Parameter Priors for Directed Acyclic Graphical Models and the Characteriration of Several Probability Distributions

By Dan Geiger, David E. Heckerman. 1999

We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n × n, n > 3, positive-definite symmetric matrix of random variables and f(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W11 - W12W22-1, is independent of {W12, W22) for every block partitioning W11, W12, W12, W22 of W. Similar characterizations of the normal and normal-Wishart distributions are provided as well. We also show how to construct a prior for every DAG model over X from the prior of a single regression model.

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Citations in the corpus

Towards Scalable Bayesian Learning of Causal DAGs (2020, methodology): “With the notation αμ and αw we adhere to the choices in the key references [9, 17].”
Variational Causal Networks: Approximate Bayesian Inference over Causal Structures (2021, methodology): “Calculate the per-sample marginal log-likelihood log p ( D|G ( i ) ) by marginalising θ to obtain the BGe Score [10, 23]. φ ← Update parameters using gradients g .”
Sparse graphical models for exploring gene expression data (2004, unknown): “and ci 1⁄4 sii kiS 1 pvðiÞki: Using standard distribution theory [8,6] we deduce the implied prior of the normal/inverse gamma form”
Belief net structure learning from uncertain interventions (2007, background): “An analogous formula can be derived for the normal-Gamma case (Geiger and Heckerman, 2002).”
Efficient Sampling and Structure Learning of Bayesian Networks (2018, methodology): “Therefore the complexity is comparable or even lower than that of structure MCMC moves (between O ( n ) and O ( n 2 ), though some rejected moves can be O (1); Giudici and Castelo, 2003).”
DiBS: Differentiable Bayesian Structure Learning (2021, methodology): “end, we employ the commonly used Bayesian Gaussian Equivalent (BGe) marginal likelihood, which scores Markov equivalent structures equally [16, 17].”
Partition MCMC for Inference on Acyclic Digraphs (2015, methodology): “We work with the BGe score (Geiger and Heckerman 1994; Heckerman and Geiger 1995; Geiger and Heckerman 2002), corrected as in Consonni and Rocca (2012) and Kuipers, Moffa, and Heckerman (2014) and sped up following Kuipers, Moffa, and Heckerman (2014).”