Limit theorems for weakly exchangeable arrays

Summary We study conditional independence relationships for random networks and their interplay with exchangeability. We show that, for finitely exchangeable network models, the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts. We then characterize all possible Markov structures for finitely exchangeable random graphs, thereby identifying a new class of Markov network models corresponding to bidirected Kneser graphs. In particular, we demonstrate that the fundamental property of dissociatedness corresponds to a Markov property for exchangeable networks described by bidirected line graphs. Finally we study those exchangeable models that are also summarized in the sense that the probability of a network depends only on the degree distribution, and we identify a class of models that is dual to the Markov graphs of Frank and Strauss. Particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of Markov properties correspond to the empty graph, the bidirected line graph of the complete graph and the complete graph.

show abstract

“…A distribution P of a random array

X = {(X_{ij})}_{i, j \in N}

over a finite node set

scriptN

is said to be weakly exchangeable (Silverman, ; Eagleson and Weber, ) if for all permutations

π \in S (N)

we have that

P false{{false(X_{italicij} = x_{italicij} false)}_{i, j \in scriptN} false} = P false{{false(X_{italicij} = x_{π false(i false) π false(j false)} false)}_{i, j \in scriptN} false} .

If the array X is symmetric—i.e. X ij = X ji , we say that it is symmetric weakly exchangeable (SWE).…”

Section: Network Models and Exchangeabilitymentioning

confidence: 99%

Random Networks, Graphical Models and Exchangeability

Lauritzen

Rinaldo

Sadeghi

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

View full text Add to dashboard Cite

show abstract

“…We refer to (1) as the exchangeability assumption. It corresponds to the condition of weakly exchangeable arrays (Eagleson and Weber [10]) applied to (I γk , Y k ) γ∈N,k∈Uγ .…”

Section: Asymptotic Framework and Assumptionsmentioning

confidence: 99%

Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population

Bonnéry¹,

Breidt²,

Coquet³

2012

Bernoulli

View full text Add to dashboard Cite

Published in at http://dx.doi.org/10.3150/11-BEJ369 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)International audienceConsider informative selection of a sample from a finite population. Responses are realized as independent and identically distributed (i.i.d.) random variables with a probability density function (p.d.f.) f, referred to as the superpopulation model. The selection is informative in the sense that the sample responses, given that they were selected, are not i.i.d. f. In general, the informative selection mechanism may induce dependence among the selected observations. The impact of such dependence on the empirical cumulative distribution function (c.d.f.) is studied. An asymptotic framework and weak conditions on the informative selection mechanism are developed under which the (unweighted) empirical c.d.f. converges uniformly, in $L_2$ and almost surely, to a weighted version of the superpopulation c.d.f. This yields an analogue of the Glivenko-Cantelli theorem. A series of examples, motivated by real problems in surveys and other observational studies, shows that the conditions are verifiable for specified designs

show abstract

“…An array ζ = (ζ i,j ; i, j ≥ 1) of random variables with values in (S, S) is called jointly (or also weakly) exchangeable, see [10] and [6], if its distribution coincides with the distribution of every array (ζ π(i),π(j) ; i, j ≥ 1). Here both indices are permuted simultaneously by any permutation π which moves only a finite number of positive integers.…”

Section: Block-factors In Random Arraysmentioning

confidence: 99%

On two–block–factor sequences and one–dependence

Matúš

1996

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The distributions of two-block-factors (f (η i , η i+1 ); i ≥ 1) arising from i.i.d. sequences (η i ; i ≥ 1) are observed to coincide with the distributions of the superdiagonals (ζ i,i+1 ; i ≥ 1) of jointly exchangeable and dissociated arrays (ζ i,j ; i, j ≥ 1). An inequality for superdiagonal probabilities of the arrays is presented. It provides, together with the observation, a simple proof of the fact that a special one-dependent Markov sequence of Aaronson, Gilat and Keane (1992) is not a two-block factor.

show abstract

Limit theorems for weakly exchangeable arrays

Cited by 35 publications

References 8 publications

Random Networks, Graphical Models and Exchangeability

Random Networks, Graphical Models and Exchangeability

Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population

On two–block–factor sequences and one–dependence

Contact Info

Product

Resources

About