  We address the problem of two-variable causal inference without intervention.
This task is to infer an existing causal relation between two random variables,
i.e. $X \rightarrow Y$ or $Y \rightarrow X$ , from purely observational data.
As the option to modify a potential cause is not given in many situations only
structural properties of the data can be used to solve this ill-posed problem.
We briefly review a number of state-of-the-art methods for this, including very
recent ones. A novel inference method is introduced, Bayesian Causal Inference
(BCI), which assumes a generative Bayesian hierarchical model to pursue the
strategy of Bayesian model selection. In the adopted model the distribution of
the cause variable is given by a Poisson lognormal distribution, which allows
to explicitly regard the discrete nature of datasets, correlations in the
parameter spaces, as well as the variance of probability densities on
logarithmic scales. We assume Fourier diagonal Field covariance operators. The
model itself is restricted to use cases where a direct causal relation $X
\rightarrow Y$ has to be decided against a relation $Y \rightarrow X$ ,
therefore we compare it other methods for this exact problem setting. The
generative model assumed provides synthetic causal data for benchmarking our
model in comparison to existing State-of-the-art models, namely LiNGAM ,
ANM-HSIC , ANM-MML , IGCI and CGNN . We explore how well the above methods
perform in case of high noise settings, strongly discretized data and very
sparse data. BCI performs generally reliable with synthetic data as well as
with the real world TCEP benchmark set, with an accuracy comparable to
state-of-the-art algorithms. We discuss directions for the future development
of BCI .
