A note on testing conditional independence for social network analysis

In social network analysis, logistic regression models have been widely used to establish the relationship between the response variable and covariates. However, such models often require the network relationships to be mutually independent, after controlling for a set of covariates. To assess the validity of this assumption, we propose test statistics, under the logistic regression setting, for three important social network drivers. They are, respectively, reciprocity, centrality, and transitivity. The asymptotic distributions of those test statistics are obtained. Extensive simulation studies are also presented to demonstrate their finite sample performance and usefulness.     

Conditional independence,conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.     

A note on marginal and conditional independence

Some statistical models imply that two random vectors are marginally independent as well as being conditionally independent with respect to another random vector. When the joint distribution of the vectors is normal, canonical correlation analysis may lead to relevant simplifications of the dependence structure. A similar application can be found in elliptical models, where linear independence does not imply statistical independence. Implications for Bayes analysis of the general linear model are discussed.     

Lacunary sequences and conditional independence

Research supported by Hungarian National Foundation for Scientific Research, Grant No. 1905.     

Cumulants on the Wiener space

We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual “graph/diagram computations” adopted in most of the probabilistic literature.     

Degree theory on Wiener space

Summary. Let (W, H, μ) be an abstract Wiener space and let Tw  =  w + u (w), where u is an H-valued random variable, be a measurable transformation on W. A Sard type lemma and a degree theorem for this setup are presented and applied to derive existence of solutions to elliptic stochastic partial differential equations. Received: 19 March 1996 / In revised form: 7 January 1997     

Potentials on abstract Wiener space

The infinite dimensional Green measure g is shown to be a product measure and this provides sufficient conditions for the existence and the differentiability of potentials. Moreover, it is shown how such conditions can be used to prove that if f is Lip and if we set u = Gf, then first D²u is Lip too and second u satisfies Δu = ?2f for a wide class of functions f with arbitrary support.     

Comparison inequalities on Wiener space

We define a covariance-type operator on Wiener space: for F$F$ and G$G$ two random variables in the Gross–Sobolev space D^1,2${\mathbb{D}}^{1,2}$ of random variables with a square-integrable Malliavin derivative, we let _ΓF,G?〈DF,−DL⁻¹G〉${\Gamma }_{F,G}?〈{DF,-DL}^{-1}G〉$, where D$D$ is the Malliavin derivative operator and L⁻¹$L^{-1}$ is the pseudo-inverse of the generator of the Ornstein–Uhlenbeck semigroup. We use Γ$\Gamma$ to extend the notion of covariance and canonical metric for vectors and random fields on Wiener space, and prove corresponding non-Gaussian comparison inequalities on Wiener space, which extend the Sudakov–Fernique result on comparison of expected suprema of Gaussian fields, and the Slepian inequality for functionals of Gaussian vectors. These results are proved using a so-called smart-path method on Wiener space, and are illustrated via various examples. We also illustrate the use of the same method by proving a Sherrington–Kirkpatrick universality result for spin systems in correlated and non-stationary non-Gaussian random media.     

Wasserstein space over the Wiener space

The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel–Üstünel. Carrying out the program of Ambrosio–Gigli–Savaré, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein–Uhlenbeck evolution equation.     

Analytic functionals on the Wiener space

We define the analyticity of Wiener functionals and study its properties and applications to oscillatory Wiener functionals. Project supported by the National Natural Science Foundation of China.     

Hausdorff measures on the Wiener space

We construct a Hausdorff measure of finite co-dimension on the Wiener space. We then extend the Federer co-area Formula to this Wiener space for functions with the sole condition that they belong to the first Sobolev space. An explicit formula for the density of the images of the Wiener measure under such functions follows naturally from this. As a corollary, this yields a new and easy proof of the Krée-Watanabe theorem concerning the regularity of the images of the Wiener measure.     

Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis

Extensions of the Nourdin-Peccati analysis to Rn-valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein's lemma), to classical analysis on Wiener space. Partial differential equations for the density of an Rn-valued centered random variable Z=(Z¹,…,Zn) are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix (−DL−1Zi|DZj), i,j=1,2,…,n, where D and L are respectively the derivative operator and the generator of the Ornstein-Uhlenbeck semigroup on Wiener space.     

A universal algebraic approach for conditional independence

In this paper we show that elementary properties of joint probability density functions naturally induce a universal algebraic structure suitable for studying probabilistic conditional independence (PCI) relations. We call this algebraic system the cain. In the cain algebra, PCI relations are represented in equational forms. In particular, we show that the cain satisfies the axioms of the graphoid of Pearl and Paz (Advances in artificial intelligence. North-Holland, Amsterdam, 1987) and the separoid of Dawid (Ann. Math. Artif. Intell. 32:335–372, 2001), these axiomatic systems being useful for general probabilistic reasoning.     

The divergence of Banach space valued random variables on Wiener space

The domain of definition of the divergence operator δ on an abstract Wiener space (W,H,μ) is extended to include W–valued and – valued “integrands”. The main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in δ’s extended domain have divergence zero. These results are then applied to the analysis of quasiinvariant flows induced by W-valued vector fields and, among other results, it turns out that these divergence-free vector fields “are responsible” for generating measure preserving flows. Mathematics Subject Classification (2000): Primary 60H07, Secondary 60H05 An erratum to this article is available at .     

Fractional order Sobolev spaces on Wiener space

Summary Fractional order Sobolev spaces are introduced on an abstract Wiener space and Donsker's delta functions are defined as generalized Wiener functionals belonging to Sobolev spaces with negative differentiability indices. By using these notions, the regularity in the sense of Hölder continuity of a class of conditional expectations is obtained.     

Superefficient drift estimation on the Wiener space

In the framework of a nonparametric functional estimation for the drift of a Brownian motion $X_t$ we construct Stein type estimators of the form $X_t+D_t\log F$ which are superefficient when $\sqrt{F}$ is a superharmonic functional on the Wiener space for the Malliavin derivative D. To cite this article: N. Privault, A. Réveillac, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.     

Logical and algorithmic properties of stable conditional independence

The logical and algorithmic properties of stable conditional independence (CI) as an alternative structural representation of conditional independence information are investigated. We utilize recent results concerning a complete axiomatization of stable conditional independence relative to discrete probability measures to derive perfect model properties of stable conditional independence structures. We show that stable CI can be interpreted as a generalization of Markov networks and establish a connection between sets of stable CI statements and propositional formulas in conjunctive normal form. Consequently, we derive that the implication problem for stable CI is coNP-complete. Finally, we show that Boolean satisfiability (SAT) solvers can be employed to efficiently decide the implication problem and to compute concise, non-redundant representations of stable CI, even for instances involving hundreds of random variables.     

Weak convergence of diffusion processes on Wiener space

Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e _i } ⊂ H be an orthonormal basis of H. Suppose that μ _n = ρ _n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e _i , then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly.