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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.

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Can we use the control variate Monte Carlo variance reduction approach to estimate variance?

The control variate technique is a super useful was of fusing low- and high-fidelity models to reduce the variance in an estimate of an expected value. Consider an expensive, high-fidelity model $f(x)$...
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How to do Monte Carlo Method for exceedingly large numbers?

For a paper I'm writing for my math class, I need to do several Monte Carlo simulations for a game I'm playing. The $p=0.6190411273$, a normal number... but the $n=2.14974(10^{10})$. I've tried to run ...
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Unbiased Metropolis within Gibbs Sampler MCMC

I am currently using MCMC to estimate the inner cells of a contingency table. The date comes from a simulation I conducted and thus I know the true values of the inner cells. However, I have the ...
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Intuition behind convergence of MCMC inference methods

I'm studying Gibbs Sampling for inference, a popular MCMC algorithm and I was stunned by its ability to fit a Gaussian Mixture just by sampling. I would like to know the intuition behind it, and why ...
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Proposal density of Metropolis algorithm

I am new to the Metropolis-Hastings algorithm and am trying to wrap my head around the key points of it. I understand that it uses a Markov Chain Monte Carlo simulation to sample points throughout a ...
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26 views

Convergence in total variation distance of Markov kernel $n$-fold composition to the stationary measure

Let $(E,\mathcal E)$ be a measurable space $\mu$ be a measure on $(E,\mathcal E)$ $p:E\to(0,\infty)$ with $$\int p\:{\rm d}\mu=1$$ and $\pi$ denote the measure with density $p$ with respect to $\mu$ $...
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How to MCMC (or other simulation) given a non-stationary distribution?

Say I was given some directed graph that satisfies the Markov property, has a stationary distribution, $\pi$, and I know the transition probabilities are functions of some unknown parameters $P_{i\to ...
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simulation of customer negotiation strategies using R

If I ask the question in the wrong forum, let me know, I will delete it. It is still difficult for me to decide the forum. I am currently studying the issue of Models for customer-supplier negotiation ...
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About function V in geometric drift condition for Markov Chain

When I read the geometric ergodicity of Markov chain in Meyn and Tweedie, I note that in the drift condition $PV(x)<\lambda V(x)+bI_{x\in C}$, where $V(x)\ge 1$ is required. Why $V(x)\ge 1$ rather ...
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Is having a burn-in time relevant when only trying to sample from a distribution?

I'm trying to simulate - via the Metropolis-Hastings algorithm - a sample $X$ of size 10000 from a density $f$ using a proposal distribution $g$. The Markov chain $X$ obtained by this algorithm has ...
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Feynman-Kac formula in action.

Briefly speaking, the Feynman-Kac formula gives a way for constructing $C^2$ functions satisfying certain PDEs in the classical sense (at least, it's how it is explained in Oksendal's book that I am ...
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Lattice vs. Random sampling for function interpolation

Suppose $f: [0,1]\times[0,1]\rightarrow \mathbf R$ is smooth. I am to interpolate the function from the function values of $f$ at $n^2,\, n\in \mathbf N$ samples points. I have the freedom to pick ...
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51 views

Python monte carlo value at risk with non normal distribution

What is the right way to implement a monte carlo method on a currency portfolio with non normal distribution? I am using a geometric brownian motion with normal random variables but I would like to ...
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1answer
25 views

How to discard negative values while adopting Monte Carlo?

I am trying to simulate a random variable, for a Monte Carlo simulation, which is equal to another Normal random variable, superimposed with a zero mean gaussian random variable as specified below -- ...
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1answer
43 views

Monte Carlo double integral over surface of $|x|+|y| \leq 1$

$\iint_{|x|+|y|\le1}\!x^2\,dxdy$ I am supposed to calculate this by using Monte Carlo integration. Can anyone give basic hints or directions? I know the idea behind the Monte Carlo integration ...
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23 views

Comparing two random variables with monte carlo sampling

Suppose there are two numbers X1 and X2 that are from a random continuous probability distribution with unknown range. You are given the value of X1 and you need to determine whether X1 is less than ...
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Derivation of tau time-stepping in Gillespie algorithm?

I'm trying to find the derivation of tau ($\tau$) in the Gillespie algorithm. All the papers and chapters I've found simply say, without actually showing its derivation: "Tau is given by" $\tau = \...
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665 views

Mathematics solution for Gerrymandering problem?

