Minimax posterior convergence rates and model selection consistency in high-dimensional DAG models based on sparse Cholesky factors

Abstract

In this paper, we study the high-dimensional sparse directed acyclic graph (DAG) models under the empirical sparse Cholesky prior. Among our results, strong model selection consistency or graph selection consistency is obtained under more general conditions than those in the existing literature. Compared to Cao, Khare and Ghosh (2017), the required conditions are weakened in terms of the dimensionality, sparsity and lower bound of the nonzero elements in the Cholesky factor. Furthermore, our result does not require the irrepresentable condition, which is necessary for Lasso type methods. We also derive the posterior convergence rates for precision matrices and Cholesky factors with respect to various matrix norms. The obtained posterior convergence rates are the fastest among those of the existing Bayesian approaches. In particular, we prove that our posterior convergence rates for Cholesky factors are the minimax or at least nearly minimax depending on the relative size of true sparseness for the entire dimension. The simulation study confirms that the proposed method outperforms the competing methods.MSC 2010 subject classifications: Primary 62C20; secondary 62F15, 62C12.

Consistent Bayesian sparsity selection for high-dimensional Gaussian DAG models with multiplicative and beta-mixture priors

“…By the assumptions in the previous work, it follows that the DAGs in the analysis do not include the models where the Cholesky factor has one or more non-zero elements for each column, since p/ 1 8 d n log p 1+k 2+k → ∞, as n → ∞, while in our result, each row can have at most R n ∼ n log n number of non-zero entries as indicated in Assumption 4. Hence, our strong model selection consistency results is more general than Ghosh, 2019, Lee, Lee, andLin, 2018] in the sense that the consistency holds for a larger class of DAGs.…”

“…Remark 2. It is worthwhile to point out that our assumptions on the true Cholesky factor are weaker compared to [Lee, Lee, and Lin, 2018]. In particular, Lee, Lee, and Lin [2018] introduce conditions A (2) and A(4) on the sparsity pattern of the true Cholesky factor such that the number of non-zero elements in each row as well as each column of L n 0 to be smaller than some constant s 0 , while in this paper, we are allowing the maximum number of non-zero entries in any column of L n 0 to grow at a smaller rate than n log pn (Assumption 2).…”

“…This assumption essentially states that the prior on the space of the 2 ( pn 2 ) possible models, places zero mass on unrealistically large models (see similar assumptions in [Johnson and Rossell, 2012, Shin, Bhattacharya, and Johnson, 2018, Narisetty and He, 2014 in the context of regression). Assumption 4 is also more relaxed compared with Condition (P) in [Lee, Lee, and Lin, 2018] where R n ∼ n(log p) −1 {(log n) −1 ∨ c 3 } for some constant c 3 . Note that this condition is for the hyperparameter of the prior distribution on the latent variables only, which does not affect the true parameter space.…”

“…See [Yang, Wainwright, and Jordan, 2016, Lee, Lee, and Lin, 2018, Cao, Khare, and Ghosh, 2018 for example.…”

“…However, the sparsity assumptions on the true model required to establish consistency are rather restrictive. In addition, as a result of the extremely small value of the edge probability q in the Bernoulli prior, the simulations studies always tend to favor more sparse models under smaller values of p. Lee, Lee, and Lin [2018] also explore the Cholesky factor selection consistency under the empirical sparse Cholesky (ESC) prior and α-posteriors. Compared with [Cao, Khare, and Ghosh, 2019], under relaxed conditions in terms of the dimensionality, sparsity and lower bound of the non-zero elements in the Cholesky factor, Lee, Lee, and Lin [2018] establish strong model selection consistency with the α-posterior distribution.…”