  Mutual independence is a key concept in statistics that characterizes the
structural relationships between variables. Existing methods to investigate
mutual independence rely on the definition of two competing models, one being
nested into the other and used to generate a null distribution for a statistic
of interest, usually under the asymptotic assumption of large sample size. As
such, these methods have a very restricted scope of application. In the present
manuscript, we propose to change the investigation of mutual independence from
a hypothesis-driven task that can only be applied in very specific cases to a
blind and automated search within patterns of mutual independence. To this end,
we treat the issue as one of model comparison that we solve in a Bayesian
framework. We show the relationship between such an approach and existing
methods in the case of multivariate normal distributions as well as
cross-classified multinomial distributions. We propose a general Markov chain
Monte Carlo (MCMC) algorithm to numerically approximate the posterior
distribution on the space of all patterns of mutual independence. The relevance
of the method is demonstrated on synthetic data as well as two real datasets,
showing the unique insight provided by this approach.
