Our second contribution concerns the composite maximum likelihood estimate.When the dimension of the problem is large, estimating the parameters of agiven graphical model through maximum likelihood is challenging, if notimpossible. The traditional approach to this problem has been local with theuse of composite likelihood based on local conditional likelihoods.  A more recent development is to have the components of the compositelikelihood be marginal likelihoods centred around each $v$. We first show thatthe estimates obtained by consensus through local conditional and marginallikelihoods are identical. We then study the asymptotic properties of thecomposite maximum likelihood estimate when both the dimension of the model andthe sample size $N$ go to infinity.

A local approach to estimation in discrete loglinear models