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## A General Framework for Mixed Graphical Models

### Citations

3461 |
The Elements of Statistical Learning
- Hastie, Tibshirani, et al.
- 2001
(Show Context)
Citation Context ...ssociate covariates with mixed types of responses. More general regression and predictive models such as Classification and Regression Trees have also been proposed for such mixed types of covariates =-=[17]-=-. Other approaches implicitly account for variables of mixed types in many machine learning procedures using suitable distance or entropy-based measures [17, 18]. There have also been non-parametric e... |

1629 |
Spatial interaction and the statistical analysis of lattice systems (with discussion
- Besag
- 1974
(Show Context)
Citation Context ...non-thin-tailed-continuous data type? This is the question we address in this paper. Towards this, we first briefly review here a recent line of work [46, 48, 50] (which extends earlier work by Besag =-=[5]-=-) which specified undirected graphical model distributions where the variables all belong to one data-type, but which could be any among a wide class of data-types. Their development was as follows. C... |

1585 |
Graphical Models
- Lauritzen
- 1996
(Show Context)
Citation Context ...r p nodes corresponding to the p variables {Xs}ps=1. The graphical model over X corresponding to GX is a set of distributions that satisfy Markov independence assumptions with respect to the graph GX =-=[25]-=-. By the Hammersley-Clifford theorem [11], any such distribution that is strictly positive over its domain also factors according to the graph in the following way. Let CX be a set of cliques (fully-c... |

1049 | A fast iterative shrinkage-thresholding algorithm for linear inverse problems
- Beck, Teboulle
(Show Context)
Citation Context ...node conditional distributions specified in (31). Our M -estimator, based on `1-penalized node-conditional maximum likelihood as described in Section 4, was implemented via projected gradient descent =-=[4]-=-. 24 0 0.1 0.2 0.3 0.4 0.50 0.2 0.4 0.6 0.8 1 FPR TP R n=200 n=100 n=72 n=50 (a) Gau-Ising Mixed MRF 0 0.1 0.2 0.3 0.4 0.50 0.2 0.4 0.6 0.8 1 FPR TP R n=200 n=100 n=72 n=50 (b) Gau MRF-Ising CRF 0 0.1... |

814 | Graphical models, exponential families, and variational inference
- Wainwright, Jordan
- 2008
(Show Context)
Citation Context ...del family represented by the graph GX takes the form: P[X] ∝ exp { ∑ c∈CX θcφc(Xc) } , (1) where {θc} are weights over the sufficient statistics. Popular instances of this model include Ising models =-=[44]-=- for discrete-valued qualitative variables, and Gaussian MRFs [40] for continuous-valued quantitative variables. Ising models specify joint distributions over a set of binary variables each with domai... |

730 | High-dimensional graphs and variable selection with the lasso. The Annals of Statistics 34
- Meinshausen, Bühlmann
- 2006
(Show Context)
Citation Context ...g these mixed CRFs (30); since the graph factors according to these mixed CRFs, we can estimate each one independently. We propose to do so following the node-wise neighborhood estimation approach of =-=[34, 35, 47, 48]-=-, which allows us to side-step the task of computing the logpartition function of the mixed CRFs. These neighborhood selection approaches seek to learn the 22 network structure through an `1-norm pena... |

294 | Approximate Bayesian inference for latent Gaussian models
- Martino
- 2007
(Show Context)
Citation Context ..., are hierarchical models that permit dependencies through latent variables. For example, Sammel et al. [37] propose a latent variable model for mixed continuous and count variables, while Rue et al. =-=[36]-=- propose latent Gaussian models that permit dependencies through a latent Gaussian MRF. While these methods provide statistical models for mixed data, they model dependencies between observed variable... |

273 |
Genome Atlas Research Network. Comprehensive genomic characterization defines human glioblastoma genes and core pathways. Nature
- Cancer
(Show Context)
Citation Context ...del the connections within the set of mutations and gene expression levels. For our analysis, we use publicly available data on invasive breast carcinoma available from the Cancer Genome Atlas (TCGA) =-=[6]-=-. Level III RNA-sequencing data for 806 patients was downloaded and pre-processed using techniques described in Allen and Liu [1], so that the expression levels can be well-modeled with the Poisson di... |

