Summary Graphical models oer simple and intuitive interpretations in terms of conditional indepen... more Summary Graphical models oer simple and intuitive interpretations in terms of conditional independence relationships, and these are especially valuable when large numbers of variables are involved. In some settings restrictions upon experiments, number of variables, and other forms of data collection may result in our being able to estimate only parts of a large graphical model. Consider a collectionC of submodels of a decomposable graphG. In this article, we address the problem of combining component graphical models, and a theory is derived to the eect that one can combine the collection C of decomposable graphs,G1;G2; ;Gm, into a larger decomposable graph, H, of the variables that are involved in G so that the conditional independence relationships inG1;G2; ;Gm may be preserved inH. It is also shown that theH which contains the actual graphG as a subgraph is determined uniquely.
Journal of Statistical Mechanics: Theory and Experiment, 2018
Dynamic networks are ubiquitous in the world. So far, many dynamic network models have been devel... more Dynamic networks are ubiquitous in the world. So far, many dynamic network models have been developed in search of network growth mechanisms at the node and edge levels. Especially, a number of fitness models have been employed for analysis of fitness (i.e. a node's inherent ability or characteristics) and popularity eects on growing networks. However, these models are not suitable for comparing the magnitude of the fitness and popularity eects. We propose a statistical dynamic network model called a fitness-popularity dynamic network (FPDN) model, where fitness and popularity eects are on equal footing. These eects are estimated under the FPDN model and the estimation procedure are applied to the network data, Flickr following, Facebook wallpost, and arXiv citation. The estimates of the two eects seem to represent the characters of the three networks with noteworthy interpretations. It is interesting to see that the popularity of a node negatively aects the growth of the in-degree of the node for the arXiv citation network while the eect is positive for the other networks.
Consider a rank-ordering problem, ranking a group of subjects by the conditional probability from... more Consider a rank-ordering problem, ranking a group of subjects by the conditional probability from a Bayesian network (BN) model of binary variables. The conditional probability is the probability that a subject is in a certain state given an outcome of some other variables. The classification is based on the rank order and the class levels are assigned with equal proportions. Two BN models are said to be similar to each other if they are of the same model structure but with different probability distributions each of which satisfies the positive association condition. Let \({\mathcal M}\) be a set of BN models which are similar to each other. We constructed a BN model M∗, which is similar to all the models in \({\mathcal M}\) and the best with regard to \({\mathcal M}\) in the sense of the Kullback-Leibler divergence measure. It is found by numerical experiments that, on average, the agreement rate of classifications between a model in \({\mathcal M}\) and the similar model M∗ is far larger than that by a random classification and the difference in agreement rate becomes more apparent as the class number increases.
We propose an algorithm for combining decomposable graphical models and apply it for building dec... more We propose an algorithm for combining decomposable graphical models and apply it for building decomposable graphical log-linear models which involve a large number of variables. A main idea in this algorithm is that we group the random variables that are involved in the data into several subsets of variables, build graphical log-linear models for the marginal data, and then combine the marginal models using graphs of prime separators (section 2). The application of the algorithm to a data set of 40 binary variables is very successful, yielding a model which is mostly the same as the true one.
A Bayes Shrinkage Estimation Method for Vector Autoregressive Models
Modelling, Identification and Control / 770: Advances in Computer Science and Engineering, 2012
Model Similarity and Rank-Order Based Classification of Bayesian Networks
Journal of Classification, 2013
ABSTRACT Suppose that we rank-order the conditional probabilities for a group of subjects that ar... more ABSTRACT Suppose that we rank-order the conditional probabilities for a group of subjects that are provided from a Bayesian network (BN) model of binary variables. The conditional probability is the probability that a subject has a certain attribute given an outcome of some other variables and the classification is based on the rank-order. Under the condition that the class sizes are equal across the class levels and that all the variables in the model are positively associated with each other, we compared the classification results between models of binary variables which share the same model structure. In the comparison, we used a BN model, called a similar BN model, which was constructed under some rule based on a set of BN models satisfying certain conditions. Simulation results indicate that the agreement level of the classification between a set of BN models and their corresponding similar BN model is considerably high with the exact agreement for about half of the subjects or more and the agreement up to one-class-level difference for about 90% or more.
