Multi-Domain Sampling With Applications to Structural Inference of Bayesian Networks

When a posterior distribution has multiple modes, unconditional expectations, such as the posterior mean, may not offer informative summaries of the distribution. Motivated by this problem, we propose to decompose the sample space of a multimodal distribution into domains of attraction of local modes. Domain-based representations are defined to summarize the probability masses of and conditional expectations on domains of attraction, which are much more informative than the mean and other unconditional expectations. A computational method, the multi-domain sampler, is developed to construct domain-based representations for an arbitrary multimodal distribution. The multi-domain sampler is applied to structural learning of protein-signaling networks from high-throughput single-cell data, where a signaling network is modeled as a causal Bayesian network. Not only does our method provide a detailed landscape of the posterior distribution but also improves the accuracy and the predictive power of estimated networks.

💡 Research Summary

The paper addresses a fundamental difficulty in Bayesian inference: when the posterior distribution is multimodal, unconditional summaries such as the posterior mean can be misleading or uninformative. To overcome this, the authors introduce a domain‑based representation (DR) that partitions the sample space into attraction domains of the local modes and records, for each domain, its probability mass and conditional expectations of quantities of interest. Formally, for a differentiable density p(x) on ℝ^m, the gradient flow d x(t)/dt = ∇p(x(t)) is used to define the limit point ν_k of any starting point x; the set of all points whose flow ends at ν_k constitutes the domain D_k. The DR for a function h is the collection {(μ_{h,k}, λ_k)}{k=1}^K where λ_k = ∫{D_k}p(x)dx and μ_{h,k}=E