Variational approximation methods have become a mainstay of contemporary Machine Learning methodology, but currently have little presence in Statistics. We devise an effective variational approximation strategy for fitting generalized linear mixed models (GLMM) appropriate for grouped data. It involves Gaussian approximation to the distributions of random effects vectors, conditional on the res...

In a recently developed approximation technique [1] for quantum field theory the standard one-loop result is used as a seed for a recursive formula that gives a sequence of improved Gaussian approximations for the generating functional. In this paper we work with the generic φ3+φ4 model in d = 0 dimensions. We compare the first, and simplest, approximation in the above sequence with the one-loo...

We calculate the negative integer moments of the (regularized) characteristic polynomials of N × N random matrices taken from the Gaussian Orthogonal Ensemble (GOE) in the limit as N → ∞. The results agree nontrivially with a recent conjecture of Berry & Keating motivated by techniques developed in the theory of singularity-dominated strong fluctuations. This is the first example where nontrivi...

We consider the path approximation of Bessel processes and develop a new efficient algorithm. This study is based on recent work by authors, Brownian motion, construction specific own techniques. It part family so-called $$\varepsilon$$ -strong approximations. More precisely, our approach constructs jointly sequences exit times corresponding positions some well-chosen domains, these domains bei...