  Gaussian graphical model is one of the powerful tools to analyze conditional
independence between two variables for multivariate Gaussian-distributed
observations. When the dimension of data is moderate or high, penalized
likelihood methods such as the graphical lasso are useful to detect significant
conditional independence structures. However, the estimates are affected by
outliers due to the Gaussian assumption. This paper proposes a novel robust
posterior distribution for inference of Gaussian graphical models using the
$\gamma$-divergence which is one of the robust divergences. In particular, we
focus on the Bayesian graphical lasso by assuming the Laplace-type prior for
elements of the inverse covariance matrix. The proposed posterior distribution
matches its maximum a posteriori estimate with the minimum $\gamma$-divergence
estimate provided by the frequentist penalized method. We show that the
proposed method satisfies the posterior robustness which is a kind of measure
of robustness in the Bayesian analysis. The property means that the information
of outliers is automatically ignored in the posterior distribution as long as
the outliers are extremely large, which also provides theoretical robustness of
point estimate for the existing frequentist method. A sufficient condition for
the posterior propriety of the proposed posterior distribution is also shown.
Furthermore, an efficient posterior computation algorithm via the weighted
Bayesian bootstrap method is proposed. The performance of the proposed method
is illustrated through simulation studies and real data analysis.
