
Transitive ProjectionsRNA interference (RNAi) is a genesilencing technique using small interfering RNAs. It has been applied as an experimental technique to "knockout" genes in model organisms like Drosophila melanogaster and C.elegans, for determining the function of a gene. Largescale RNAi data are available for such model organisms. It is known that gene silencing leads to a number of distinct phenotypes. Our objective is to build a gene hierarchy from the nested structure of affected phenotypes. To reconstruct pathway features from nested structure of affected downstream genes Markowetz et al. 2005 develop a score based model on the marginal likelihood P(effectsmodel). Mathematically, a model is a transitively closed graph with the nodes corresponding to genes. However with large number of genes exhaustive enumeration of all models become impractical. To tackle this we introduce an edgewise strategy to learn a model where by for each pair of genes (A,B) we decide between the four models: "no edge between A and B", "an edge from A to B", "an edge from B to A", "double arrowed edge between A and B". In learning so we encounter with the problem that transitivity of the model may be lost. Does the figure show a false positive edge or a false negative edge? The challenge therefore is to find the nearest transitively closed graphs which we call "transitive projections" of the graph. In other words, for a given edgewise learned graph using minimum number of edge additions, deletions or both, we aim to find transitively closed graphs. To address this problem we propose an algorithm based on stepwise building of transitively closed graphs starting with an empty graph. We use the fact that by adding a new edge to a graph the distance to a fixed graph either increases or decreases by one. 