Gerrymandering is a practice intended to establish a political advantage for a particular party or group by manipulating district boundaries, and can create large disproportions in voting results as ...
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Monte Carlo Markov Chain - Metropolis-Hastings - Estimation of parameters

I have 6 parameters to estimate : $p=(\theta=[a,b]$, $\nu=[r_0,c_0,\alpha,\beta])$ with Bayesian and MCMC methods : $$\text{PSF}(r,c) = \bigg(1 + \dfrac{r^2 + c^2}{\alpha^2}\bigg)^{-\beta}$$ and the ...
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Why is the Monte Carlo integration dimensionally independent?

Suppose that the random variables $x_1, x_2, ... x_N$ are drawn independently from the probability density function $p(x)$. Now the convergence rate of a deterministic numerical integration method ...
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1answer
68 views

Why does the Monte-Carlo Method Work?

I've been reading about the Monte-Carlo Method and how it is much simpler for computers to use the Monte-Carlo Method to guesstimate solutions to complex problems like the Standard Model. It is ...
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Random numbers associated in Monte Carlo Simulation

An Outcome has a probability of 35% of occurring. To Simulate this outcome, which integer random numbers on a scale of 0-99 should be associated with it for Monte Carlo Simulation?
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Approximating the first moment of h(x) where $x$~Lognomal($\mu, \sigma$)

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$). So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
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How can we measure how good a Metropolis-Hastings estimator of an integral is?

Let $(E,\mathcal E,\mu)$ be a $\sigma$-finite measure space $Q$ be a Markov kernel on $(E,\mathcal E)$ with $$Q(x,\;\cdot\;)=q(x,\;\cdot\;)\mu\;\;\;\text{for all }x\in E\tag1$$ for some $\mathcal E\...
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Markov kernel for the simulated tempering algorithm

Let $(E,\mathcal E,\mu)$ be a measure space $I$ be a set with $|I|\in\mathbb N$ and $\zeta$ denote the counting measure on $(I,2^I)$ $w_t:E\to[0,1]$ be $\mathcal E$-measurable for $t\in I$ with $$\...
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How to find Minimum of Function with Many Local Minima

I'm trying to find the smallest local minima within a given boundary condition, e.g., a circle. Currently, I'm using a monte carlo algorithm to approximate the minimum, followed by a gradient descent ...
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73 views

Why is Monte Carlo integration randomly sampled?

As I understand, Monte Carlo integration uses stochastic sampling to sample points. Obviously, you would need many samples to reach an accurate result, but why does this process have to be random? ...
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Optimal importance sampling distribution

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\mathcal E,\mu)$ be a measure space $f:E\to\mathbb R$ be $\mathcal E$-measurable $p:E\to[0,\infty)$ be $\mathcal E$-measurable ...
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Quickly estimating the optimal bin size for the Jackknife re-sampling method

I am trying to apply the Jackknife method to compute confidence intervals for data collected from Monte Carlo simulations. My understanding is that to counter the effect of autocorrelations I should ...
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Construct a new confidence interval (CI) from a published asymmetric CI which was determined using Monte Carlo simulation.

Background: NOAA publishes precipitation frequency estimates for the U.S. where the authors use a Monte Carlo simulation technique to generate 1,000 synthetic data sets to estimate a 90% confidence ...
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essential understanding issue in monte carlo simulation

lets look at my problem with this example: $$ \int_0^1 f(x) dx=\mathbb{E}[f(U)]$$ where $U$ is uniformly distributed on $[0,1]$. The Monte Carlo estimator would be $\displaystyle M:=\frac{1}{n} \sum_{...
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Monte Carlo Integration - unbounded domain

Monte Carlo Integration - function np.sin(θ)^24) / (θ^2) where 0<θ<∞ . How can we integrate this function over a bounded domain and get an accurate result and what bounds should you choose to ...
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39 views

Markov Chain Monte Carlo thermolization time estimation (not by eye)

For a given MCMC algorithm, there are two important time(=step) scale. $\tau_{thermolization}$ also known as burn-in time, intialization time. $\tau_{indenpendent}$ the time scale to make $X_i$ and $...
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168 views