223 |
Graphical models for associations between variables, some of which are qualitative and some quantitative.
- Lauritzen, Wermuth
- 1989
(Show Context)
Citation Context ...onal data. Due in part to its importance, there has been some recent set of proposals towards such direct parametric statistical models, building on some seminal earlier work by Lauritzen and Wermuth =-=[26]-=-. We review these in the next sub-section after first providing some background on Markov Random Fields (MRFs). As we will show in the next sub-section, these are however largely targeted to the case ... |

172 | Propagation of probabilities, means and variances in mixed graphical association models.
- Lauritzen
- 1992
(Show Context)
Citation Context ...X2s 2σ2s } . (2) 1.2.2 Conditional Gaussian Models We now review conditional Gaussian models, which were the first proposed class of mixed graphical models, introduced in [26], and further studied in =-=[16, 24, 27, 25]-=-. Let Y := (Y1, . . . , Yp) ∈ Rp be a continuous response random vector, and let X := (X1, . . . , Xq) ∈ {1, . . . , k}q be a discrete covariate random vector. Taken together, (X,Y ) is then a mixed r... |

116 | Estimating high-dimensional directed acyclic graphs with the PC-algorithm
- Kalisch, Bühlmann
- 2007
(Show Context)
Citation Context ...ncies even if the true underlying BDMRF is unknown; further theoretical and empirical investigations that build on these are needed. Also, while there are several popular algorithms for learning DAGs =-=[20]-=-, little is known about estimating MRFs with mixed directed and undirected edges, an open area of future research. Overall, we have proposed a novel class of distributions for mixed variables, that ha... |

103 |
Gaussian Markov distributions over finite graphs.
- Speed, Kiiveri
- 1986
(Show Context)
Citation Context ...{ ∑ c∈CX θcφc(Xc) } , (1) where {θc} are weights over the sufficient statistics. Popular instances of this model include Ising models [44] for discrete-valued qualitative variables, and Gaussian MRFs =-=[40]-=- for continuous-valued quantitative variables. Ising models specify joint distributions over a set of binary variables each with domain X = {0, 1}, with the form P[X] ∝ exp { ∑ (s,t)∈EX θstXsXt } , 3 ... |

92 |
The p53 tumour suppressor gene.
- AJ, Momand, et al.
- 1991
(Show Context)
Citation Context ...covers several novel connections to be investigated and further validated in future work; these include: the TP53 mutation is linked to ADAM6 expression; TP53 29 is a well known tumor suppressor gene =-=[30]-=- and ADAM6 is a long non-coding RNA over-expressed in breast cancer [38]. The FGF3 mutation is linked to CCND1 expression; FGF3 regulates estrogen expanding breast cancer stem cells [15] and CDN1 lead... |

92 | The nonparanormal: Semiparametric estimation of high dimensional undirected graphs.
- Liu, Lafferty, et al.
- 2009
(Show Context)
Citation Context ...procedures using suitable distance or entropy-based measures [17, 18]. There have also been non-parametric extensions of probabilistic graphical models using copulas [13, 33] or rank-based estimators =-=[45, 31]-=-, which could potentially be used for mixed data; non-parametric methods, however, may suffer from a loss of statistical efficiency when compared to parametric families, especially under very highdime... |

62 |
Latent variable models for mixed discrete and continuous outcomes
- Sammel, Ryan, et al.
- 1997
(Show Context)
Citation Context ...r such mixed data, the most popular, especially in survey statistics and spatial statistics [12], are hierarchical models that permit dependencies through latent variables. For example, Sammel et al. =-=[37]-=- propose a latent variable model for mixed continuous and count variables, while Rue et al. [36] propose latent Gaussian models that permit dependencies through a latent Gaussian MRF. While these meth... |

58 |
Highdimensional ising model selection using `1-regularized logistic regression.
- Ravikumar, Wainwright, et al.
- 2010
(Show Context)
Citation Context ...g these mixed CRFs (30); since the graph factors according to these mixed CRFs, we can estimate each one independently. We propose to do so following the node-wise neighborhood estimation approach of =-=[34, 35, 47, 48]-=-, which allows us to side-step the task of computing the logpartition function of the mixed CRFs. These neighborhood selection approaches seek to learn the 22 network structure through an `1-norm pena... |

55 | Markov random fields in statistics
- Clifford
- 1990
(Show Context)
Citation Context ...s {Xs}ps=1. The graphical model over X corresponding to GX is a set of distributions that satisfy Markov independence assumptions with respect to the graph GX [25]. By the Hammersley-Clifford theorem =-=[11]-=-, any such distribution that is strictly positive over its domain also factors according to the graph in the following way. Let CX be a set of cliques (fully-connected subgraphs) of the graph GX , and... |

