Maximum a posteriori density estimation and the sparse grid combination technique

We study a novel method for maximum a posteriori (MAP) estimation of the probability density function of an arbitrary, independent and identically distributed \(d\)-dimensional data set. We give an interpretation of the MAP algorithm in terms of regularised maximum likelihood. We also present numerical experiments using a sparse grid combination technique and the `opticom' method. The numerical results demonstrate the viability of parallelisation for the combination technique. References H. J. Bungartz, M. Griebel, D. Roschke and C. Zenger. Pointwise convergence of the combination technique for the Laplace equation. East-West J. Numer. Math, 2:21--45 (1994). http://zbmath.org/?q=an:00653220 J. Garcke. Regression with the optimised combination technique. In Proceedings of the 23rd international conference on Machine learning, ICML '06, pages 321--328 (2006). doi:10.1145/1143844.1143885 J. Garcke. Sparse grid tutorial. Technical report (2011). http://page.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf M. Griebel and M. Hegland. A finite element method for density estimation with Gaussian process priors. SIAM J. Numer. Anal., 47:4759--4792 (2010). doi:10.1137/080736478 M. Griebel, M. Schneider and C. Zenger. A combination technique for the solution of sparse grid problems. In Iterative methods in linear algebra (Brussels, 1991), pages 263--281. North-Holland, Amsterdam (1992). M. Hegland. Adaptive sparse grids. ANZIAM J., 44:C335--C353 (2003). http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/685 M. Hegland. Approximate maximum a posteriori with Gaussian process priors. Constr. Approx., 26:205--224 (2007). doi:10.1007/s00365-006-0661-4 M. Hegland, J. Garcke, and V. Challis. The combination technique and some generalisations. Linear Algebra Appl., 420:249--275 (2007). doi:10.1016/j.laa.2006.07.014 C. T. Kelley. Solving nonlinear equations with Newton's method. Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003). H. Kobayashi, B.L. Mark, and W. Turin. Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance. Cambridge University Press (2012). C. Pflaum and A. Zhou. Error analysis of the combination technique. Numerische Mathematik, 84:327--350 (1999). doi:10.1007/s002110050474 D. W. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization. John Wiley and Sons (2004). C. Zenger. Sparse grids. Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, 31 (1990)