Monte Carlo to evaluate infinite integral on R

I am using Monte Carlo method to evaluate the integral above: $$\int_0^\infty \frac{x^4sin(x)}{e^{x/5}} \ dx $$ I transformed variables using $u=\frac{1}{1+x}$ so I have the following finite integral: ...
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Antithetic sampling and Monte Carlo simulation

Consider: \begin{align} f(x) = \left\{ \begin{array}{ll} 0, & 0 < x < 0.9 \\ 100, & 0.9 < x < 0.91 \\ 0, & \textrm{otherwise} \\ \end{array}\right. \end{align} Determine ...
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Show that the metropolis matrix is irreducible

Let $E$ be a countable set equipped with the discrete topology and $\mathcal E:=\mathcal B(E)$ $\pi$ be a probability measure on $(E,\mathcal E)$ with $$\pi(x):=\pi(\left\{x\right\})>0\;\;\;\text{...
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127 views

How to calculate the transition matrix in Markov sampling (example)?

Let's say you can simulate a discrete uniform distribution $\{0,1\}$ (like a coin toss). With $P\{1\} = P\{2\} = 0.5$. Now we would like to simulate a distribution $S = \{1,2,3\}$ with $P_Z\{1\} = 0....
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33 views

using monte carlo to determine uncertainty.

In my thesis there are some uncertainties (for example in geometry: the diameter of cylinder, the height ,.. or the temperature of inlet fluid ) and I want to know what is the effect of them in my ...
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69 views

Approximating pi rate of convergence

I have been reading about a method for approximating $\pi$ using two uniform distributions and the ratio of points that lie within the circle compared to the square formed by the two uniform ...
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difference between the the means of subsampling from an unbiased sample

Suppose I have a distribution 𝔽 with mean M. Also, assume we have a set of i.i.d samples of size n denoted by X=$\{x_1,x_2,...,x_n\}$ from 𝔽. As a result, all $x_1,...,x_n$ are independent with ...
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why monte carlo method use random sampling? and not a specific numbers? [duplicate]

in calculating the area of a circle in a square we use random points to calculate the fraction of circle! but why we dont assume a simple grid and put our points in the center of it. this seems more ...
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1answer
44 views

Monte-Carlo approximation with small samples

Let me suppose I have one function of $y$ given $x$ : $f(y\mid x)$ and $N$ samples of $x$ : $\{x_i\}_{i=1}^N$. Here, I’d like to create a distribution over the space of $y$ based on this function $f$ ...
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47 views

Generating samples of a Gaussian vector with independent components above a hyperplane

Let $(Z_1,Z_2,\dots,Z_N)$ be a Gaussian vector with iid components, zero mean, and unit variance. Consider a hyperplane in $\mathbb{R}^N$. I am interested in generating samples of this Gaussian vector ...
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Generating samples from certain regions of a Gaussian distribution

Let $X$ be a Gaussian random variable where for simplicity, $X \sim N(0,1)$. I am interested in generating samples of $X$ in the following cases: $X \mid X \in (a,b)$ where $a < b$ or $X \mid X >...
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statistical errors associated to Monte Carlo sampling

I have $n$ successive observation $A_\mu $ of a quantity $A$ and I need to understand how the expectation values of the square of the statistical error depends from the autocorrelation time but a ...
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Markov Processes and Detailed Balance

Section 2.2.3 of this book: http://itf.fys.kuleuven.be/~fpspXIII/material/Barkema_FPSPXIII.pdf discusses the detailed balance condition in the context of Markov chain Monte Carlo algorithms. First ...
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To Generating a random variable from a non conventional density function i.e. the time dependent diffusion equation

I am interested in generating a random variable, this is, to obtaining a point (x,y)) by a non conventional density function i.e. the time dependent diffusion equation over an irregular domain on $R^2$...
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214 views

Variance of Estimator (Monte Carlo Integration)

Copying this question directly from another stack-exchange site, since didn't receive any good answers there. So I was reading this paper by Lafortune, "Mathematical Models and Monte Carlo algorithms"...
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monte carlo simulation of stock prices - time step

In my quant finance textbook, a geometric brownian motion is simulated, using a user-defined time horizon $T$ with a number of time steps, which the author denotes as $NSteps$. He then calculates the $...
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Monte Carlo sampling for Fourier reconstruction in high-dimension

I have a problem that involves the (discrete) Fourier transform of a high dimensional function. I can somehow compute and store all the coefficients, but I don't wish to sum all the terms (n^d where n ...