Collapsibility For Chain Graphs

In recent years,graphical models, which can well explain a lot of problem such as multivariate causal relationships,develop dramatically. Graphical models including inde-pendence graphs,directed acyclic graphs (DAG)and Bayesian networks have been applied widely to many fields,such as data mining,pattern recognition,artificial intelligence and causal discovery. Graphical models can be used to cope with uncertainty for a large sys-tem with a great number of variables.Markov network is a graph model whose variables are simple symmetrical correla-tive and consist of undirected edges only and Bayesian network has variables which have causal relationships with each other and consist of directed edges only.In fact,variables often couldn’t predicate their causal relationships due to lack of prior distribution in-formation,so we have another statistical graph model which includes both two kinds of relationships,we call it chain graphs.chain graphs have wider application than undirected graphs and directed acyclic graphs even though it is more complex.Collapsibility for statistical graph model means that the inference outcome will not changed marginalization over some variables.It is much more efficient when graph models satisfy this kind of characteristic.There are three kinds of collapsibility in chain graphs,estimate collapsibility,conditional independence collapsibility and model collapsibility. Estimates collapsibility means that the maximum likelihood estimates for variables from subgraph are equivalent to the maximum likelihood estimates from whole graph.For a variable V,we have: P(XV\{v})=PGV\{v}(XV\{v}),Estimates collapsibility is stringent comparing with other two types of collapsi-bility,it not only needs variables have same conditional independence but also numerical precisely equivalence.Conditional independence collapsibility require variables have same conditional independence restriction in whole variable set and subset.If we chose Ⅰ(GV) to represent the conditional independence restrict of GV,then conditional independence collapsibility can be written as: I(GV)V\{v}=I(GV\{v}),.In aspect of removable,conditional independence collapsibility is relate to estimate col-lapsibility,and the condition of conditional independence collapsibility is weaker than estimate collapsibility.we also mention that model collapsibility is another collapsibility in statistical graph model,model collapsibility means the distributions from whole set and subset are same,i.e. for p(x)∈M, we have p(xB)∈MB,we can write as: M(GV)B=M(GB).If we have model collapsibility we also have conditional independence collapsibility and if we restrict the distribution to contingency tables or Gaussian distribution,this two kinds of collapsibility are equivalence.In this essay,we use three parts to introduce the collapsibility of chain graph model.In part2,we introduce estimate collapsibility for chain block.In section§2.1,we intro-duce the background of estimate collapsibility.In section§2.2,we represent some concepts and notions of estimate collapsibility of chain block.In section§2.3,we give the theory of estimate collapsibility for chain graph and prove that its necessary and sufficient condi-tion by drawing the concept of c-removable into collapsibility.In part3,we discuss the other two kinds of collapsibility.We also discuss the rela-tionships among three kinds of collapsibility of chain block and give a algorithm to find minimum block set which estimate are collapsible.In section§3.1we introduce the back-ground of two collapsibility for chain block and In§3.2section we represent some concepts and notions of estimate collapsibility of chain block.In§3.3section, we discuss conditional independence collapsibility for chain graph.In this section, we first redefine a new kind of removable characteristic, t-removable,which is weaker than c-removable and we use t-removable to prove conditional independence collapsibility for chain graph block. And then we introduce the concept of model collapsibility for chain graph and its conditions.In§3.4section,we discuss the relationships of estimate collapsibility,conditional indepen- dence collapsibility and model collapsibility.And for estimate collapsibility, we show a algorithm to look for a minimum chain block set which meet estimate collapsibility.