Follow-up to October 6 meeting

Current work in graphical models:
- Mitra et al. (2013)
- Airoldi et al. (2009) (stochastic block models)
Background:
- Besag (1974) (used by Mitra et al)
- Airoldi (2007) (introduction for biologists)
Computing with graphical models:
- Højsgaard, Edwards, and Lauritzen (2012) (pdf is on google drive)

Mitra et al. (2013)

Goal: infer complex biological network of histone modifications’ interactions (via ChIP-Seq data analysis)
Challenge: evaluating a normalization constant in an autologistic model
Approach: MC evaluation of ratios of normalization constants

Use of results in Besag (1974)

Besag (1974) gives “a convenient parameterization of all possible joint probability models for a multivariate binary vector \(e = (e_1, ..., e_m) \in \lbrace 0,1\rbrace^m\) with a MRF conditional independence structure”
- What is the conditional independence structure of a MRF?
- What is Besag’s parameterization?
- Why is his parameterization useful & convenient?

Autologistic model

\[p(e|\beta, G)= p(0|\beta, G)\exp\lbrace \sum_i \beta_ie_i + \sum_{i < j}\beta_{ij}e_ie_j + \sum_{i < j < k}\beta_{ijk}e_ie_je_k + \dots + \beta_{1\dots m}e_1\dots e_m\rbrace \]
- all “interaction” terms among \(m\) variables
conditional independence (implied by G) leads to restrictions on \(\beta\)
what is the source of the name “autologistic”?
- how does it relate to a logistic model, in, for example, a regression setting?

Clique

clique: a set of vertices of which all pairs of vertices are connected
- \(\lbrace i_1, i_2\rbrace\in E\) (where \(E\) is the collection of edges)
a collection of vertices \(\lbrace i_1, \dots i_k\rbrace\) that is not connected means that the corresponding “interaction” (\(\beta\)) term \(\beta_{i_1\dots i_k} = 0\)
Mitra et al. (2013) assumes that all interactions of at least 3 nodes is zero
- Is this a reasonable assumption? Why?
- Yuan mentioned in our meeting of October 6 that Bayesian networks can accommodate higher order interactions (while it seems that gaussian graphical models can’t, or have limited capacity to do so)

Parameterization of autologistic model & implications on MCMC

Mitra et al. (2013) state that others say that the “centered” version of autologistic model improves mixing of MCMC posterior simulation
- what are the options for parameterizing autologistic models?
- why would distinct parameterizations impact MCMC mixing?
- how do the reports cited by Mitra et al. (2013) support their statements?
- Mitra et al. (2013) states that their simulation results agree with the results from the cited works (Caragea & Kaiser 2009 and Hughes et al 2011)

Autologistic model: Centered vs. Not centered models

Equations 3 (not centered) & 4 (centered)
if \(e_i\) has no edges, then \[p(e_i|\beta, G) = p(0|\beta, G) e^{\beta_ie_i}\]
no parameters added in going from not centered to centered

Besag (1974)

complete conditional posterior for \(\beta\) requires evaluation of normalization constant \(p(0|\beta, G)\) in the centered model (eqn 4)
- involves a sum over all \(2^m\) possible \(m\)-vectors in \(\lbrace 0,1\rbrace^m\)
- Mitra et al. (2013) discuss later a MCMC posterior simulation method (avoids cmoputing the constant) and an importance sampling method

Next steps

Work through remainder of section 4 & all of section 5 in Mitra et al. (2013)
Can I email Riten to ask for the R package from this study?
Review Kam’s course materials on MCMC & importance sampling
Find a brief note on “autologistic models”
Work through chapter 1 of Højsgaard, Edwards, and Lauritzen (2012) (graphical models with R book)
stochastic block models (Airoldi et al. (2009)) as a possible way to accommodate nodes of two distinct categories??

References

Airoldi, Edoardo M. 2007. “Getting Started in Probabilistic Graphical Models.” PLoS Computational Biology 3 (12). Public Library of Science: e252.

Airoldi, Edoardo M, David M Blei, Stephen E Fienberg, and Eric P Xing. 2009. “Mixed Membership Stochastic Blockmodels.” In Advances in Neural Information Processing Systems, 33–40.

Besag, Julian. 1974. “Spatial Interaction and the Statistical Analysis of Lattice Systems.” Journal of the Royal Statistical Society. Series B (Methodological). JSTOR, 192–236.

Højsgaard, Søren, David Edwards, and Steffen Lauritzen. 2012. Graphical Models with R. Springer.

Mitra, Riten, Peter Müller, Shoudan Liang, Lu Yue, and Yuan Ji. 2013. “A Bayesian Graphical Model for Chip-Seq Data on Histone Modifications.” Journal of the American Statistical Association 108 (501). Taylor & Francis: 69–80.

Introduction to graphical models