Scalable iterative methods for sampling from massive Gaussian random vectors

Sampling from Gaussian Markov random fields (GMRFs), that is multivariate Gaussian ran- dom vectors that are parameterised by the inverse of their covariance matrix, is a fundamental problem in computational statistics. In this paper, we show how we can exploit arbitrarily accu- rate approximations to a GMRF to speed up Krylov subspace sampling methods. We also show that these methods can be used when computing the normalising constant of a large multivariate Gaussian distribution, which is needed for both any likelihood-based inference method. The method we derive is also applicable to other structured Gaussian random vectors and, in particu- lar, we show that when the precision matrix is a perturbation of a (block) circulant matrix, it is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure

The amyloid precursor protein: A biochemical enigma in brain development, function and disease

AbstractFor 20years the amyloid cascade hypothesis of Alzheimer disease (AD) has placed the amyloid-β peptide (Aβ), formed from the amyloid precursor protein (APP), centre stage in the process of neurodegeneration. However, no new therapeutic agents have reached the clinic through exploitation of the hypothesis. The APP metabolites, including Aβ, generated by its proteolytic processing, have distinct physiological functions. In particular, the cleaved intracellular domain of APP (AICD) regulates expression of several genes, including APP itself, the β-secretase BACE-1 and the Aβ-degrading enzyme, neprilysin and this transcriptional regulation involves direct promoter binding of AICD. Of the three major splice isoforms of APP (APP695, APP751, APP770), APP695 is the predominant neuronal form, from which Aβ and transcriptionally-active AICD are preferentially generated by selective processing through the amyloidogenic pathway. Despite intensive research, the normal functions of the APP isoforms remain an enigma. APP plays an important role in brain development, memory and synaptic plasticity and secreted forms of APP are neuroprotective. A fuller understanding of the physiological and pathological actions of APP and its metabolic and gene regulatory network could provide new therapeutic opportunities in neurodegeneration, including AD