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We present baycn (BAYesian Causal Network), a novel approximate Bayesian method for inferring probabilities of edge directions and edge absence, while allowing flexible, user-specified priors to encode sparsity and an input graph to incorporate biological knowledge. For inference, we develop a Metropolis-Hastings-like sampler over graph structures based on a pseudo-posterior with a plug-in likelihood, which eliminates potentially high-dimensional nuisance parameters. This formulation substantially improves computational efficiency while yielding posterior probabilities that reflect Markov equivalence. We apply baycn to two genomic applications: distinguishing direct from indirect target genes of a shared genetic variant, and inferring combinatorial binding of transcription factors during tissue differentiation in Drosophila embryos. Both applications involve discrete and continuous data types that are common in genomics. 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