An outline of quasi-probability*: Why quasi-Monte-Carlo methods are statistically valid and how their errors can be estimated statistically
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John H. Halton
The classical model of probability theory, due principally to Kolmogorov, defines probability as a totally-one measure on a sigma-algebra of subsets (events) of a given set (the sample space), and random variables as real-valued functions on the sample space, such that the inverse images of all Borel sets are events. From this model, all the results of probability theory are derived. However, the assertion that any given concrete situation is subject to probability theory is a scientific hypothesis verifiable only experimentally, by appropriate sampling, and never totally certain. Furthermore classical probability theory allows for the possibility of “outliers”—sampled values which are misleading. In particular, Kolmogorov's Strong Law of Large Numbers asserts that, if, as is usually the case, a random variable has a finite expectation (its integral over the sample space), then the average value of N independently sampled values of this function converges to the expectation with probability 1 as N tends to infinity. This implies that there may be sample sequences (belonging to a set of total probability 0) for which this convergence does not occur.
It is proposed to derive a large and important part of the classical probabilistic results, on the simple basis that the sample sequences are so constructed that the corresponding average values do converge to the mathematical expectation as N tends to infinity, for all Riemann-integrable random variables. A number of important results have already been proved, and further investigations are proceeding with much promise. By this device, the stochastic nature of some concrete situations is no longer a likely scientific hypothesis, but a proven mathematical fact, and the problem of outliers is eliminated. This model may be referred-to as “quasi-probability theory”; it is particularly appropriate for the large class of computations that are referred-to as “quasi-Monte-Carlo”.
© de Gruyter 2004
Articles in the same Issue
- Foreword
- An outline of quasi-probability*: Why quasi-Monte-Carlo methods are statistically valid and how their errors can be estimated statistically
- QMC techniques for CAT bond pricing*
- Parallel Quasi-Monte Carlo Methods for Linear Algebra Problems
- Algorithm of statistical simulation of dynamic systems with distributed change of structure*
- Frequency Analysis of Semiconductor Devices Using Full-Band Cellular Monte Carlo Simulations
- The ⊝-Maruyama scheme for stochastic functional differential equations with distributed memory term*
- Subdiffusion and Superdiffusion in Lagrangian Stochastic Models of Oceanic Transport
- Approximations of functional integrals with respect to measures generated by solutions of stochastic differential equations
- Normalization of the Spectral Test in High Dimensions
- A spectral Monte Carlo method for the Poisson equation
- Convergence rate for spherical processes with shifted centres*
- On the Power of Quantum Algorithms for Vector Valued Mean Computation
- Parallel Quasirandom Walks on the Boundary
- Trajectory Splitting by Restricted Replication
- Upper Bounds for Bermudan Style Derivatives
- Stochastic Eulerian model for the flow simulation in porous media. Unconfined aquifers*
- Solution of the Space-dependent Wigner Equation Using a Particle Model
- Subgrid Modeling of Filtration in a Porous Medium with Multiscale Log-Stable permeability
- Comparison of Quasi-Monte Carlo-Based Methods for Simulation of Markov Chains
- Monte Carlo methods for fissured porous media: a gridless approach*
- Smoothed Transformed Density Rejection*
- Adaptive adjoint Monte Carlo simulation for the uncertainty
- A Nuclear Measurement Technique of Water Penetration in Concrete Barriers
- System availability and reliability analysis by direct Monte Carlo with biasing
- On the Scrambled Halton Sequence
- Monte-Carlo simulation of the chord length distribution function across convex bodies, non-convex bodies and random media
- Discrepancy of sequences generated by dynamical system
- Operator-Split Method for Variance Reduction in Stochastic Solutions of the Wigner Equation
- Two variants of a stochastic Euler method for homogeneous balance differential equations*
- Full band Monte Carlo simulation - beyond the semiclassical approach
