Nonnegative Bayesian nonparametric factor models with completely random measures

We present a Bayesian nonparametric Poisson factorization model for modeling dense network data with an unknown and potentially growing number of overlapping communities. The construction is based on completely random measures and allows the number of communities to either increase with the number of nodes at a specified logarithmic or polynomial rate, or be bounded. We develop asymptotics for the number and size of the communities of the network and derive a Markov chain Monte Carlo algorithm for targeting the exact posterior distribution for this model. The usefulness of the approach is illustrated on various real networks.

On Itô formulas for jump processes

A well-known Itô formula for finite-dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite-dimensional Itô formula for continuous semimartingales from Krylov (Probab Theory Relat Fields 147:583–605, 2010) to a class of $$L_p$$ L p -valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.

Random measure algebras

In this article, we introduce algebras of random measures. Algebra is a vector space V V over a field F F with a multiplication satisfying the property: 1) distribution and 2) c ( x ⋅ y ) = ( c x ) ⋅ y = x ⋅ ( c y ) c(x\cdot y) = (cx)\cdot y = x\cdot (cy) for every c ∈ F c \in F and x , y ∈ V x, y \in V . The first operation is a trivial addition operation. For the second operation, we present three different methods 1) a convolution by covariance method, 2) O-dot product, 3) a convolution of bimeasures by Morse-Transue integral. With those operations, it is possible to build three different algebras of random measures.

From additive to second-order processes

The familiar Poisson process is a member of a class of stochastic processes known as additive processes. This broad class also contains the birth-death processes. Second-order processes are processes with two moments finite. The class of second-order processes includes the well-known weakly stationary as well as harmonizable processes. A natural evolution of concepts linking the class of additive processes and the class of second-order processes will be detailed. The connection arises via stable processes and random measures