## The Annals of Statistics

### A new prior for discrete DAG models with a restricted set of directions

#### Abstract

In this paper, we first develop a new family of conjugate prior distributions for the cell probability parameters of discrete graphical models Markov with respect to a set $\mathcal{P}$ of moral directed acyclic graphs with skeleton a given decomposable graph $G$. This family, which we call the $\mathcal{P}$-Dirichlet, is a generalization of the hyper Dirichlet given in [Ann. Statist. 21 (1993) 1272–1317]: it keeps the directed strong hyper Markov property for every DAG in $\mathcal{P}$ but increases the flexibility in the choice of its parameters, that is, the hyper parameters.

Our second contribution is a characterization of the $\mathcal{P}$-Dirichlet, which yields, as a corollary, a characterization of the hyper Dirichlet and a characterization of the Dirichlet also. Like the characterization of the Dirichlet given in [Ann. Statist. 25 (1997) 1344–1369], our characterization of the $\mathcal{P}$-Dirichlet is based on local and global independence of the probability parameters and also a separability property explicitly defined here but implicitly used in that paper through the choice of two particular DAGs. Another advantage of our approach is that we need not make the assumption of the existence of a positive density function. We use the method of moments for our proofs.

#### Article information

Source
Ann. Statist., Volume 44, Number 3 (2016), 1010-1037.

Dates
Revised: September 2015
First available in Project Euclid: 11 April 2016

https://projecteuclid.org/euclid.aos/1460381685

Digital Object Identifier
doi:10.1214/15-AOS1396

Mathematical Reviews number (MathSciNet)
MR3485952

Zentralblatt MATH identifier
1341.62153

#### Citation

Massam, Hélène; Wesołowski, Jacek. A new prior for discrete DAG models with a restricted set of directions. Ann. Statist. 44 (2016), no. 3, 1010--1037. doi:10.1214/15-AOS1396. https://projecteuclid.org/euclid.aos/1460381685

#### References

• [1] Andersson, S. A., Madigan, D. and Perlman, M. D. (1997). A characterization of Markov equivalence classes for acyclic digraphs. Ann. Statist. 25 505–541.
• [2] Bobecka, K. and Wesołowski, J. (2007). The Dirichlet distribution and process through neutralities. J. Theoret. Probab. 20 295–308.
• [3] Bobecka, K. and Wesołowski, J. (2009). Moments method approach to characterizations of Dirichlet tables through neutralities. Publ. Math. Debrecen 74 321–339.
• [4] Chang, W.-Y., Gupta, R. D. and Richards, D. St. P. (2010). Structural properties of the generalized Dirichlet distributions. In Algebraic Methods in Statistics and Probability II (M. A. G. Viana and H. P. Wynn, eds.). Contemp. Math. 516 109–124. Amer. Math. Soc., Providence, RI.
• [5] Darroch, J. N. and Ratcliff, D. (1971). A characterization of the Dirichlet distribution. J. Amer. Statist. Assoc. 66 641–643.
• [6] Dawid, A. P. and Lauritzen, S. L. (1993). Hyper-Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21 1272–1317.
• [7] Diaconis, P. and Ylvisaker, D. (1979). Conjugate priors for exponential families. Ann. Statist. 7 269–281.
• [8] Fabius, J. (1973). Two characterizations of the Dirichlet distribution. Ann. Statist. 1 583–587.
• [9] Geiger, D. and Heckerman, D. (1997). A characterization of the Dirichlet distribution through global and local parameter independence. Ann. Statist. 25 1344–1369.
• [10] Heckerman, Geiger, D. and Chickering, D. M. (1995). Learning Bayesian networks: The combination of knowledge and statistical data. Mach. Learn. 20 197–243.
• [11] James, I. R. and Mosimann, J. E. (1980). A new characterization of the Dirichlet distribution through neutrality. Ann. Statist. 8 183–189.
• [12] Járai, A. (1998). Regularity property of the functional equation of the Dirichlet distribution. Aequationes Math. 56 37–46.
• [13] Járai, A. (2005). Regularity Properties of Functional Equations in Several Variables. Advances in Mathematics (Springer) 8. Springer, New York.
• [14] Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Oxford Univ. Press, New York.
• [15] Massam, H. and Wesołowski, J. (2015). Supplement to “A new prior for discrete DAG models with a restricted set of directions.” DOI:10.1214/15-AOS1396SUPP.
• [16] Ng, K. W., Tian, G.-L. and Tang, M.-L. (2011). Dirichlet and Related Distributions. Wiley, Chichester.
• [17] Rajaratnam, B., Massam, H. and Carvalho, C. M. (2008). Flexible covariance estimation in graphical Gaussian models. Ann. Statist. 36 2818–2849.
• [18] Ramamoorthi, R. V. and Sangalli, L. M. (2007). On a characterization of the Dirichlet distribution. In Bayesian Statistics and Its Applications (S. K. Upadhyay, U. Singh and D. K. Dey, eds.) 385–397. Anamaya Publishers, New Delhi.
• [19] Sakowicz, A. and Wesołowski, J. (2014). Dirichlet distribution through neutralities with respect to two partitions. J. Multivariate Anal. 129 1–15.
• [20] Studený, M. and Vomlel, J. (2009). A reconstruction algorithm for the essential graph. Internat. J. Approx. Reason. 50 385–413.

#### Supplemental materials

• Proofs and some detailed examples for “A new prior for discrete DAG models with a restricted set of directions”. Supplement A contains proofs and examples. We provide the proof of Lemma 2.1 and give a simple example of $\mathfrak{p}$-perfect ordering of the cliques and vertices as given in (2.1). We also provide the proofs of Theorems 4.1, 5.1, 5.2 and 6.1. We give the details of the derivation of the four $\mathcal{P}$-Dirichlet families in Example 3.1 continued. We also illustrate, with two examples, a possible extension of the $\mathcal{P}$-Dirichlet distribution to arbitrary DAGs through the use of essential graphs.