Morpho-statistical description of networks through graph modelling and Bayesian inference

Collaborative graphs are relevant sources of information to understand behavioural tendencies of groups of individuals. Exponential Random Graph Models (ERGMs) are commonly used to analyze such social processes including dependencies between members of the group. Our approach considers a modified version of ERGMs, modeling the problem as an edge labelling one. The main difficulty is inference since the normalising constant involved in classical Markov Chain Monte Carlo (MCMC) approaches is not available in an analytic closed form. The main contribution is to use the recent ABC Shadow algorithm. This algorithm is built to sample from posterior distributions while avoiding the previously mentioned drawback. The proposed method is illustrated on real data sets provided by the Hal platform and provides new insights on self-organised collaborations among researchers

A bounded distribution derived from the shifted Gompertz law

A two-parameter probability distribution with bounded support is derived from the shifted Gompertz distribution. It is shown that this model corresponds to the distribution of the minimum of a random number with shifted Poisson distribution of independent random variables having a common power function distribution. Some statistical properties are written in closed form, such as the moments and the quantile function. To this end, the incomplete gamma function and the Lambert W function play a central role. The shape of the failure rate function and the mean residual life are studied. Analytical expressions are also provided for the moments of the order statistics and the limit behavior of the extreme order statistics is established. Moreover, the members of the new family of distributions can be ordered in terms of the hazard rate order. The parameter estimation is carried out by the methods of maximum likelihood, least squares, weighted least squares and quantile least squares. The performance of these methods is assessed by means of a Monte Carlo simulation study. Two real data sets are used to illustrate the usefulness of the proposed distribution

Algorithmic Randomness and Capacity of Closed Sets

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m-random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an effectively closed set with capacity q

The interplay of classes of algorithmically random objects

We study algorithmically random closed subsets of $2^\omega$, algorithmically random continuous functions from $2^\omega$ to $2^\omega$, and algorithmically random Borel probability measures on $2^\omega$, especially the interplay between these three classes of objects. Our main tools are preservation of randomness and its converse, the no randomness ex nihilo principle, which say together that given an almost-everywhere defined computable map between an effectively compact probability space and an effective Polish space, a real is Martin-L\"of random for the pushforward measure if and only if its preimage is random with respect to the measure on the domain. These tools allow us to prove new facts, some of which answer previously open questions, and reprove some known results more simply. Our main results are the following. First we answer an open question of Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that $\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if it is the set of zeros of a random continuous function on $2^\omega$. As a corollary we obtain the result that the collection of random continuous functions on $2^\omega$ is not closed under composition. Next, we construct a computable measure $Q$ on the space of measures on $2^\omega$ such that $\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if $\mathcal{X}$ is the support of a $Q$-random measure. We also establish a correspondence between random closed sets and the random measures studied by Culver in previous work. Lastly, we study the ranges of random continuous functions, showing that the Lebesgue measure of the range of a random continuous function is always contained in $(0,1)$

Effective Capacity and Randomness of Closed Sets

We investigate the connection between measure and capacity for the space of nonempty closed subsets of {0,1}*. For any computable measure, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain measures an effectively closed set with positive capacity and with Lebesgue measure zero

Translating the Cantor set by a random

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructive dimension