## Generating Mixed Hierarchical Interaction Models by Selection (1999)

Citations: | 2 - 1 self |

### BibTeX

@TECHREPORT{Lauritzen99generatingmixed,

author = {Steffen L. Lauritzen},

title = {Generating Mixed Hierarchical Interaction Models by Selection},

institution = {},

year = {1999}

}

### OpenURL

### Abstract

: This note is concerned with the class of hierarchical interaction models for mixed discrete and continuous variables as defined by Edwards (1990) and modified by Lauritzen (1996). In particular it is shown that any hierarchical log-linear interaction model can be generated by selection on a set of response variables in a directed Markov model over what we have termed the selection graph of the model. An inequality is established for the entries in the concentration matrix of any Gaussian undirected Markov distribution obtained by conditioning on the values of the response variables in the selection graph, thus demonstrating that not all such distributions can be generated in this way. Finally it is shown that in the mixed case only hierarchical models of the type defined by Edwards (1990) can be generated by selection as above. KEYWORDS: Bayesian networks; Conditional Gaussian distribution; Covariance selection; Gaussian graphical models; Log-linear interaction models; Recursive mode...

### Citations

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Citation Context ...less readily interpreted than undirected graphical models (Darroch, Lauritzen and Speed, 1980) or recursive graphical models (Wermuth and Lauritzen, 1983) also known as Bayesian networks (Pearl 1986, =-=Jensen 1996-=-). In the following we shall investigate the possibility of interpreting hierarchical models through a selection process in simple recursive models. We begin by describing the basic elements of Condit... |

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Citation Context ...they remain less readily interpreted than undirected graphical models (Darroch, Lauritzen and Speed, 1980) or recursive graphical models (Wermuth and Lauritzen, 1983) also known as Bayesian networks (=-=Pearl 1986-=-, Jensen 1996). In the following we shall investigate the possibility of interpreting hierarchical models through a selection process in simple recursive models. We begin by describing the basic eleme... |

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Citation Context ...ith k fl = 0 unless flsin G. Equivalently, Y fl and Ysare assumed to be conditionally independent unless flsin G. An undirected graphical Gaussian model is also known as a covariance selection model (=-=Dempster, 1972-=-). In analogy of the discrete case we shall investigate to which extent an arbitrary covariance selection model can be obtained from a recursive Gaussian graphical model by selection, i.e. by conditio... |

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Citation Context ... demands: 4. If / d () flfl is restricted to be zero, then so is / d () fl be for all . We shall refer to models given by restrictions that satisfy 1 through 4 as MIM models, because the program MIM (=-=Edwards, 1995-=-) is based upon such models. However, as noted in the discussion of Edwards (1990) there seems to be no immediate justification for the conditon 4 above and indeed there are sensible models that are n... |

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Citation Context ... with a directed acyclic graph (DAG) as such a graph can be thought of as representing a generating mechanism (Cox and Wermuth, 1996) or it can be associated with a causal theory among the variables (=-=Pearl, 1995-=-). Below we shall investigate to what extent hierarchical interaction models can be generated from simple recursive models by selection on one or more response variables. Generating Mixed Hierarchical... |

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Citation Context ...terpretable graphical models are the recursive models , i.e. the Markov models associated with a directed acyclic graph (DAG) as such a graph can be thought of as representing a generating mechanism (=-=Cox and Wermuth, 1996-=-) or it can be associated with a causal theory among the variables (Pearl, 1995). Below we shall investigate to what extent hierarchical interaction models can be generated from simple recursive model... |

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