- Coin Tossing Algorithms for Integral Equations and Tractability
- Optimal Korobov Coefficients for Good Lattice Points in Quasi Monte Carlo Algorithms
- A theoretical view on transforming low-discrepancy sequences from a cube to a simplex
- Weighted simulation of steady-state transport within the standard Monte Carlo paradigm
- Dynamic probabilistic method of numerical modeling of multidimensional hydrometeorological fields
- Quasirandom Sequences in Branching Random Walks*
- Discrete random walk on large spherical grids generated by spherical means for PDEs*
- Reusing paths in radiosity and global illumination
- Measures of Uniform Distribution in Wavelet Based Image Compression
- Random Walk Algorithms for Estimating Effective Properties of Digitized Porous Media*
- Security of Pseudo-random Generator and Monte Carlo Method
- Randomization of Quasi-Monte Carlo Methods for Error Estimation: Survey and Normal Approximation*
- Monte Carlo Simulation of Narrow-Width SOI Devices: Incorporation of the Short Range Coulomb Interaction
- Dagger-sampling variance reduction in Monte Carlo reliability analysis
Articles in the same Issue
- Foreword
- An outline of quasi-probability*: Why quasi-Monte-Carlo methods are statistically valid and how their errors can be estimated statistically
- QMC techniques for CAT bond pricing*
- Parallel Quasi-Monte Carlo Methods for Linear Algebra Problems
- Algorithm of statistical simulation of dynamic systems with distributed change of structure*
- Frequency Analysis of Semiconductor Devices Using Full-Band Cellular Monte Carlo Simulations
- The ⊝-Maruyama scheme for stochastic functional differential equations with distributed memory term*
- Subdiffusion and Superdiffusion in Lagrangian Stochastic Models of Oceanic Transport
- Approximations of functional integrals with respect to measures generated by solutions of stochastic differential equations
- Normalization of the Spectral Test in High Dimensions
- A spectral Monte Carlo method for the Poisson equation
- Convergence rate for spherical processes with shifted centres*
- On the Power of Quantum Algorithms for Vector Valued Mean Computation
- Parallel Quasirandom Walks on the Boundary
- Trajectory Splitting by Restricted Replication
- Upper Bounds for Bermudan Style Derivatives
- Stochastic Eulerian model for the flow simulation in porous media. Unconfined aquifers*
- Solution of the Space-dependent Wigner Equation Using a Particle Model
- Subgrid Modeling of Filtration in a Porous Medium with Multiscale Log-Stable permeability
- Comparison of Quasi-Monte Carlo-Based Methods for Simulation of Markov Chains
- Monte Carlo methods for fissured porous media: a gridless approach*
- Smoothed Transformed Density Rejection*
- Adaptive adjoint Monte Carlo simulation for the uncertainty
- A Nuclear Measurement Technique of Water Penetration in Concrete Barriers
- System availability and reliability analysis by direct Monte Carlo with biasing
- On the Scrambled Halton Sequence
- Monte-Carlo simulation of the chord length distribution function across convex bodies, non-convex bodies and random media
- Discrepancy of sequences generated by dynamical system
- Operator-Split Method for Variance Reduction in Stochastic Solutions of the Wigner Equation
- Two variants of a stochastic Euler method for homogeneous balance differential equations*
- Full band Monte Carlo simulation - beyond the semiclassical approach
- Coin Tossing Algorithms for Integral Equations and Tractability
- Optimal Korobov Coefficients for Good Lattice Points in Quasi Monte Carlo Algorithms
- A theoretical view on transforming low-discrepancy sequences from a cube to a simplex
- Weighted simulation of steady-state transport within the standard Monte Carlo paradigm
- Dynamic probabilistic method of numerical modeling of multidimensional hydrometeorological fields
- Quasirandom Sequences in Branching Random Walks*
- Discrete random walk on large spherical grids generated by spherical means for PDEs*
- Reusing paths in radiosity and global illumination
- Measures of Uniform Distribution in Wavelet Based Image Compression
- Random Walk Algorithms for Estimating Effective Properties of Digitized Porous Media*
- Security of Pseudo-random Generator and Monte Carlo Method
- Randomization of Quasi-Monte Carlo Methods for Error Estimation: Survey and Normal Approximation*
- Monte Carlo Simulation of Narrow-Width SOI Devices: Incorporation of the Short Range Coulomb Interaction
- Dagger-sampling variance reduction in Monte Carlo reliability analysis
