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Title: Bayesian Model Selection for Identifying Markov Equivalent Causal Graphs

Paper content:
Many approaches to causal discovery are limited by their inability to discriminate between Markov equivalent graphs given only observational data.
We formulate causal discovery as a marginal likelihood based Bayesian model selection problem.
We adopt a parameterization based on the notion of the independence of causal mechanisms which renders Markov equivalent graphs distinguishable.
We complement this with an empirical Bayesian approach to setting priors so that the actual underlying causal graph is assigned a higher marginal likelihood than its alternatives.
Adopting a Bayesian approach also allows for straightforward modeling of unobserved confounding variables, for which we provide a variational algorithm to approximate the marginal likelihood, since this desirable feat renders the computation of the marginal likelihood intractable.
We believe that the Bayesian approach to causal discovery both allows the rich methodology of Bayesian inference to be used in various difficult aspects of this problem and provides a unifying framework to causal discovery research.
We demonstrate promising results in experiments conducted on real data, supporting our modeling approach and our inference methodology.
Causal networks (CNs) are special Bayesian networks where all edges reflect causal relations (Pearl, 2009 ).
The aim of causal structure learning is identifying the CN underlying the observed data.
In this paper, we focus on the problem of scoring causal graphs using marginal likelihood in a way that identifies the unique causal generative graph.
Succeeding to do so is very valuable, since once the correct CN is selected, various causal inference tasks such as estimating causal effects or examining confounder distributions becomes straightforward in a Bayesian framework.
A central challenge in such an attempt, however, is adopting a prior selection policy that not only allows discriminating between Markov equivalent graphs but also assigns higher marginal likelihood score to the actual underlying CN.
The key notion underlying our solution to first part of this challenge is the widely accepted principle of independence of the cause-effect mechanisms (Janzing et al., 2012) , that is, the natural mechanisms that generate the cause and the effect (based on cause) must be independent of each other.
We embody this assumption by assuming the mutual independence of the parameters pertaining to cause and effect distributions in a Bayesian model, a line of reasoning that is natural to this modeling perspective, where parameters are modeled as random variables (Spiegelhalter et al., 1993; Heckerman et al., 1995; Geiger et al., 1997; Blei et al., 2003) .
By assigning independent priors to the cause and effect variables, we render them statistically independent.
Critically, this assignment of independent priors also breaks the likelihood equivalence between Markov equivalent graphs.
This is contrast to other ways of selecting independent priors such as the BDeu prior, which leads to assigning equal marginal likelihood to Markov equivalent graphs (Heckerman et al., 1995) .
As mentioned above, though breaking likelihood equivalence does not necessarily lead to assigning a higher marginal likelihood to the actual underlying CN, it is a prerequisite for doing so 1 .
The second part of the problem is adapting a prior selection policy that leads to assigning a higher marginal likelihood to the actual CN compared to its alternatives.
In this work, we use an empirical Bayesian approach in selecting the hyperparameters of the independent priors described above, as we learn the priors that lead to assigning higher marginal likelihood to the actual CN from labeled data.
The current approach is in the intersection of various other approaches in the literature, thereby combining many of their respective advantages (Spirtes and Zhang, 2016; Glymour et al., 2019) .
It is based on the notion of mechanism independence similar to Janzing et al. (2012) ; Zhang et al. (2015) , does not assume causal sufficiency similar to Silva et al. (2006) ; Shimizu et al. (2009) ; Janzing et al. ( , 2012 ; Zhang et al. (2015) ; Schölkopf et al. (2016) , can theoretically work on arbitrary graph structures that possibly include latent variables similar to Spirtes et al. (1993) , and can discriminate between Markov equivalent structures similar to Shimizu et al. (2006) ; Zhang and Hyvärinen (2008); Hoyer et al. (2009); Janzing et al. (2012); Zhang et al. (2015) .
Our approach diverges from other Bayesian methods (Stegle et al., 2010; Shimizu and Bollen, 2014; Zhang et al., 2016) in various dimensions such as by being able to distinguish between Markov equivalent causal graphs, using marginal likelihood (or approximations thereof) instead of surrogate scores such as BIC, or being able to model non-linear relationships.
In Section 2, we introduce an example model for continuous observations and latent categorical confounders.
To approximate the marginal likelihood in graphs which include latent confounders, we present a variational inference algorithm in Section 3.
After testing our approach on various real data sets in Section 4, we present our conclusions and further avenues of research in Section 5.
Overall, we show that Bayesian model selection is a promising framework that can facilitate causal research significantly both through conceptual unification and increased performance.
Given that Bayesian modeling is agnostic to specific variable types, conditional distributions, and to approximate inference methodology, the value of a successful Bayesian modeling approach for causal research is immense.
Though our empirical Bayesian approach to setting priors can be useful in various contexts (e.g. in data sets where only some of the bivariate causal directions are known), finding other principled ways of assigning (or integrating out) priors that do not require labeled data is an important direction for future research.
Conducting causal discovery with different variable types, and/or different distributions would also be beneficial for demonstrating current approach's viability in various contexts.

assistant: text: We cast causal structure discovery as a Bayesian model selection in a way that allows us to discriminate between Markov equivalent graphs to identify the unique causal graph.