40 |
Cyclin D1 in breast cancer pathogenesis.
- Arnold, Papanikolaou
- 2005
(Show Context)
Citation Context ...t cancer [38]. The FGF3 mutation is linked to CCND1 expression; FGF3 regulates estrogen expanding breast cancer stem cells [15] and CDN1 leads to over-expression of hormone receptors in breast cancer =-=[3]-=-. The PIK3CA mutation is linked to CLEC3A expression and NAT1 expression; PIK3CA is a known oncogene [10], CLEC3A affects tumor metastasis [42], and NAT1 is a potential marker for the estrogen recepto... |

40 | Statistical Estimation of Correlated Genome Associations to a Quantitative Trait Network. PLoS Genetics 5
- Kim, Xing
- 2009
(Show Context)
Citation Context ... learning models [2]. These are especially popular in expression quantitative trait loci (eQTL) analyses, which seek to link changes in functional gene expression levels to specific genomic mutations =-=[22]-=-. Recent approaches [51] further allow these multiple regression models to associate covariates with mixed types of responses. More general regression and predictive models such as Classification and ... |

40 |
The ADAMs family of metalloproteases: multidomain proteins with multiple functions.
- Seals, Courtneidge
- 2003
(Show Context)
Citation Context ...d in future work; these include: the TP53 mutation is linked to ADAM6 expression; TP53 29 is a well known tumor suppressor gene [30] and ADAM6 is a long non-coding RNA over-expressed in breast cancer =-=[38]-=-. The FGF3 mutation is linked to CCND1 expression; FGF3 regulates estrogen expanding breast cancer stem cells [15] and CDN1 leads to over-expression of hormone receptors in breast cancer [3]. The PIK3... |

38 | Stability approach to regularization selection (stars) for high dimensional graphical models. Arxiv preprint arXiv:1006.3316,
- Liu, Roeder, et al.
- 2010
(Show Context)
Citation Context ...ng et al. [49] for further details. Node-wise neighborhood selection as described in Section 4 was employed to learn the edge structure of the network. Stability selection, as described in Liu et al. =-=[32]-=-, was used with parameter level β = 0.01 to determine the optimal level of regularization. Our estimated network is presented in Figure 7 where blue nodes denote gene expression biomarkers and yellow ... |

36 |
Decomposition of maximum likelihood in mixed graphical interaction models.
- Frydenberg, Lauritzen
- 1989
(Show Context)
Citation Context ...X2s 2σ2s } . (2) 1.2.2 Conditional Gaussian Models We now review conditional Gaussian models, which were the first proposed class of mixed graphical models, introduced in [26], and further studied in =-=[16, 24, 27, 25]-=-. Let Y := (Y1, . . . , Yp) ∈ Rp be a continuous response random vector, and let X := (X1, . . . , Xq) ∈ {1, . . . , k}q be a discrete covariate random vector. Taken together, (X,Y ) is then a mixed r... |

31 | Copula gaussian graphical models and their application to modeling functional disability data. The Annals of Applied Statistics
- Dobra, Lenkoski
- 2011
(Show Context)
Citation Context ...ed types in many machine learning procedures using suitable distance or entropy-based measures [17, 18]. There have also been non-parametric extensions of probabilistic graphical models using copulas =-=[13, 33]-=- or rank-based estimators [45, 31], which could potentially be used for mixed data; non-parametric methods, however, may suffer from a loss of statistical efficiency when compared to parametric famili... |

27 | Heterogeneous multitask learning with joint sparsity constraints - Yang - 2009 |

20 | Penalized likelihood methods for estimation of sparse highdimensional directed acyclic graphs
- Shojaie, Michailidis
(Show Context)
Citation Context ...iables within each block. Learning the structure of even solely directed edges of directed graphical models is known to be an NP-hard problem [25], unless a partial ordering of the variables is known =-=[39]-=-. Accordingly, in this paper, we assume that the partial ordering of the blocked DAG is known a-priori, based on domain knowledge. This assumption is especially relevant in areas such as high-throughp... |

