From 460e74c2915bf0f13ae680e513adf0558470a31d Mon Sep 17 00:00:00 2001 From: Michael T Smith Date: Tue, 27 Jun 2017 14:03:42 +0100 Subject: [PATCH] massive comment quoting literature --- GPy/core/model.py | 70 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 70 insertions(+) diff --git a/GPy/core/model.py b/GPy/core/model.py index 1e4a9955..d9786a9e 100644 --- a/GPy/core/model.py +++ b/GPy/core/model.py @@ -53,6 +53,76 @@ class Model(ParamzModel, Priorizable): def CCD(self): """ Code is based on implementation within GPStuff, INLA and the original Sanchez and Sanchez paper (2005) + + + + CCD = Central Composite Design, pick hyperparameters around the MAP estimate to allow us to estimate the + integral over them. + + Quoting https://arxiv.org/pdf/1206.5754.pdf (section 5.4) which + describes GPStuff, which this work is based upon. + + "Rue et al. (2009) suggest a central composite design (CCD) for choosing + the representative points from the posterior of the parameters with + the aim of finding points that allow one to estimate the curvature + of the posterior distribution around the mode. The design used here + copies GPstuff's fractional factorial design (Sanchez and Sanchez, 2005) + augmented with a center point and a group of star points." + + "The design points are all on the surface of a d-dimensional sphere + and the star points consist of 2d points along each axis. The + integration is then a finite sum with special weights (Vanhatalo et al., + 2010)." + + Quoting that article: + + "The integration weights can then be determined from the statistics of a + standard Gaussian variable, + + E[z^T z] = d + E[z] = 0 + E[1] = 1 + + where d is the dimensionality of \theta. + sphere has radius \sqrt{d}f_0. + The integration weights are equal for the points on the sphere. + + This results in integration weights, + + \Delta = [ (n_p-1) e^(-df_0^2/2) (f_0^2-1) ]^{-1} + + n_p = number of points on sphere. + + + "CCD integration speeds up the computations considerably compared to the + grid search or Monte Carlo integration since the number of the design + points grows very moderately." + + "Since CCD is based on the assumption that the posterior of the parameter + is (close to) Gaussian, the densities at the points on the circumference + should be monitored in order to detect serious discrepancies from this + assumption. These densities are identical if the posterior is Gaussian + and we have located the mode correctly, and thereby great variability on + their values indicates that CCD has failed." + + TODO: From the description of GPStuff: "The posterior of the + parameters may be far from a Gaussian distribution but for a suitable + transformation, which is made automatically in the toolbox..." -- is + this the same transformation we perform below? + + TODO: Implement the above weights in the summation. + + + References: + Sanchez, Susan M., and Paul J. Sanchez. "Very large fractional factorial and central composite designs." ACM Transactions on Modeling and Computer Simulation (TOMACS) 15.4 (2005): 362-377. + http://calhoun.nps.edu/bitstream/handle/10945/35346/SanchezSanchezACM_TOMACS_05.pdf?sequence=1 + + Rue, Håvard, Sara Martino, and Nicolas Chopin. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations." Journal of the royal statistical society: Series b (statistical methodology) 71.2 (2009): 319-392. + http://www.jstor.org/stable/40247579 + + Vanhatalo, Jarno, Ville Pietiläinen, and Aki Vehtari. "Approximate inference for disease mapping with sparse Gaussian processes." Statistics in medicine 29.15 (2010): 1580-1607. + http://lib.tkk.fi/Diss/2010/isbn9789526033815/article4.pdf + """ modal_params = self.optimizer_array[:].copy() num_free_params = modal_params.shape[0]