Characterizing the Minimal Essential Graphs of Maximal Ancestral Graphs

Learning ancestor graph is a typical NP-hard problem. We consider the problem to represent a Markov equivalence class of ancestral graphs with a compact representation. Firstly, the minimal essential graph is defined to represent the equivalent class of maximal ancestral graphs with the minimum number of invariant arrowheads. Then, an algorithm is proposed to learn the minimal essential graph of ancestral graphs based on the detection of minimal collider paths. It is the first algorithm to use necessary and sufficient conditions for Markov equivalence as a base to seek essential graphs. Finally, a set of orientation rules is presented to orient edge marks of a minimal essential graph. Theory analysis shows our algorithm is sound, and complete in the sense of recognizing all minimal collider paths in a given ancestral graph. And the experiment results show we can discover all invariant marks by these orientation rules.

On Causal Identification under Markov Equivalence

In this work, we investigate the problem of computing an experimental distribution from a combination of the observational distribution and a partial qualitative description of the causal structure of the domain under investigation. This description is given by a partial ancestral graph (PAG) that represents a Markov equivalence class of causal diagrams, i.e., diagrams that entail the same conditional independence model over observed variables, and is learnable from the observational data. Accordingly, we develop a complete algorithm to compute the causal effect of an arbitrary set of intervention variables on an arbitrary outcome set.

Ancestral Graph with Bias in Gene Conversion

Gene conversion is a genetic mechanism by which one gene is ‘copied and pasted’ onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within a d-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with a d-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.

Ancestral Graph with Bias in Gene Conversion

Gene conversion is a genetic mechanism by which one gene is ‘copied and pasted’ onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within a d-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with a d-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.