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Estimation of a Structural Vector Autoregression Model Using NonGaussianity
"... Analysis of causal effects between continuousvalued variables typically uses either autoregressive models or structural equation models with instantaneous effects. Estimation of Gaussian, linear structural equation models poses serious identifiability problems, which is why it was recently proposed ..."
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Analysis of causal effects between continuousvalued variables typically uses either autoregressive models or structural equation models with instantaneous effects. Estimation of Gaussian, linear structural equation models poses serious identifiability problems, which is why it was recently proposed to use nonGaussian models. Here, we show how to combine the nonGaussian instantaneous model with autoregressive models. This is effectively what is called a structural vector autoregression (SVAR) model, and thus our work contributes to the longstanding problem of how to estimate SVAR’s. We show that such a nonGaussian model is identifiable without prior knowledge of network structure. We propose computationally efficient methods for estimating the model, as well as methods to assess the significance of the causal influences. The model is successfully applied on financial and brain imaging data.
Latent Composite Likelihood Learning for the Structured Canonical Correlation Model: Supplementary Material
"... We present four pieces of supplementary material: first, an approach for the CDN inference problem of computing likelihood functions, which for our purposes we believe it is simpler to implement than other approaches presented in the literature; second, a discussion of the convergence of LEARNSTRUCT ..."
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We present four pieces of supplementary material: first, an approach for the CDN inference problem of computing likelihood functions, which for our purposes we believe it is simpler to implement than other approaches presented in the literature; second, a discussion of the convergence of LEARNSTRUCTUREDCCAII; third, brief comments on identification and initialization; fourth, details on the preprocessing of the NHS data. 1 SIMPLER CDN INFERENCE An efficient procedure for transforming CDFs into PMFs is given in detail by Huang et al. (2010), which is particularly sophisticated and seemingly hard to implement. However, one can reduce the problem of computing PMFs from CDFs following the structure of Equation (5) – itself just a rearrangement of the general formulation (Joe, 1997) for binary variables: just introduce “pseudo ” random variables corresponding to the difference indicatorsZand construct the corresponding factor graph. Notice that the term (−1) ∑p i=1 zi is itself a product of univariate factors over the pseudo set Z. Equation (5) is the “marginal ” of a pseudo distributionP(Z,Y) and can be found by any standard exact method of inference. We used junction trees. Figure 1 shows an example of reducing the problem of computing the PMF of graph Y1 ↔ Y2 ↔ Y3. The result is analogous in the continuous case: one just have to create indicator variables that pick which factors are being derived and which are not. This simple link is not mentioned in previous papers, to the best of our knowledge. In any case, the customized method described by Huang et al. (2010) readily includes details on how to generate parameter gradients, and it is useful as a framework for developing approximate algorithms (as already hinted by Huang and Frey, 2008): in our case, the
Discovering Cyclic Causal Models with Latent Variables: A General SATBased Procedure
"... We present a very general approach to learning the structure of causal models based on dseparation constraints, obtained from any given set of overlapping passive observational or experimental data sets. The procedure allows for both directed cycles (feedback loops) and the presence of latent varia ..."
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We present a very general approach to learning the structure of causal models based on dseparation constraints, obtained from any given set of overlapping passive observational or experimental data sets. The procedure allows for both directed cycles (feedback loops) and the presence of latent variables. Our approach is based on a logical representation of causal pathways, which permits the integration of quite general background knowledge, and inference is performed using a Boolean satisfiability (SAT) solver. The procedure is complete in that it exhausts the available information on whether any given edge can be determined to be present or absent, and returns “unknown ” otherwise. Many existing constraintbased causal discovery algorithms can be seen as special cases, tailored to circumstances in which one or more restricting assumptions apply. Simulations illustrate the effect of these assumptions on discovery and how the present algorithm scales. 1
Learning Directed Graphical Models from Nonlinear and NonGaussian Data Data Analysis Project for Master of Science in Machine Learning
"... Traditional constraintbased and scorebased methods for learning directed graphical models from continuous data have two significant limitations: (i) they require (in practice) assuming dependencies are linear with Gaussian noise; (ii) they cannot distinguish between Markov equivalent structures. M ..."
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Traditional constraintbased and scorebased methods for learning directed graphical models from continuous data have two significant limitations: (i) they require (in practice) assuming dependencies are linear with Gaussian noise; (ii) they cannot distinguish between Markov equivalent structures. More recent structure learning methods avoid both limitations by directly exploiting characteristics of the observed data distribution resulting from nonlinear effects and nonGaussian noise. We review these methods and focus on the additive noise model approach, which while more general than traditional approaches also suffers from two major limitations: (i) it is invertible for certain distribution families, i.e. linear Gaussians, and thus not useful for structure learning in these cases; (ii) it was originally proposed for the two variable case with a multivariate extension that requires enumerating all possible DAGs, which is ususally intractable. To address these two limitations, we introduce weakly additive noise models, which extends the additive noise model framework to cases where additive noise models are invertible and noise is not additive.
Sparse Linear Identifiable Multivariate Modeling
"... In this paper we consider sparse and identifiable linear latent variable (factor) and linear Bayesian network models for parsimonious analysis of multivariate data. We propose a computationally efficient method for joint parameter and model inference, and model comparison. It consists of a fully Bay ..."
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In this paper we consider sparse and identifiable linear latent variable (factor) and linear Bayesian network models for parsimonious analysis of multivariate data. We propose a computationally efficient method for joint parameter and model inference, and model comparison. It consists of a fully Bayesian hierarchy for sparse models using slab and spike priors (twocomponent δfunction and continuous mixtures), nonGaussian latent factors and a stochastic search over the ordering of the variables. The framework, which we call SLIM (Sparse Linear Identifiable Multivariate modeling), is validated and benchmarked on artificial and real biological data sets. SLIM is closest in spirit to LiNGAM (Shimizu et al., 2006), but differs substantially in inference, Bayesian network structure learning and model comparison. Experimentally, SLIM performs equally well or better than LiNGAM with comparable computational complexity. We attribute this mainly to the stochastic search strategy used, and to parsimony (sparsity and identifiability), which is an explicit part of the model. We propose two extensions to the basic i.i.d. linear framework: nonlinear dependence on observed variables, called SNIM (Sparse Nonlinear Identifiable Multivariate modeling) and allowing for correlations between latent variables, called CSLIM (Correlated SLIM), for the temporal and/or spatial data. The source code and scripts are available from