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Questions tagged [monte-carlo]

Questions on Monte Carlo methods, methods that require the repeated generation of pseudo- or quasi-random numbers for computing their results.

When solving problems that depend on a large number of variables, an analytical solution may not exist or may be too hard to compute variable-by-variable because of the curse of dimensionality. In these situations, Monte Carlo methods sample the space of variables in a random way to obtain numerical approximations. The choice and distribution of sampling points is critical to obtaining accurate results – for this reason, quasi-random sampling (e.g. the Sobol or Halton sequences) is often employed.

The method was named in 1946 by Stanislaw Ulam in reference to the Monte Carlo casino.

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Why does Monte-Carlo integration work better than naive numerical integration in high dimensions?

Can anyone explain simply why Monte-Carlo works better than naive Riemann integration in high dimensions? I do not understand how chosing randomly the points on which you evaluate the function can yield a more precise result than distributing these…
MonteCarloCasino
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Probability that a stick randomly broken in two places can form a triangle

Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is $\frac14$ (coordinatize the stick form $0$ to $1$, call the breaking points $x$ and $y$,…
Isaac
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How to sample from a copula?

It is quite clear in many cases how to construct random vectors having specified copulas, e.g. the Gaussian copula, for example starting from a multivariate normal random vector (obtained for example with the Choleski factorization, etc.), and then…
RandomGuy
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How can I intuit the role of the central limit theorem in breaking the curse of dimensionality for Monte Carlo integration

I would like to more intuitively understand where the power of Monte Carlo integration comes from for large-dimensional domains of integration. Other questions on this site have referenced the proof that the scaling of the error in a MC integration…
kleingordon
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Expected Value and Variance of Monte Carlo Estimate of $\int_{0}^{1}e^{-x}dx$

Given the integral: $I=\int_{0}^{1}e^{-x}dx$, use standard Monte Carlo with 1000 random numbers and repeat the simulation 1000 times. (a) What is the expected value and variance of the simple Monte Carlo estimate of I ? I wrote the following to…
Chris
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Monte-Carlo for the Wasserstein metric

Let $(X,d)$ be some metric space and assume that $d\leq 1$. Further, let $\mu, $ $\nu$ be two Borel probability measures on $X$ and let $$ \Gamma(\mu,\nu) = \{\gamma - \text{measure on }X\times X:\gamma(A,X) = \mu(A),\gamma(X,A) = \nu(A) \} $$ to…
SBF
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Is there a natural way to multiply measures?

Given two measures $\mu$ and $\nu$ on some measurable space $X$, is there a way to multiply them to get $\mu \cdot \nu$, another measure on $X$ (and not $X \times X$, as for the usual notion of product measure)? Here's a case where I know how to…
Will
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Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov Chain $(x_0,x_1,...)$ with the stationary…
Ufuk Can Bicici
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Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone please check my results and my proof below? I would…
Paul
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integration method

I want to calculate an integral by using the hit and miss method. I can not understand how this method works. I would be grateful if someone could explain me and help me to calculate the value, with a realistic and simple example as $I=\int_{0}^{1}…
johan paul
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Trouble grasping the idea of MCMC

I'm currently trying to learn Monte Carlo Markov chain, and I'm having troubles to grasp what it is good for and hoping that someone could help me. I want to get some help of general understanding of MCMC, therefore the question are a bit…
uoiu
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An example of high dimension (financial) integrals?

Introduction This question mainly arises out of the context of [Quasi Monte Carlo integration][1]. Which uses "quasi-random" numbers, (i.e. deterministic) with low discrepancy to reduce the variance in Monte Carlo integration. This reduction in…
Bhoris Dhanjal
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Intuitive explanation of the spectral gap in context of Markov Chain Monte Carlo (MCMC)

I'm learning about Markov Chains Monte Carlo methods and mixing times, and could use some help understanding the concept of the Spectral Gap and why / how it relates to the mixing times. Thus far what I have learned is this: given an ergodic…
user3280193
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Simulating from a Multivariate Gaussian without Cholesky

I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive definite covariance matrix. The standard way to…
dan_x
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Monte-Carlo integration

Let a function $f$ to be $x\in \left[a,b\right],\:0\le f\left(x\right)\le c$. We want to calculate the approximation of the definite integral of the function in the range $[a,b]$, we can suppose that the exact integral is very difficult to calculate…
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