This post is partly inspired by this post. Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix
While it is well-known that two basic categories of graph models are widely studied in statistics
- Undirected graph representation for Hidden Markov chain models;
- Directed graph representation for hierarchical Bayesian models
see [wiki1] [Zoubin]. Treatments of asymptotics of joint likelihood of these two categories of models seem to be separately discussed.
To be more specific, the asymptotics of Bayesian network is usually based on the knowledge of intermediate hierarchy structure like seminal work of [Geiger et.al] or marginal structure like more recent work of [Rusakov&Geiger]. However the asymptotics of hidden markov chain is based on the properties of transition kernels (irreducible, ergodic etc.) and associated stochastic processes like [Bickel et.al] and [de Gunst&Shcherbakova].
Question
Since both statistical models given sample size $N$ can be represented as random graphs or corresponding $N\times N$ random matrices, is there a treatment/reference for asymptotic result (increasing size $N$) that applies to the random matrices and hence the graph models?
Or more precisely, how can we translate/understand the asymptotic/limiting results $N\rightarrow \infty$ for a sequence of random matrices $A_{G_N}$ back to the sequence of random graphs $G_N$ (represented by the random matrices $A_{G_N}$) as the sample/observation size $N\rightarrow \infty$ for general statistical models of these two categories?
Any reference will be appreciated.
Reference
[wiki1]https://en.wikipedia.org/wiki/Bayesian_network
[Zoubin] Ghahramani, Zoubin. "An introduction to hidden Markov models and Bayesian networks." International journal of pattern recognition and artificial intelligence 15.01 (2001): 9-42. http://mlg.eng.cam.ac.uk/zoubin/papers/ijprai.pdf
[Geiger et.al]Geiger, Dan, David Heckerman, and Christopher Meek. "Asymptotic model selection for directed networks with hidden variables." Learning in graphical models. Springer Netherlands, 1998. 461-477.
[Rusakov&Geiger]Rusakov, Dmitry, and Dan Geiger. "Asymptotic model selection for naive Bayesian networks." Journal of Machine Learning Research 6.Jan (2005): 1-35.
[Bickel]Bickel, Peter J., Ya’acov Ritov, and Tobias Ryden. "Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models." The Annals of Statistics 26.4 (1998): 1614-1635.
[de Gunst&Shcherbakova]de Gunst, M. CM, and O. Shcherbakova. "Asymptotic behavior of Bayes estimators for hidden Markov models with application to ion channels." Mathematical Methods of Statistics 17.4 (2008): 342-356.