19 | Adaptive multitask Lasso: With application to eQTL detection,” in
- Lee, Zhu, et al.
- 2010
(Show Context)
Citation Context ...tions, affect gene expression levels. Note that the standard method for finding connections between mutations and expression levels, commonly called expression Quantitative Trait Loci (eQTL) analysis =-=[29]-=-, uses regression models to find connections between the two different types of biomarkers but cannot model the connections within the set of mutations and gene expression levels. For our analysis, we... |

19 | Graphical models via generalized linear models.
- Yang, Allen, et al.
- 2012
(Show Context)
Citation Context ...inuous, or belonged to some other non-categorical non-thin-tailed-continuous data type? This is the question we address in this paper. Towards this, we first briefly review here a recent line of work =-=[46, 48, 50]-=- (which extends earlier work by Besag [5]) which specified undirected graphical model distributions where the variables all belong to one data-type, but which could be any among a wide class of data-t... |

14 | Discovery and validation of breast cancer subtypes
- Kapp, Jeffrey, et al.
(Show Context)
Citation Context ...s that we identify are well-known breast cancer biomarkers: the GATA3 mutation is linked to SLC39A6 expression; the ratio of these gene’s expressions levels are used to define breast cancer sub-types =-=[21]-=- and both of these biomarkers have been previously implicated in breast cancer [43]. The FGFR1 mutation is linked to PEG3 expression; the former regulates growth factors that are known to be amplified... |

11 |
Pontil M. Multi-task feature learning. Advances in neural information processing systems
- Evgeniou
- 2007
(Show Context)
Citation Context ...together. Instead, they relate a set of multivariate response variables of one type to multivariate covariate variables of another type, using multiple regression models or multi-task learning models =-=[2]-=-. These are especially popular in expression quantitative trait loci (eQTL) analyses, which seek to link changes in functional gene expression levels to specific genomic mutations [22]. Recent approac... |

10 |
On graphical models via univariate exponential family distributions.
- Yang, Ravikumar, et al.
- 2015
(Show Context)
Citation Context ...g these mixed CRFs (30); since the graph factors according to these mixed CRFs, we can estimate each one independently. We propose to do so following the node-wise neighborhood estimation approach of =-=[34, 35, 47, 48]-=-, which allows us to side-step the task of computing the logpartition function of the mixed CRFs. These neighborhood selection approaches seek to learn the 22 network structure through an `1-norm pena... |

6 |
Hierarchical clustering of mixed data based on distance hierarchy,"
- Hsu, Chen, et al.
- 2007
(Show Context)
Citation Context ...roposed for such mixed types of covariates [17]. Other approaches implicitly account for variables of mixed types in many machine learning procedures using suitable distance or entropy-based measures =-=[17, 18]-=-. There have also been non-parametric extensions of probabilistic graphical models using copulas [13, 33] or rank-based estimators [45, 31], which could potentially be used for mixed data; non-paramet... |

6 |
GATA-3 expression in breast cancer has a strong association with estrogen receptor but lacks independent prognostic value
- Voduc, Cheang, et al.
(Show Context)
Citation Context ...inked to SLC39A6 expression; the ratio of these gene’s expressions levels are used to define breast cancer sub-types [21] and both of these biomarkers have been previously implicated in breast cancer =-=[43]-=-. The FGFR1 mutation is linked to PEG3 expression; the former regulates growth factors that are known to be amplified in breast cancer [19] while the latter modulates the related process of cancer pro... |

6 |
On poisson graphical models.
- Yang, Ravikumar, et al.
- 2013
(Show Context)
Citation Context ...ial family distributions can be used to specify these homogeneous pairwise EBDMRFs. A particularly interesting class of these could be the variants of the Poisson distribution proposed by Yang et al. =-=[49]-=- to build Poisson graphical models that permit both positive and negative conditional dependencies. Within our EBDMRFs, these could be used to expand the possible formulations of mixed Poisson graphic... |

5 | High-dimensional mixed graphical models. ArXiv e-prints, arXiv:1304.2810, - Cheng, Levina, et al. - 2013 |

5 |
Stable graphical model estimation with random forests for discrete, continuous, and mixed variables. arXiv preprint arXiv:1109.0152
- FELLINGHAUER, BÜHLMANN, et al.
- 2011
(Show Context)
Citation Context ...rom a loss of statistical efficiency when compared to parametric families, especially under very highdimensional sampling regimes. Others have proposed to build network models based on random forests =-=[14]-=- which are able to handle mixed types of variables, but these do not correspond to a multivariate density. 2 Among parametric statistical modeling approaches for such mixed data, the most popular, esp... |

