Related papers: Improving Monte Carlo randomized approximation sch…
Space filling designs are central to studying complex systems in various areas of science. They are used for obtaining an overall understanding of the behaviour of the response over the input space, model construction and uncertainty…
We introduce a general Monte Carlo method based on Nested Sampling (NS), for sampling complex probability distributions and estimating the normalising constant. The method uses one or more particles, which explore a mixture of nested…
For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced by approximations to a distribution's inverse cumulative distribution function. These approximations are…
We review the method of stochastic error correction which eliminates the truncation error associated with any subspace diagonalization. Monte Carlo sampling is used to compute the contribution of the remaining basis vectors not included in…
Monte Carlo experiments produce samples in order to estimate features of a given distribution. However, simultaneous estimation of means and quantiles has received little attention, despite being common practice. In this setting we…
An introduction to the basics of Monte Carlo is given. The topics covered include, sample space, events, probabilities, random variables, mean, variance, covariance, characteristic function, chebyshev inequality, law of large numbers,…
This paper considers a new approach to using Markov chain Monte Carlo (MCMC) in contexts where one may adopt multilevel (ML) Monte Carlo. The underlying problem is to approximate expectations w.r.t. an underlying probability measure that is…
Monte Carlo methods are widely used for approximating complicated, multidimensional integrals for Bayesian inference. Population Monte Carlo (PMC) is an important class of Monte Carlo methods, which utilizes a population of proposals to…
Estimating predictive uncertainty is crucial for many computer vision tasks, from image classification to autonomous driving systems. Hamiltonian Monte Carlo (HMC) is an sampling method for performing Bayesian inference. On the other hand,…
Simple Monte Carlo is a versatile computational method with a convergence rate of $O(n^{-1/2})$. It can be used to estimate the means of random variables whose distributions are unknown. Bernoulli random variables, $Y$, are widely used to…
We design and implement a novel algorithm for computing a multilevel Monte Carlo (MLMC) estimator of the cumulative distribution function of a quantity of interest in problems with random input parameters or initial conditions. Our approach…
Simulating samples from arbitrary probability distributions is a major research program of statistical computing. Recent work has shown promise in an old idea, that sampling from a discrete distribution can be accomplished by perturbing and…
We establish epigraphical and uniform laws of large numbers for sample-based approximations of law invariant risk functionals. These sample-based approximation schemes include Monte Carlo (MC) and certain randomized quasi-Monte Carlo…
Rejection Sampling is a fundamental Monte-Carlo method. It is used to sample from distributions admitting a probability density function which can be evaluated exactly at any given point, albeit at a high computational cost. However,…
In a recent paper Beskos et al (2011), the Sequential Monte Carlo (SMC) sampler introduced in Del Moral et al (2006), Neal (2001) has been shown to be asymptotically stable in the dimension of the state space d at a cost that is only…
Monte Carlo and Quasi-Monte Carlo methods present a convenient approach for approximating the expected value of a random variable. Algorithms exist to adaptively sample the random variable until a user defined absolute error tolerance is…
Hamiltonian Monte Carlo and underdamped Langevin Monte Carlo are state-of-the-art methods for taking samples from high-dimensional distributions with a differentiable density function. To generate samples, they numerically integrate…
Monte Carlo sampling techniques have broad applications in machine learning, Bayesian posterior inference, and parameter estimation. Often the target distribution takes the form of a product distribution over a dataset with a large number…
A key limitation of sampling algorithms for approximate inference is that it is difficult to quantify their approximation error. Widely used sampling schemes, such as sequential importance sampling with resampling and Metropolis-Hastings,…
Partial differential equation is a powerful tool to characterize various physics systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper, we present a…