Hyper-EM for large recursive models of categorical variables
Computational Statistics & Data Analysis, 2000
ABSTRACT When a recursive model is of a manageable size on a computing machine to use, whether it... more ABSTRACT When a recursive model is of a manageable size on a computing machine to use, whether it involves latent variables or not matters little. Application of an EM algorithm for the model is straightforward. But when the model is large enough to reach or exceed the storage space and contains latent variables, parameter estimation for the model looks almost infeasible. In this paper, a new EM approach is proposed for large recursive models and its convergence is proved. A key idea behind it is (1) that we partition a model into several submodels in such a way that the variables of submodel A, say, are conditionally independent of the other variables in the model given that the values of the variables of submodel A which are involved in any other submodels are known and (2) that the likelihood function for the whole model is factorized by the submodels.
Computational Statistics & Data Analysis, 2009
The Pearson's chi-squared statistic (X 2 ) does not in general follow a chi-square distribution w... more The Pearson's chi-squared statistic (X 2 ) does not in general follow a chi-square distribution when it is used for goodness-of-fit testing for a multinomial distribution based on sparse contingency table data. We explore properties of Zelterman's (1987) D 2 statistic and compare them with those of X 2 and we also compare these two statistics and the statistic (L r ) which is proposed by Maydeu-Olivares and Joe (2005) in the context of power of the goodness-of-fit testing when the given contingency table is very sparse. We show that the variance of D 2 is not larger than the variance of X 2 under null hypotheses where all the cell probabilities are positive, that the distribution of D 2 becomes more skewed as the multinomial distribution becomes more asymmetric and sparse, and that, as for the L r statistic, the power of the goodness-of-fit testing depends on the models which are selected for the testing. A simulation experiment strongly recommends to use both D 2 and L r for goodness-of-fit testing with large sparse contingency table data.
Structure learning for Bayesian networks has been made in a heuristic mode in search of an optima... more Structure learning for Bayesian networks has been made in a heuristic mode in search of an optimal model to avoid an explosive computational burden. In the learning process, a structural error which occurred at a point of learning may deteriorate its subsequent learning. We proposed a remedial approach to this error-for-error process by using marginal model structures. The remedy is made by fixing local errors in structure in reference to the marginal structures. In this sense, we call the remedy a marginally corrective procedure. We devised a new score function for the procedure which consists of two components, the likelihood function of a model and a discrepancy measure in marginal structures. The proposed method compares favourably with a couple of the most popular algorithms as shown in experiments with benchmark data sets.
We explore the properties of subgraphs (called Markovian subgraphs) of a decomposable graph under... more We explore the properties of subgraphs (called Markovian subgraphs) of a decomposable graph under some condition. For a decomposable graph G and a collection γ of its Markovian subgraphs, we show that the set χ(G) of the intersections of all the neighboring cliques of G contains ∪g∈γχ(g). We also show that χ(G) = ∪g∈γχ(g) holds for a certain type of G which we call a maximal Markovian supergraph of γ.
Suppose that we rank-order the conditional probabilities for a group of subjects that are provide... more Suppose that we rank-order the conditional probabilities for a group of subjects that are provided from a Bayesian network (BN) model of binary variables. The conditional probability is the probability that a subject has a certain attribute given an outcome of some other variables and the classification is based on the rankorder. Under the condition that the class sizes are equal across the class levels and that all the variables in the model are positively associated with each other, we compared the classification results between models of binary variables which share the same model structure. In the comparison, we used a BN model, called a similar BN model, which was constructed under some rule based on a set of BN models satisfying certain conditions. Simulation results indicate that the agreement level of the classification between a set of BN models and their corresponding similar BN model is considerably high with the exact agreement for about half of the subjects or more and the agreement up to one-class-level difference for about 90% or more.
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Papers by Sung-Ho Kim