Partial Order MCMC for Structure Discovery in Bayesian Networks

We present a new Markov chain Monte Carlo method for estimating posterior probabilities of structural features in Bayesian networks. The method draws samples from the posterior distribution of partial orders on the nodes; for each sampled partial order, the conditional probabilities of interest are computed exactly. We give both analytical and empirical results that suggest the superiority of the new method compared to previous methods, which sample either directed acyclic graphs or linear orders on the nodes.Peer reviewe

Joint Multiple Testing Procedures for Graphical Model Selection with Applications to Biological Networks

Gaussian graphical models have become popular tools for identifying relationships between genes when analyzing microarray expression data. In the classical undirected Gaussian graphical model setting, conditional independence relationships can be inferred from partial correlations obtained from the concentration matrix (= inverse covariance matrix) when the sample size n exceeds the number of parameters p which need to estimated. In situations where n \u3c p, another approach to graphical model estimation may rely on calculating unconditional (zero-order) and first-order partial correlations. In these settings, the goal is to identify a lower-order conditional independence graph, sometimes referred to as a ‘0-1 graphs’. For either choice of graph, model selection may involve a multiple testing problem, in which edges in a graph are drawn only after rejecting hypotheses involving (saturated or lower-order) partial correlation parameters. Most multiple testing procedures applied in previously proposed graphical model selection algorithms rely on standard, marginal testing methods which do not take into account the joint distribution of the test statistics derived from (partial) correlations. We propose and implement a multiple testing framework useful when testing for edge inclusion during graphical model selection. Two features of our methodology include (i) a computationally efficient and asymptotically valid test statistics joint null distribution derived from influence curves for correlation-based parameters, and (ii) the application of empirical Bayes joint multiple testing procedures which can effectively control a variety of popular Type I error rates by incorpo- rating joint null distributions such as those described here (Dudoit and van der Laan, 2008). Using a dataset from Arabidopsis thaliana, we observe that the use of more sophisticated, modular approaches to multiple testing allows one to identify greater numbers of edges when approximating an undirected graphical model using a 0-1 graph. Our framework may also be extended to edge testing algorithms for other types of graphical models (e.g., for classical undirected, bidirected, and directed acyclic graphs)

Event log visualisation with conditional partial order graphs: from control flow to data

Process mining techniques rely on event logs: the extraction of a process model (discovery) takes an event log as the input, the adequacy of a process model (conformance) is checked against an event log, and the enhancement of a process model is performed by using available data in the log. Several notations and formalisms for event log representation have been proposed in the recent years to enable efficient algorithms for the aforementioned process mining problems. In this paper we show how Conditional Partial Order Graphs (CPOGs), a recently introduced formalism for compact representation of families of partial orders, can be used in the process mining field, in particular for addressing the problem of compact and easy-to-comprehend visualisation of event logs with data. We present algorithms for extracting both the control flow as well as the relevant data parameters from a given event log and show how CPOGs can be used for efficient and effective visualisation of the obtained results. We demonstrate that the resulting representation can be used to reveal the hidden interplay between the control and data flows of a process, thereby opening way for new process mining techniques capable of exploiting this interplay.Peer ReviewedPostprint (author's final draft