Nihat Ay

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References

J. Rauh, N. Ay, 2011. Robustness and Conditional Independence Ideals. arXiv

Nihat Ay and David C. Krakauer. 2007. Geometric robustness theory and biological networks. Theory in Biosciences 125 (2): 93-121.

Nihat Ay, Jessica C. Flack, and David C. Krakauer. 2007. Robustness and Complexity Co-constructed in Multimodal Signaling Networks. Philosophical Transactions of the Royal Society. London. 362: 441-447. http://www.jstor.org/stable/20209856

Nihat Ay and Daniel Polani. 2006. Information Flows in Causal Networks santa fe institute workingpaper

Wolfgang Löhr and Nihat Ay. 2008. On the Generative Nature of Prediction santa fe institute workingpaper

Nihat Ay. 2008. A Refinement of the Common Cause Principle santa fe institute workingpaper

Causal Graphs Laboratory paper: http://www.santafe.edu/education/schools/complex-systems-summer-schools/2011-program-info/

Prokopenko, M.; Nihat ay; Obst, Olivier; Polani, D. 2010. Phase transitions in least-effort communications Journal of Statistical Mechanics – Theory and Experiment, November 2010

Main References for Causal Graphs

Lauritzen, Steffen. 1996. Graphical Models.

Ch.3

Markovian parents - Unidirectional Graph

p36 Main theorem

p46

p48 3.25 D-separation (note Union...)^m <- m designates "moral" graph (modal?) parents have to marry

(see Lauritzen slides below, 85% through)

ancestral graph... do all parents (some operation => D-separation

p51 Interaction theory = D-sep in Markov graph

...=> Global => Local => Pairwise --- related to Factorization F

product Str p35 Density

Phi $\phi$ the interaction => physics literature

interaction graphs => Interaction potential $\phi$

Cowell, Robert G., Philip Dawid, Steffen L. Lauritzen, and D. J. Spiegelhalter. 1999. Probabilistic Networks and Expert Systems: Exact Computational Methods for Bayesian Networks (Information Science and Statistics). Berlin: Springer - Verlag

index: see interaction potential p.28 3.1 p.86-89, 6.2 - 6.2.1 6.333 Pi $\pi$ used by physicists (product) refers to interaction

for the energy thing - books on graphical models. E,g., Winkler is one of the many p61.

Lauritzen Steffen L. and Speigelhalter, D.J. (1988) Local computation with probabilities on graphical structures and their application to expert systems (with discusssion) J.R.Statist. B, 50, 157 - 224. Pearl. Berlin: Springer - Verlag.

Supplementary

Gerhard Winkler. 1995. Image Analysis, Random Fields, and Dynamic Monte Carlo Methods. Berlin: Springer - Verlag

p61 Corollary 3.3.2 Gibbs field potential - physical lanugage - he makes explicit the physics(physical) connection

normalized vacuum potential

Size = the interaction potential

Part 2,p62: last paragraph Gibbs fields and Dynamic Monte Carlo Methods. Gibbs Sampler chapter. Ising model

Jordan, Michael, ed. 2001. Learning in Graphical Models. Cambridge, MA: MIT Press.

his chapter, et al.: "An Introduction to variational methods for Graphical Models."

Summary

Jordan, Michael (UC Berkeley) - has a new book coming out,

History and overview

Lauritzen, Steffen. 2011. Graphical Models. PPT in PDF. Graduate Lectures, Oxford.

slides 2/3rd the way thru:

A particular successful development is associated with BUGS, (Gilks et al., 1994) (WinBUGS, OpenBUGS).

it enables a Bayesian analyst to focus on substantive modelling whereas the technical model specification and computational side is taken care of automatically,

exploiting modularity, factorization, and MCMC methodology, including the Gibbs and Metropolis–Hastings sample

Conforming with Bayesian paradigm, parameters and observations are explicitly represented in model as nodes in graph, all being observables;

Linear regression (next slide) model { }

for( i in 1 : N ) { Y[i] ~ dnorm(mu[i],tau) mu[i] <- alpha + beta * (x[i] - xbar)

} tau ~ dgamma(0.001,0.001) sigma <- 1 / sqrt(tau) alpha ~ dnorm(0.0,1.0E-6) beta ~ dnorm(0.0,1.0E-6)

Data and BUGS model for pumps

The number of failures Xi is assumed to follow a Poisson distribution with parameter θi ti , i = 1, . . . , 10

where θi is the failure rate for pump i and ti is the length of operation time of the pump (in 1000s of hours). The data are shown below.

Pump 1 2 3 4 5 6 7 8 9 10 ti 94.5 15.7 62.9 126 5.24 31.4 1.05 1.05 2.01 10.5 xi 5 1 5 14 3 19 1 1 4 22

A gamma prior distribution is adopted for the failure rates: θi ∼ Γ(α,β),i = 1,...,10

BUGS program for pumps

With suitable priors the program becomes

for (i in 1 : N) { theta[i] ~ dgamma(alpha, beta) lambda[i] <- theta[i] * t[i] x[i] ~ dpois(lambda[i])

} alpha ~ dexp(1) beta ~ dgamma(0.1, 1.0)model {model {

}

Moral graph example

The DAG is used for modular specification of the model, and the moral graph for local computatio

Is a huge conceptual extension of so-called Bayesian hierarchical model

distinction prior/likelihood and parameter/random variable less well define

If founder nodes in network are considered fixed and unknown, no reason not to consider models in Fisherian paradigm.

SEM

In Pearl 2009 p27 1.4.1 Structural Equations

1.40 Noise, pa stands for parents that directly determine the value of X and U represent errors ("disturbances") due to omitted factors

1.41 Linear --> Fit data assuming interaction. p28 price demand income wages, p29 do operator, Predictions, Interventions, Counterfactuals, as types of queries

p1 use of probabilities, p2 eliminating paradoxes

• Energy : Gibbs, W. (1902). Elementary Principles of Statistical Mechanics. NewHaven, Connecticut: Yale University Press. (ref in Lauritzen 2011)

Bartlett 1935

Darroch 1980. "Papers setting the scene include Darroch et al. (1980), Wermuth and Lauritzen (1983), and Lauritzen and Wermuth (1989)." (ref in Lauritzen 2011)

Darroch, J. N., Steffen L. Lauritzen, and T. P. Speed (1980). Markov fields and log-linear interaction models for contingency tables. The Annals of Statistics 8, 522–539.

• Wright 1921, 1923 extends from Gaussian case by Pearl 2009:27-38 "Structural equations" and functional causal models.
• Wold, Herman O. A. 1954. Causality and Econometrics. Econometrica 22: 162-177.
• Nihat says that the SEM is deterministic since the functional form = f(pa_i, u_i) in SEM (and modified in Hal White's papers) can be integrated and turned into a probabilistic form, similar to regression: x_i= SUM_k.ne.1a_ikx_k + u_i, ..., n.

as in Lauritsen, Lauritsen doesn't believe the deterministic form is needed, although Pearl provides for it (actually both), as in 2009p27: 1.40 and 1.41.