Monte Carlo integration with a growing number of control variates

It is well known that Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model. A central limit theorem is established for the integration error if the number of control variates tends to infinity. The integration error is scaled by the standard deviation of the error term in the regression model. If the linear span of the control variates is dense in a function space that contains the integrand, the integration error tends to zero at a rate which is faster than the square root of the number of Monte Carlo replicates. Depending on the situation, increasing the number of control variates may or may not be computationally more efficient than increasing the Monte Carlo sample size.Comment: 22 pages. Numerical experiments in earlier versio

Simulation Experiments in Practice: Statistical Design and Regression Analysis

In practice, simulation analysts often change only one factor at a time, and use graphical analysis of the resulting Input/Output (I/O) data. The goal of this article is to change these traditional, naïve methods of design and analysis, because statistical theory proves that more information is obtained when applying Design Of Experiments (DOE) and linear regression analysis. Unfortunately, classic DOE and regression analysis assume a single simulation response that is normally and independently distributed with a constant variance; moreover, the regression (meta)model of the simulation model’s I/O behaviour is assumed to have residuals with zero means. This article addresses the following practical questions: (i) How realistic are these assumptions, in practice? (ii) How can these assumptions be tested? (iii) If assumptions are violated, can the simulation's I/O data be transformed such that the assumptions do hold? (iv) If not, which alternative statistical methods can then be applied?metamodel;experimental design;jackknife;bootstrap;common random numbers;validation

Testing subspace Granger causality

The methodology of multivariate Granger non-causality testing at various horizons is extended to allow for inference on its directionality. Empirical manifestations of these subspaces are presented and useful interpretations for them are provided. Simple vector autoregressive models are used to estimate these subspaces and to find their dimensions. The methodology is illustrated by an application to empirical monetary policy, where a conditional form of Okun’s law is demonstrated as well as a statistical monetary policy reaction function to oil price changes

Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function

This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-\Loeve expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Lo\`{e}ve coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Lo\`{e}ve expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters' uncertainty in the inference formulation.Comment: 34 pages, 17 figure

Past and present cosmic structure in the SDSS DR7 main sample

We present a chrono-cosmography project, aiming at the inference of the four dimensional formation history of the observed large scale structure from its origin to the present epoch. To do so, we perform a full-scale Bayesian analysis of the northern galactic cap of the Sloan Digital Sky Survey (SDSS) Data Release 7 main galaxy sample, relying on a fully probabilistic, physical model of the non-linearly evolved density field. Besides inferring initial conditions from observations, our methodology naturally and accurately reconstructs non-linear features at the present epoch, such as walls and filaments, corresponding to high-order correlation functions generated by late-time structure formation. Our inference framework self-consistently accounts for typical observational systematic and statistical uncertainties such as noise, survey geometry and selection effects. We further account for luminosity dependent galaxy biases and automatic noise calibration within a fully Bayesian approach. As a result, this analysis provides highly-detailed and accurate reconstructions of the present density field on scales larger than $\sim~3$ Mpc$/h$, constrained by SDSS observations. This approach also leads to the first quantitative inference of plausible formation histories of the dynamic large scale structure underlying the observed galaxy distribution. The results described in this work constitute the first full Bayesian non-linear analysis of the cosmic large scale structure with the demonstrated capability of uncertainty quantification. Some of these results will be made publicly available along with this work. The level of detail of inferred results and the high degree of control on observational uncertainties pave the path towards high precision chrono-cosmography, the subject of simultaneously studying the dynamics and the morphology of the inhomogeneous Universe.Comment: 27 pages, 9 figure