In this paper, we study the log-behavior of a new sequence {S n } ∞ n=0 , which was defined by Z-W Sun. We find that the sequence is log-convex by using the interlacing method. Additionally, we consider ratio log-behavior of {S n } ∞ n=0 and find the sequences {S n+1 /S n } ∞ n=0 and { n √ S n } ∞ n=1 are log-concave. Our results give an affirmative answer to a conjecture of Z-W Sun on the ratio monotonicity of this new sequence.

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This paper generalizes the structural Markov properties for undirected decomposable graphs to arbitrary ones. This helps us to exploit the conditional independence properties of joint prior laws to analyze and compare multiple graphical structures, while being able to take advantage of the common conditional independence constraints. This work provides a theoretical support for full Bayesian posterior updating about the structure of a graph using data from a certain distribution. We further investigate the ratio of graph law so as to simplify the acceptance probability of the Metropolis–Hastings sampling algorithms.

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