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...y, we have no elegant tricks that allow a closed form solution. Therefore, we provide a solution which uses our closed form solution of Eq. (6) as a subroutine in a Markov Chain Monte Carlo algorithm =-=[15]-=-. 4.1 The basic algorithm We introduce a uniform prior over orderings ffi , and define ffi to be of the same nature as the priors we used in the previous section. It is important to note that ...

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...ring ffi . In other words, we restrict attention to structures where if )! #%$ "! then #$ffi . This assumption was a standard one in the early work on learning Bayesian networks from data =-=[4]-=-. 3.1 Computing marginal likelihood We first consider the problem of computing the probability of the data given the ordering: ffiss ffiss(5) Note that this summation...

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... each instantiation " to its parents [8]. In Gaussian networks, we might use a Wishart prior [9]. For our purpose, we need only require that the prior satisfies two basic assumptions, as presented in =-=[10]-=-: 9 Global parameter independence: Let & fi ffi >@?=AffBC fiED be the parameters specifying the behavior of the variable ! given the various instantiations to its parents. Then we require that ...

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...ight want to use Bayesian network learning to understand the mechanisms by which genes in a cell produce proteins, which in turn cause other genes to express themselves, or prevent them from doing so =-=[6]-=-. In this case, our main goal is to discover the causal and dependence relations between the expression levels of different genes [12]. The common approach to discovering structure is to use learning ...

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...rk structures, and to a non-Bayesian bootstrap approach. 1 Introduction In the last decade there has been a great deal of research focused on the problem of learning Bayesian networks (BNs) from data =-=[3, 8]-=-. An obvious motivation for this problem is to learn a model that we can then use for inference or decision making, as a substitute for a model constructed by a human expert. In certain cases, however...

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...orderings, this enumeration is possible for as many as 10 variables, but for structures, we are limited to domains with 5–6 variables. We took two data sets — Vote and Flare — from the UCI repository =-=[16]-=- and selected five variables from each. We generated datasets of sizes ands , and computed the full Bayesian averaging posterior for these datasets using both methods. Figure 1 compares the res...

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...onal constraints that restrict the space (as in [11]). Alternatively, this summation can be approximated by considering only a subset of possible structures. Several approximations have been proposed =-=[13, 14]-=-. One theoretically well-founded approach is to use Markov Chain Monte Carlo (MCMC) methods: we define a Markov chain over structures whose stationary distribution is the posterior s, we then...

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...onal constraints that restrict the space (as in [11]). Alternatively, this summation can be approximated by considering only a subset of possible structures. Several approximations have been proposed =-=[13, 14]-=-. One theoretically well-founded approach is to use Markov Chain Monte Carlo (MCMC) methods: we define a Markov chain over structures whose stationary distribution is the posterior s, we then...

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...he exact computation for both types of features. We then considered larger datasets, where exhaustive enumeration is not an option. For this purpose we used synthetic data generated from the Alarm BN =-=[1]-=-, a network with 37 nodes. Here, our computational tricks are necessary. We used the following settings: (max. number of parents in a family) ; (max. number of potential parents) s ; (...

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...e families for a node ! is at most * - . 6 - . Hence, the total cost of computing Eq. (6) is at most 66 - 6 - . We note that the decomposition of Eq. (6) was first mentioned by Buntine =-=[2]-=-, but the ramifications for Bayesian model averaging were not pursued. The concept of Bayesian model averaging using a closed-form summation over an exponentially large set of structures was proposed ...

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...rk structures, and to a non-Bayesian bootstrap approach. 1 Introduction In the last decade there has been a great deal of research focused on the problem of learning Bayesian networks (BNs) from data =-=[3, 8]-=-. An obvious motivation for this problem is to learn a model that we can then use for inference or decision making, as a substitute for a model constructed by a human expert. In certain cases, however...

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...r model for the structure of the domain. Indeed, in small domains with a substantial amount of data, it has been shown that the highest scoring model is orders of magnitude more likely than any other =-=[11]-=-. In such cases, model selection is a good approximation. Unfortunately, there are many domains of interest where this situation does not hold. In our gene expression example, it is now possible to me...

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...tes. We also present results showing that our approach accurately captures dominant features even with sparse data, and that it outperforms both MCMC over structures and the non-Bayesian bootstrap of =-=[5]-=-. 2 Bayesian learning of Bayesian networks Consider the problem of analyzing the distribution over some set of random variables , based on a fully observed data set ss ...

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...e other genes to express themselves, or prevent them from doing so [6]. In this case, our main goal is to discover the causal and dependence relations between the expression levels of different genes =-=[12]-=-. The common approach to discovering structure is to use learning with model selection to provide us with a single high-scoring model. We then use that model (or its equivalence class) as our model fo...

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.... In discrete networks, the standard assumption is a Dirichlet prior over &flfiffisfor each node )! and each instantiation " to its parents [8]. In Gaussian networks, we might use a Wishart prior =-=[9]-=-. For our purpose, we need only require that the prior satisfies two basic assumptions, as presented in [10]: 9 Global parameter independence: Let & fi ffi >@?=AffBC fiED be the parameters specifyi...

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...Bayesian model averaging were not pursued. The concept of Bayesian model averaging using a closed-form summation over an exponentially large set of structures was proposed (in a different setting) in =-=[17]-=-. The computation of ffi is useful in and of itself; as we show in the next section, computing the probability ffi is a key step in our MCMC algorithm. 3.2 Probabilities of features Fo...

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...his move with probabilitys ffi H * ffi ffi Hs ffi * ffi H ffis It is well known that the resulting chain is reversible and has the desired stationary distribution =-=[7]-=-. We consider several specific constructions for the proposal distribution, based on different neighborhoods in the space of orderings. In one very simple construction, we consider only operators that...