5 |
Estrogen expands breast cancer stem-like cells through paracrine FGF/Tbx3 signaling
- Fillmore
- 2010
(Show Context)
Citation Context ...ppressor gene [30] and ADAM6 is a long non-coding RNA over-expressed in breast cancer [38]. The FGF3 mutation is linked to CCND1 expression; FGF3 regulates estrogen expanding breast cancer stem cells =-=[15]-=- and CDN1 leads to over-expression of hormone receptors in breast cancer [3]. The PIK3CA mutation is linked to CLEC3A expression and NAT1 expression; PIK3CA is a known oncogene [10], CLEC3A affects tu... |

5 |
Targeting fibroblast growth factor receptors blocks PI3K/AKT signaling, induces apoptosis, and impairs mammary tumor outgrowth and metastasis
- Dey, Bianchi, et al.
- 2010
(Show Context)
Citation Context ... biomarkers have been previously implicated in breast cancer [43]. The FGFR1 mutation is linked to PEG3 expression; the former regulates growth factors that are known to be amplified in breast cancer =-=[19]-=- while the latter modulates the related process of cancer progression [41]. The STAT3 mutation is linked to ERBB2 expression; these are known to be amplified in HERB2 sub-types [9]. Our estimated netw... |

5 | Mixed graphical association models [with discussion and reply]. - Lauritzen, Andersen, et al. - 1989 |

5 |
Learning mixed graphical models. arXiv preprint arXiv:1205.5012,
- Lee, Hastie
- 2012
(Show Context)
Citation Context ...onents are zero, so that gd(X) = 0, hd(X) = 0,Kd(X) = 0. Recently, there have been several proposals for estimating the graph structure of these CG models in high-dimensional settings. Lee and Hastie =-=[28]-=- consider a specialization of CG models involving only pairwise interactions between any two variables and propose sparse node-wise estimators for graph selection. Cheng et al. [8] further consider th... |

4 |
PIK3CA mutation impact on survival in breast cancer patients and in ERα, PR and ERBB2-based subgroups. Breast Cancer Res
- Cizkova, Susini, et al.
(Show Context)
Citation Context ... cancer stem cells [15] and CDN1 leads to over-expression of hormone receptors in breast cancer [3]. The PIK3CA mutation is linked to CLEC3A expression and NAT1 expression; PIK3CA is a known oncogene =-=[10]-=-, CLEC3A affects tumor metastasis [42], and NAT1 is a potential marker for the estrogen receptor positive sub-type [23]. Overall, this genomics example demonstrates the direct applicability of our cla... |

4 | Mixed graphical models via exponential families.
- Yang, Baker, et al.
- 2014
(Show Context)
Citation Context ...inuous, or belonged to some other non-categorical non-thin-tailed-continuous data type? This is the question we address in this paper. Towards this, we first briefly review here a recent line of work =-=[46, 48, 50]-=- (which extends earlier work by Besag [5]) which specified undirected graphical model distributions where the variables all belong to one data-type, but which could be any among a wide class of data-t... |

3 |
A local poisson graphical model for inferring networks from sequencing data
- Allen, Liu
- 2013
(Show Context)
Citation Context ...nvasive breast carcinoma available from the Cancer Genome Atlas (TCGA) [6]. Level III RNA-sequencing data for 806 patients was downloaded and pre-processed using techniques described in Allen and Liu =-=[1]-=-, so that the expression levels can be well-modeled with the Poisson distribution. For the aberration data, we used Level II nonsilent somatic mutations and Level III copy number variation data for 95... |

3 |
Statistics for spatial data, volume 900
- Cressie, Cassie
- 1993
(Show Context)
Citation Context ...but these do not correspond to a multivariate density. 2 Among parametric statistical modeling approaches for such mixed data, the most popular, especially in survey statistics and spatial statistics =-=[12]-=-, are hierarchical models that permit dependencies through latent variables. For example, Sammel et al. [37] propose a latent variable model for mixed continuous and count variables, while Rue et al. ... |

3 |
CNTools: Convert segment data into a region by sample matrix to allow for other high level computational analyses. R package (Version 1.6.0
- Zhang
(Show Context)
Citation Context ...oisson distribution. For the aberration data, we used Level II nonsilent somatic mutations and Level III copy number variation data for 951 patients. The later was segmented using standard techniques =-=[52]-=- and merged with the mutation data at the gene level to form a binary matrix, indicating whether a mutation or copy number aberration is present or absent in the coding region of the gene. This leaves... |

