## Compatible Prior Distributions for DAG models (2002)

Citations: | 7 - 2 self |

### BibTeX

@MISC{Roverato02compatibleprior,

author = {Alberto Roverato and Guido Consonni},

title = {Compatible Prior Distributions for DAG models},

year = {2002}

}

### OpenURL

### Abstract

The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given DAG (Directed Acyclic Graph) model, using a conditioning approach. We define a class of parameterisations consistent with the modular structure of the DAG and derive a procedure, invariant within this class, which we name reference conditioning.

### Citations

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Citation Context ...ed graphs; for the definition of perfect DAG and decomposable graph see Lauritzen (1996, pp.7-8). Any perfect DAG D is Markov equivalent to its undirected version D∼; moreover D∼ is decomposable (see =-=Lauritzen, 1996-=-, p.52). Conversely, if G = (V,E) is a decomposable undirected graph, then there exists a perfect DAG D Markov equivalent to G with D∼ = G (see Lauritzen, 1996, p.18). In Example 1 modelMmay be descri... |

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Citation Context ...t be the same as prior beliefs on η2 conditional on MD2 as well as on MD02 . Accordingly, the prior distribution for η2 should be the same under MD and MD0 (this property is named prior modularity by =-=Heckerman et al., 1995-=-) and the Bayes factor to compare MD2 and MD02 should be identically one. In Example 1 the prior distribution for η under MD is such that η1⊥ η2 with η2 ∼ a22/χ2δ (see Dawid and Lauritzen, 2000); neve... |

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Citation Context ...or DAG models having no causal interpretation. Example 4 For the problem in Example 2 let Σ ∼ IW (δ,A) as in Example 1. It can be easily checked from standard result for the Wishart distribution (see =-=Muirhead, 1982-=-, Theorem 3.2.10) that (φ11, φ12)⊥ φ22 and φ211 ∼ χ2δ+1/a11·2 φ12|φ11 ∼ N ( −φ11a12/a22, a−122 ) φ222 ∼ χ2δ/a22 where a11·2 = a11 − a212/a22. Applying reference conditioning, i.e. conditioning on {φ12... |

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Citation Context ...g chain graph model is a generalisation of the directed and undirected graphical model. Two graphs are said to be Markov equivalent if they encode the same set of conditional independence assertions (=-=Frydenberg, 1990-=-; Andersson et al. 1997). Markov equivalence is an equivalence relation and we denote by [D] the set of all DAGs Markov equivalent to D. Frydenberg (1990) provided necessary and sufficient conditions ... |

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Citation Context ...ult of a structural assignment system, as described in Lauritzen and Richardson (2001, eqn. 4). It is worth pointing out that in this case the distribution of XV is causally Markov with respect to D (=-=Lauritzen, 2001-=-, Theorem 2.20) and so the orientation of the arrows is crucial for the interpretation of the model. In the causal interpretation of DAG models, the local conditional families MDi s represent the prim... |

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Citation Context ...l is a generalisation of the directed and undirected graphical model. Two graphs are said to be Markov equivalent if they encode the same set of conditional independence assertions (Frydenberg, 1990; =-=Andersson et al. 1997-=-). Markov equivalence is an equivalence relation and we denote by [D] the set of all DAGs Markov equivalent to D. Frydenberg (1990) provided necessary and sufficient conditions for the Markov equivale... |

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Citation Context ...the variables so that pa(i) ⊆ {i + 1, . . . , v}. If Φ denotes the upper triangular matrix with entries φij and zero elsewhere, then ΦTΦ = Σ−1, which represents the Cholesky decomposition of Σ−1 (see =-=Wermuth, 1980-=-; Wermuth and Cox, 2000 and Roverato, 2000, 2001). For an interpretation of the elements φij notice that βij and βrj represent the unit-change effect of Xj on variables Xi and Xr respectively. The sta... |

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Citation Context ... . , v}. If Φ denotes the upper triangular matrix with entries φij and zero elsewhere, then ΦTΦ = Σ−1, which represents the Cholesky decomposition of Σ−1 (see Wermuth, 1980; Wermuth and Cox, 2000 and =-=Roverato, 2000-=-, 2001). For an interpretation of the elements φij notice that βij and βrj represent the unit-change effect of Xj on variables Xi and Xr respectively. The standardised versions φij and φrj allow a dir... |

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Citation Context ...by MG the Gaussian graphical model with graph G, that is the family of v-dimensional Gaussian distributions, with zero mean, satisfying the pairwise Markov property with respect to G (Dempster, 1972; =-=Wermuth, 1976-=-). If D and D0 are perfect DAGs then G = D∼ and G0 = D∼0 are decomposable undirected graphs. Furthermore G and D∼, as well as G0 and D∼0 , are Markov equivalent, so that MD ≡ MG and MD0 ≡ MG0 , and th... |

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(Show Context)
Citation Context ...-diagonal entries. Recall that σii = 1/σii·V \{i}, where σii·V \{i} is the variance of the conditional distribution of Xi given the remaining variables XV \{i} (see Whittaker, 1990 p.143). Example 1 (=-=Dawid and Lauritzen, 2000-=-) LetM be a bivariate normal model with zero mean and letM0 be the submodel with X1⊥X2. This constraint can be expressed in several alternative ways, depending on the parameterisation adopted for the ... |

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(Show Context)
Citation Context ..., Cr = V \{r} and Cs = V \{s}, and one separator, S = V \{r, s}, and the determinants of H1(Σ−11 ) and H0(Σ−10 ) are |H1(Σ−11 )| ∝ |Σ1|v+1 and |H0(Σ−10 )| ∝ |ΣCrCr |v|ΣCsCs |v |ΣSS |v−1 respectively (=-=Roverato and Whittaker, 1998-=-). Recalling that |Σ0| = |ΣCrCr ||ΣCsCs ||ΣSS |−1 (see Lauritzen, 1996 p.145), |ΣCrCr | = σrr·S |ΣSS | and |ΣCsCs | = σss·S |ΣSS | (see Lauritzen, 1996 equation (B.1)) and that σrr·S = 1/σrr0 and σss·... |

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(Show Context)
Citation Context ...o that pa(i) ⊆ {i + 1, . . . , v}. If Φ denotes the upper triangular matrix with entries φij and zero elsewhere, then ΦTΦ = Σ−1, which represents the Cholesky decomposition of Σ−1 (see Wermuth, 1980; =-=Wermuth and Cox, 2000-=- and Roverato, 2000, 2001). For an interpretation of the elements φij notice that βij and βrj represent the unit-change effect of Xj on variables Xi and Xr respectively. The standardised versions φij ... |

4 |
Graphical Models in Applied Multivariate
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(Show Context)
Citation Context ... \{i,j} = −σij/ √ σiiσjj in the off-diagonal entries. Recall that σii = 1/σii·V \{i}, where σii·V \{i} is the variance of the conditional distribution of Xi given the remaining variables XV \{i} (see =-=Whittaker, 1990-=- p.143). Example 1 (Dawid and Lauritzen, 2000) LetM be a bivariate normal model with zero mean and letM0 be the submodel with X1⊥X2. This constraint can be expressed in several alternative ways, depen... |

3 | Order-invariant group reference priors for natural exponential families having a simple quadratic variance function - Consonni, Veronese, et al. - 2000 |

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