Causal inference is perhaps a more interesting topic than statistical inference, where not only correlations but also causations are studied. We can illustrate the causal structure of a group of variables using directed acyclic graphs (DAGs) that represent the causal directions and conditional independencies. However, the graphical representation has some limitations if an effect is the result of an interaction, so an alternative standard analytical representation is using structural equation models (SEMs), where each variable is defined by its parents and some noise. An important task in causal inference is to discover the causal structure with only observational data. This is in general impossible since variables with different causal structures can have exactly the same joint probability distribution and the discovery can only be limited to Markov equivalent classes even in the best case. However, under the assumption that all relevant variables are observed, the task is possible in some cases, one of which is if we assume the causal relations to be linear and the noises to have equal variance. We consider the bivariate case under this assumption. We propose priors for the underlying causal direction, the equal but unknown variance, and the non-zero causal effect of the correct direction. We also discussed some criteria for setting proper prior hyperparameters. The conjugate priors we set allow us to derive a closed-form posterior distribution and further summarize it with credible regions (CRs). The posterior will be a mixture of a point mass at 0 and a continuous distribution, so we especially discussed the conditions of 0 being included in the CR. We suggest three types of CRs, i.e., the equal-tailed interval (ETI), the highest density region (HDR), and the “threshold CR”. The third type is proposed by us, where the decision to include 0 is made by comparing the density at 0 with a self-defined threshold. Our model is tested and compared with both simulated data and benchmarks. In the experiments, we figure out that our model tends to be too certain about predicting the graphical structure, so we use a bootstrap average to reduce the effect of extreme decisions. We implement our model with R and the results show comparable coverage rates and smaller average widths.      «

Causal inference is perhaps a more interesting topic than statistical inference, where not only correlations but also causations are studied. We can illustrate the causal structure of a group of variables using directed acyclic graphs (DAGs) that represent the causal directions and conditional independencies. However, the graphical representation has some limitations if an effect is the result of an interaction, so an alternative standard analytical representation is using structural equation mo...     »