2 |
Selection and estimation for mixed graphical models. ArXiv e-prints, arXiv:1311.0085,
- Chen, Witten, et al.
- 2013
(Show Context)
Citation Context ...e distributions for variables of varied data-types, they are nonetheless specified for the setting where all the variables belong to the same type. Accordingly, there have been some recent extensions =-=[50, 7]-=- of the above for the more general setting of interest in this paper, where each variable belongs to a potentially different type. Their construction was as follows, and can be seen to be an extension... |

2 |
Promoter hypomethylation of the N-acetyltransferase 1 gene in breast cancer
- Kim, Kang, et al.
- 2008
(Show Context)
Citation Context ...n is linked to CLEC3A expression and NAT1 expression; PIK3CA is a known oncogene [10], CLEC3A affects tumor metastasis [42], and NAT1 is a potential marker for the estrogen receptor positive sub-type =-=[23]-=-. Overall, this genomics example demonstrates the direct applicability of our class of BDMRF models for learning relevant connections both within and between cancer biomarkers of different types. 7 Di... |

2 |
et al. High-dimensional semiparametric gaussian copula graphical models. The Annals of Statistics
- Liu, Han, et al.
(Show Context)
Citation Context ...ed types in many machine learning procedures using suitable distance or entropy-based measures [17, 18]. There have also been non-parametric extensions of probabilistic graphical models using copulas =-=[13, 33]-=- or rank-based estimators [45, 31], which could potentially be used for mixed data; non-parametric methods, however, may suffer from a loss of statistical efficiency when compared to parametric famili... |

1 |
Stat3 activation in her2-overexpressing breast cancer promotes epithelial-mesenchymal transition and cancer stem cell traits
- Chung, Giehl, et al.
(Show Context)
Citation Context ...d in breast cancer [19] while the latter modulates the related process of cancer progression [41]. The STAT3 mutation is linked to ERBB2 expression; these are known to be amplified in HERB2 sub-types =-=[9]-=-. Our estimated network also discovers several novel connections to be investigated and further validated in future work; these include: the TP53 mutation is linked to ADAM6 expression; TP53 29 is a w... |

1 |
a nontransforming cancer progression gene, is a positive regulator of cancer aggressiveness and angiogenesis
- Peg-3
- 1999
(Show Context)
Citation Context ...R1 mutation is linked to PEG3 expression; the former regulates growth factors that are known to be amplified in breast cancer [19] while the latter modulates the related process of cancer progression =-=[41]-=-. The STAT3 mutation is linked to ERBB2 expression; these are known to be amplified in HERB2 sub-types [9]. Our estimated network also discovers several novel connections to be investigated and furthe... |

1 |
Matrilysin (mmp-7) cleaves c-type lectin domain family 3 member a (clec3a) on tumor cell surface and modulates its cell adhesion activity
- Tsunezumi, Higashi, et al.
(Show Context)
Citation Context ... to over-expression of hormone receptors in breast cancer [3]. The PIK3CA mutation is linked to CLEC3A expression and NAT1 expression; PIK3CA is a known oncogene [10], CLEC3A affects tumor metastasis =-=[42]-=-, and NAT1 is a potential marker for the estrogen receptor positive sub-type [23]. Overall, this genomics example demonstrates the direct applicability of our class of BDMRF models for learning releva... |

1 |
et al. Regularized rank-based estimation of high-dimensional nonparanormal graphical models
- Xue, Zou
(Show Context)
Citation Context ...procedures using suitable distance or entropy-based measures [17, 18]. There have also been non-parametric extensions of probabilistic graphical models using copulas [13, 33] or rank-based estimators =-=[45, 31]-=-, which could potentially be used for mixed data; non-parametric methods, however, may suffer from a loss of statistical efficiency when compared to parametric families, especially under very highdime... |

1 |
Conditional random fields via univariate exponential families
- Yang, Ravikumar, et al.
(Show Context)
Citation Context ...inuous, or belonged to some other non-categorical non-thin-tailed-continuous data type? This is the question we address in this paper. Towards this, we first briefly review here a recent line of work =-=[46, 48, 50]-=- (which extends earlier work by Besag [5]) which specified undirected graphical model distributions where the variables all belong to one data-type, but which could be any among a wide class of data-t... |