Maximum likelihood estimation in Gaussian models under total positivity

Lauritzen, Steffen; Uhler, Caroline; Zwiernik, Piotr

doi:10.1214/17-aos1668

Cited by 48 publications

(54 citation statements)

References 36 publications

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“…By [20,Theorem 2], all partial correlations of the associated Gaussian distribution are nonnegative. Following [23], this is precisely what it means for a distribution to be MTP 2 . Hence, all a ij|K are nonnegative for our matrix Σ.…”

Section: Oriented Gaussoids and Positivitymentioning

confidence: 98%

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The Geometry of Gaussoids

et al. 2018

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A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lněnička and Matúš are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models. a 12|3 maps to σ 12 σ 33 − σ 13 σ 23 whereas a 13|2 maps to −(σ 12 σ 23 − σ 13 σ 22 ). Maintaining this sign convention is important to keep the algebra consistent with its statistical interpretation.Let J n denote the ideal generated by all homogeneous polynomials in the kernel of the map above. This defines an irreducible variety V (J n ) of dimension n+1 2 in the projective space P 2 n−2 (4+( n 2 ))−1 whose coordinates are P ∪ A. There is a natural projection from LGr(n, 2n) onto V (J n ), obtained by deleting all minors that are neither principal nor almost-principal. This is analogous to [30, Observation III.12], where the focus was on principal minors p I .Proposition 1.1. The degree of the projective variety of principal and almost-principal minors coincides with the degree of the Lagrangian Grassmannian. For n ≥ 2, it equals degree(V (J n )) = degree(LGr(n, 2n)) = n+1 2 ! 1 n · 3 n−1 · 5 n−2 · · · (2n − 1).

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Section: Oriented Gaussoids and Positivitymentioning

confidence: 98%

“…In Theorem 5.6 we prove the same fact for gaussoids. Positive gaussoids are important for statistics, because they correspond to the MTP 2 distributions, which have received a lot of attention in the recent literature [11,23].…”

Section: Oriented Gaussoids and Positivitymentioning

confidence: 99%

The Geometry of Gaussoids

et al. 2018

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“…In this context, a relevant role is played by the family of Gaussian totally positive distributions of order two (MTP 2 ). Fallat et al (2017) and Lauritzen et al (2019) studied the multivariate association structure of MTP 2 distributions and showed that they satisfy useful properties within the undirected graphical model framework. Interestingly, a Gaussian distribution is MTP 2 if and only if all the entries of R are either zero or positive (Karlin and Rinott, 1983;Fallat et al, 2017).…”

Section: Interpretation Of Covariance Path Weightsmentioning

confidence: 99%

“…It follows immediately that these distributions have the additional property that, for any path, both the weight and all the partial weights are positive. More generally, a random variable X V has a signed MTP 2 distribution if there exists a diagonal matrix ∆ = {δ vv } v∈V with δ vv = ±1 such that ∆X V is MTP 2 (Karlin and Rinott, 1981;Lauritzen et al, 2019). In the Gaussian case, also for this wider family of distributions the weights of the paths are feasible of a clear interpretation because it follows from Lemma 6.1 of Section 6 that in Gaussian signed MTP 2 distributions for any pair x, y ∈ V the paths Π xy are either all positive or all negative.…”

Section: Interpretation Of Covariance Path Weightsmentioning

confidence: 99%

Path weights in concentration graphs

Roverato

Castelo

2020

Biometrika

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A graphical model provides a compact and efficient representation of the association structure of a multivariate distribution by means of a graph. Relevant features of the distribution are represented by vertices, edges and other higher-order graphical structures, such as cliques or paths. Typically, paths play a central role in these models because they determine the independence relationships among variables. However, while a theory of path coefficients is available in models for directed graphs, little has been investigated about the strength of the association represented by a path in an undirected graph. Essentially, it has been shown that the covariance between two variables can be decomposed into a sum of weights associated with each of the paths connecting the two variables in the corresponding concentration graph. In this context, we consider concentration graph models and provide an extensive analysis of the properties of path weights and their interpretation. More specifically, we give an interpretation of covariance weights through their factorisation into a partial covariance and an inflation factor. We then extend the covariance decomposition over the paths of an undirected graph to other measures of association, such as the marginal correlation coefficient and a quantity that we call the inflated correlation. The usefulness of these finding is illustrated with an application to the analysis of dietary intake networks.

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“…In fact, most notions of positive dependence are implied by MTP 2 ; see for example [CSS05] for a recent overview. The special case of Gaussian MTP 2 distributions was studied by Karlin and Rinott [KR83] and in [SH15,LUZ17] from the perspective of MLE and optimization. Unfortunately, maximum likelihood estimation under MTP 2 is ill-defined, since the likelihood function is unbounded.…”

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confidence: 99%

Maximum likelihood estimation for totally positive log‐concave densities

Robeva

Sturmfels

Tran

et al. 2020

Scandinavian J Statistics

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We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP2) distributions and log-L -concave (LLC ) distributions. In both cases we also assume logconcavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors in R d from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n ≥ 3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0, 1} d or in R 2 under MTP2, and for samples in Q d under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.MSC 2010 subject classifications: Primary 62G07, 62H12; secondary 52B55

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Maximum likelihood estimation in Gaussian models under total positivity

Cited by 48 publications

References 36 publications

The Geometry of Gaussoids

The Geometry of Gaussoids

Path weights in concentration graphs

Maximum likelihood estimation for totally positive log‐concave densities

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