chain event graphs [RSS Midlands seminar]

Last evening, I attended the RSS Midlands seminar here in Warwick. The theme was chain event graphs (CEG), As I knew nothing about them, it was worth my time listening to both speakers and discussing with Jim Smith afterwards. CEGs are extensions of Bayes nets with originally many more nodes since they start with the probability tree involving all modalities of all variables. Intensive Bayesian model comparison is then used to reduce the number of nodes by merging modalities having the same children or removing variables with no impact on the variable of interest. So this is not exactly a new Bayes net based on modality dummies as nodes (my original question). This is quite interesting, esp. in the first talk illustration of using missing value indicators as a supplementary variable (to determine whether or not data is missing at random). I also wonder how much of a connection there is with variable length Markov chains (either as a model or as a way to prune the tree). A last vague idea is a potential connection with lumpable Markov chains, a concept I learned from Kemeny & Snell (1960): a finite Markov chain is lumpable if by merging two or more of its states it remains a Markov chain. I do not know if this has ever been studied from a statistical point of view, i.e. testing for lumpability, but this sounds related to the idea of merging modalities of some variables in the probability tree…

Hierarchical vs. graphical models

An email from a reader:

I’m studying your book  The Bayesian Choice (2007). I know that you mentioned graphics models in the Notes of Section 10. But I’m still confused on what is Hierarchical Bayesian Models and Graphical Models. It seems to me that Hierarchical models are just special cases of graphical models. The Bayesian network that corresponds to a graphical model is a DAG in general and the Bayesian network that corresponds to a hierarchical model is just a chain of directed edges.

Would you please take a few seconds to point me some references or discussions so that I can understand their differences better?

To which I can only reply that hierarchical models are indeed special cases of graphical models for which the edges have some kind of causal interpretation and where some conditional independence relations are imposed by the hierarchy, which is not always the case for graphical models. As probabilistic objects and as distributions, both structures belong to the same family. Inference on those objects may be different though, in that [presence or absence of] edges may be examined individually in graphical models, less so in hierarchical models where they come [and go] in batches respecting the hierarchy (see e.g. random effect models).

Anathem

One colleague of mine in Dauphine gave me Anathem to read a few weeks ago. I had seen it in a bookstore once and planned to read it, so this was a perfect opportunity. I read through it slowly at first and then with more and more eagerness as the story built on, spending a fair chunk of the past evenings (and Metro rides) into finishing it. Anathem is a wonderful book, especially for mathematicians, and while it could still qualify as a science-fiction book, it blurs the frontiers between the genres of science-fiction, speculative fiction, documentary writings and epistemology… Just imagine any other sci’fi’ book being reviewed in Nature! Still, the book was awarded the 2009 Locus SF Award. So it has true sci’fi’ characteristics, including Clarke-ian bouts of space opera with a Rama-like vessel popping out of nowhere. But this is not the main feature that makes Anathem so unique and fascinating.

“The Adrakhonic theorem, which stated that the square of a right triangle hypotenuse was equal to the sum of the squares of the other two sides…” (p. 128)

Le Monde puzzle [46]

This week puzzle in Le Monde does not make much sense, unless I miss the point: in an undirected graph with 2011 nodes, each node is linked with at least 1005 other nodes. Is there always a node that is linked with all the nodes? If I take the first two nodes, 1 and 2, if there were no common node, the 1005 nodes linked with 1 would differ from the 1005 nodes linked with 2 and would not include 2. This corresponds to 1+1005+1005+1=2012 nodes… Sounds too easy, doesn’t it?!

Update (111810): Easy, too easy! Robin pointed out to me yesterday that this proves that 1 and 2 share a neighbour, nothing more. His counterexample to the existence of a common neighbour is to create nodes from node i to any node but (i-1) mod n and (i+1) mod n, which makes a graph with n-3 edges leaving from each node and still no common neighhbour! Conclusion: I should not try to quick-solve puzzles at 4:38am…

CoRe in CiRM [3]

Still drudging along preparing the new edition of Bayesian Core. I am almost done with the normal chapter, where I also changed the Monte Carlo section to include specific tools (bridge) for evidence/Bayes factor approximation. Jean-Michel has now moved to the new hierarchical model chapter and analysed longitudinal  datasets that will constitute the core of the chapter, along with the introduction of DAGs. (meaning some time “wasted” on creating DAG graphs, I am afraid!) He is also considering including a comparison with OpenBUGS and JAGS implementations, with the convincing motive that the hierarchical models are used in settings where practitioners have no time/ability to derive Gibbs samplers for a whole collection of models they want to compare… And we are  vaguely wondering whether or not using a pun in the title, from Bayesian Core to Bayesian CoRe, in order to emphasise the R link. This morning, it took me half an hour to figure out how resetting new counters (like our exercise environment) in LaTeX,

\makeatletter
\@addtoreset{exx}{chapter}
\makeatother

but I eventually managed it, thanks to the UK TeX Archive.