Being Bayesian about Network Structure (1301.3856v1)

Abstract: In many domains, we are interested in analyzing the structure of the underlying distribution, e.g., whether one variable is a direct parent of the other. Bayesian model-selection attempts to find the MAP model and use its structure to answer these questions. However, when the amount of available data is modest, there might be many models that have non-negligible posterior. Thus, we want compute the Bayesian posterior of a feature, i.e., the total posterior probability of all models that contain it. In this paper, we propose a new approach for this task. We first show how to efficiently compute a sum over the exponential number of networks that are consistent with a fixed ordering over network variables. This allows us to compute, for a given ordering, both the marginal probability of the data and the posterior of a feature. We then use this result as the basis for an algorithm that approximates the Bayesian posterior of a feature. Our approach uses a Markov Chain Monte Carlo (MCMC) method, but over orderings rather than over network structures. The space of orderings is much smaller and more regular than the space of structures, and has a smoother posterior `landscape'. We present empirical results on synthetic and real-life datasets that compare our approach to full model averaging (when possible), to MCMC over network structures, and to a non-Bayesian bootstrap approach.

• The paper presents an innovative method that efficiently sums over variable orderings to compute feature posteriors in Bayesian networks.
• It employs MCMC sampling over orderings instead of network structures, resulting in faster convergence and more consistent inference.
• Empirical results on synthetic and real datasets demonstrate improved edge detection and robust performance in high-dimensional applications.

Bayesian Network Structure Inference with MCMC over Orderings

The paper "Being Bayesian about Network Structure" by Daphne Koller and Nir Friedman tackles the challenge of inferring Bayesian network structures in cases where data availability is modest relative to model complexity. Traditional Bayesian model selection seeks the Maximum A Posteriori (MAP) model but falters when numerous models possess significant posterior probabilities. The authors propose an innovative approach, focusing on computing feature posteriors, such as the existence of an edge, across the landscape of potential network structures.

Efficient Calculation over Orderings

A central contribution of this work is the methodology for efficiently summing over an exponential number of networks that align with a specific variable ordering. This strategy allows calculation of both the data's marginal probability and the feature's posterior for a given ordering. The novel aspect here lies in exploiting the tractability offered by the fixed ordering, which, while simplifying the problem, also shifts the inferencing challenge to the space of orderings.

MCMC over Orderings

To address the broader challenge of model averaging across all possible networks, the authors introduce an MCMC technique that diverges from prior work by sampling over orderings instead of network structures directly. This approach leverages the smaller and more regular space of variable orderings, which possesses a smoother posterior probability landscape. Consequently, the Markov Chain over orderings mixes more efficiently, leading to more reliable posterior estimates for structural features.

Empirical Evaluation and Comparison

Empirical results demonstrate the efficacy of this MCMC approach within both synthetic and real-world datasets. Compared to full model averaging and other approximation techniques, such as MCMC over network structures and non-Bayesian bootstrap methods, the ordering-based MCMC consistently exhibited faster convergence and higher consistency across different runs. The paper illustrates that, especially when data is sparse, traditional MCMC methods over network structures may suffer from convergence issues due to the peaked nature of the posterior landscape, whereas orderings offer a more robust alternative.

Theoretical and Practical Implications

Theoretically, the paper contributes to the understanding of Bayesian structure learning by transforming the inferencing paradigm from focusing solely on network structures to taking a broader view of variable orderings. Practically, this facilitates the stable identification of dominant features even in complex, high-dimensional domains, such as gene expression analysis, where traditional data-driven structure learning may not yield reliable results.

Future Directions in Bayesian Inference

The approach proposed opens avenues for improvement in networked data analysis, particularly in fields requiring cause-and-effect linkage discoveries with limited data. Future work could incorporate partial ordering knowledge or prior domain constraints to further guide the search through MCMC sampling. Expansions to handle cases with missing data, by integrating the method with data imputation techniques within the MCMC framework, are also promising directions that could enhance model predictions in real-world applications.

In summary, this paper presents a compelling shift from structure-focused Bayesian network learning approaches to a more nuanced perspective involving the ordering of network variables, demonstrating improved convergence and feature detection capabilities through a novel application of MCMC sampling.