M5MS09. Graphical Modelling

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1 Course: MMS09 Setter: Walden Checker: Ginzberg Editor: Calderhead External: Wood Date: April, 0 MSc EXAMINATIONS (STATISTICS) May-June 0 MMS09 Graphical Modelling Setter s signature Checker s signature Editor s signature MMS09 Graphical Modelling (0) Page of

2 MSc EXAMINATIONS (STATISTICS) May-June 0 MMS09 Graphical Modelling Date: Friday st May 0 Time: Credit will be given for all questions attempted but extra credit will be given for complete or nearly complete answers. Calculators may not be used. c 0 Imperial College London MMS09 Page of

3 . (a) Consider X = [X, X, X, X ] T having pdf f X (x) = C exp(x + x x + x x x ) on the four dimensional cube 0 < x j <, j =,..., where the constant C ensures integration to unity. (i) Using the factorization theorem draw the corresponding conditional independence graph G. What are the cliques of the graph? Explain their relationship to the factorization of f X (x) according to G. (b) (i) Let G = (V, E) be an undirected graph and consider ( apple j < k apple p). Let A be the adjacency matrix such that A jk = if (j, k) E and zero otherwise. Define the Gaussian graphical model M(G). In the context of a Gaussian graphical model describe what is meant by the datagenerating distribution being faithful to the graph. (c) Let X,..., X N be a sample from a multivariate normal distribution N (0, ) in a Gaussian graphical model M(G) where G = (V, E) is an undirected graph with a single missing edge (, ). The sample covariance matrix and its inverse are b = 0 0 ; b = (i) Carefully describe the single-step Wermuth-Scheidt algorithm and use it to find the modified covariance matrix T such that T jk = 0 if (j, k) 6 E, where as usual T jk is the (j, k)th element of T. Where constraints are imposed by more than one missing edge how is the singlestep Wermuth-Scheidt algorithm modified? (Equations are not required, a short description is su cient). (d) Briefly describe the lasso-type regularization approach to estimating a Gaussian graphical model proposed by Yuan and Lin. What is the key property of the lasso-type approach? The BIC formula is used for determining an appropriate value for the constraint parameter. How are the degrees of freedom calculated in the BIC formula? MMS09 Graphical Modelling (0) Page of

4 Figure : DAG G!.. (a) (i) State the well-numbered pairwise directed Markov property determined by any DAG graph G!. (iii) Write down the five pairwise conditional independencies implied by the wellnumbered pairwise directed Markov property for the DAG G! in Fig.. Use the conditional independencies from to find the recursive factorization identity for the joint density f X,X,X,X,X in simplified form. Fully explain each step of the simplification. (b) (i) Consider a chain consisting of the vertices i 0, i,..., i m of a DAG G!. Define a collider and a noncollider. Define what it means for two of the vertices of the DAG G! to be d-separated. (iii) State the global directed Markov property of the DAG G!. (iv) Derive the five pairwise conditional independencies of (a) for the DAG G! in Fig. using the global directed Markov property, giving full justifications. MMS09 Graphical Modelling (0) Page of

5 . (a) (i) Define the undirected partial correlation graph for p-vector-valued stationary time series. In the building of a partial correlation graph researchers typically make use of partial coherence or the spectral matrix and its inverse. Why are such approaches favoured over use of the partial cross-covariance sequence? (b) Describe, in a step-by-step fashion, the building of a partial correlation graph for p- vector-valued stationary time series using the Kullback-Leibler divergence and multiple hypothesis testing. It is not necessary to go into full mathematical details regarding the exact form of the Kullback-Leibler divergence or the form of Matsuda s test statistic. The steps and statistical ideas/terms being used must be clearly explained/defined. MMS09 Graphical Modelling (0) Page of