Bayesian Stopping Criteria

The blog Why Add Q to MC?, explained the advantages of carefully chosen, low discrepancy sampling sites for approximating multivariate integrals, or equivalently, expectations of functions of multivariate random variables \begin{align} \mu: &= \int_{[0,1]^d} f(\boldsymbol{x}) \, \text{d}{\boldsymbol{x}} = {\mathbb{E}}[f(\boldsymbol{x})], \; \text{where } \boldsymbol{x} \sim \mathcal{U}[0,1]^d, \\ \mu \approx \widehat{\mu}_n  &: = \frac 1n \sum_{i=1}^n f(\boldsymbol{x}_i),Continue reading “Bayesian Stopping Criteria”

Quasi-Monte Carlo Software Article

We recently uploaded an article on Quasi-Monte Carlo Software to https://arxiv.org/abs/2102.07833. Abstract: Practitioners wishing to experience the efficiency gains from using low discrepancy sequences need correct, well-written software. This article, based on our MCQMC 2020 tutorial, describes some of the better quasi-Monte Carlo (QMC) software available. We highlight the key software components required to approximateContinue reading “Quasi-Monte Carlo Software Article”

What Makes a Sequence “Low Discrepancy”?

The first blog post, “Why add Q to MC?”, introduced the concept of evenly spread points, which are commonly referred to as low discrepancy (LD) points. This is in contrast to independent and identically distributed (IID) points. Consider two sequences, $\boldsymbol{T}_1, \boldsymbol{T}_2, \ldots \overset{\text{IID}}{\sim} \mathcal{U}[0,1]^d$ \[\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots \overset{\text{LD}}{\sim} \mathcal{U}[0,1]^d.\] Both sequences are expected toContinue reading “What Makes a Sequence “Low Discrepancy”?”