# 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.

758 questions
Filter by
Sorted by
Tagged with
4 views

### How to calculate the varinace of MCMC estimator (updated)

I am trying to calcualte the variance of the MCMC estimator shown below (assuming unbiased estimator $E[\theta] = \mu$): $$\theta = \frac{1}{N}\sum_{1}^{n}f(X^{n})$$ However, it is assumed that the ...
• 135
19 views

### How to calculate the varinace of MCMC estimator

I am trying to calcualte the variance of the MCMC estimator shown below (assuming unbiased estimator $E[\theta] = \mu$): $$\theta = \frac{1}{N}\sum_{1}^{n}f(X^{n})$$ However, it is assumed that the ...
• 135
11 views

### How do i compute the important sampling hence obtaining the mean and variance of the importance density. Is there a book with these type of questions?

Suppose it is desired to sample from a Student-t3 with 3 degrees of freedom given by a function $$\frac{4\sqrt3}{\pi}(1+\frac{x^2}{3})^{-2}$$ Importance sampling is used with an importance density ...
1 vote
33 views

### Monte Carlo simulation for economic growth

internet. For the institution I work for, I need to investigate economic data. What I was asked for is that based on any historical economic data of the country, I need to build "base","...
1 vote
61 views

### How to combine antithetic variable and control variate methods in mote carlo integration?

I want to use R to estimate the integral $\theta=\int_{0}^{1} e^{x^2 }\,dx$ by monte carlo integration with variance reduction. The variance reduction I want to use is combining the antithetic ...
• 13
1 vote
35 views

### Sampling Spin Configurations in Ising Models

Consider an instance of the Ising model, with $N$ number of spins on a 2D square lattice (or any other 2D structure) wrapped into a torus to avoid boundary conditions (in other words, periodic ...
121 views

### How to fit the Expectation?

Imagine I have the following approximation to the expectation of a random variable $Y$ $$E[Y]\simeq \alpha x^{-\beta}$$ for $x,\alpha>0$ and $0<\beta<1$. Now, imagine I have some average ...
• 3,027
50 views

### Dynamic Monte Carlo in a random walk

I am simulating a random walk of two particles connected by a spring that has spring energy $E(x)=\frac{1}{2}C(x-a)^2$, where $C$ is a spring constant and $a$ the equilibrium length of the spring, and ...
1 vote
37 views

### Convergence of Monte Carlo estimation of two-variable function's expectation $\mathbb{E}[f(X, Y)]$

I'd like to estimate the expectation value estimation of two-variable function $\mathbb{E}[f(X, Y)]$ by using "nested" Monte Carlo integration, where $X$ and $Y$ are independent and could ...
1 vote
29 views

### Variance of Variance -> Confidence Interval?

let's consider some random variables collected in the vector $X$ following the distribution $f_X(X)$. We want to compute the probability that: $$p = \textrm{Pr} [G(X) < 0]$$ where $G(X)$ is some ...
• 11
46 views

### Are Monte Carlo methods considered Bayesian or Frequentist? [closed]

I went down the Frequentist vs Bayesian rabbit hole. I was searching for specific examples and I came across this question. Is sampling with Monte Carlo techniques a Bayesian or a Frequentist approach ...
24 views

### Metropolis - Hasting sampling: sampled from target distribution but shapes of histogram (accepted samples) is off

The target distribution is of the form: $p(x) = x^{-6}.e^{\frac{-2.475}{x}}$ with a support in the interval $[0.0, 2.0]$. This gives a plot like Now, to choose a proposal kernel, I think a lognormal ...
• 5,957
23 views

### Wolfram Mathematica Monte Carlo for integrals approximation

I wanted to implement the Monte Carlo method for multiple integrals approximation in Wolfram Mathematica. Namely I wanted to let the user insert as input the dimension of the integral and the number ...
• 1
1 vote
91 views

### Why the Markov chain transition matrix doesn't have an eigenvalue $\lambda=1$?

Given an instance of the transverse Ising model, I am trying to sample from the Boltzmann probability distribution $\mu(s)=\frac{e^{-E(s)/T}}{Z}$ ($Z$ is the partition function, $s$ is a spin ...
59 views

### Metropolis-Hastings not clearly understood

I am trying to understand how the Metropolis-Hastings algorithm works and, if possible, to build a small example myself (to be sure that I have understood correctly). Unfortunately, there are still a ...
• 423
39 views

### How to find a lognormal distribution which mimics the shape of a given normal distribution?

Question: If I give you the mean vector and the covariance matrix of a multivariate gaussian distribution (call it G1), is there a way to find a lognormal distribution which mimics the shape of G1? I ...
• 101
1 vote
34 views

### Monte Carlo integral over infinite discrete set

In many articles (e.g. in Wikipedia https://en.wikipedia.org/wiki/Monte_Carlo_integration ), Monte Carlo integration is introduced as an numerical approximation of an integration over $n$ dimensional ...
15 views

### Understanding Simulation in the Study

I'm trying to understand a simulation study by Cuevas, Febrero-Bande and Fraiman (2004) "An ANOVA test for functional data". In the paper they explain how they conducted the simulation. ...
• 13
19 views

### Help with formula on linear nested conditioned expectations

I am working on a nested integration problem and want to develop an efficient estimator for said problem. The problem has the form: $\mathbb{E}_x\left[F(x,\mathbb{E}_y\left[G(y, x))\right]\right]$ , ...
• 3
37 views

### Under what assumption 0/0 =1 (Monte Carlo)

The following is taken from some lecture notes on Monte Carlo methods: I have never seen in anywhere else that we can treat $0/0=1$. Why it is okay to do it here?
• 339
1 vote
45 views

### What is the point density of a point sampling pattern in this context?

I'm trying to understand the following assertion which is made in this paper on p. 3: In $d$-dimensional space the point density, defined as the number of points per unit volume, is inversely ...
• 13.5k
49 views

• 111
1 vote
12 views

### Volume of a sample in stratified sampling

Let $(X_n)_{n\in\mathbb N}$ be a time-homogeneous Markov chain with stationary distribution $\mu$. Assume $\mu$ has a density with respect to some reference measure $p$ and let $f$ be an $\mu$-...
• 13.5k
40 views

155 views

### Randomized algorithm to estimate $\pi$

I was looking for an algorithm to create a PI estimator, and I ran across this: https://stackoverflow.com/questions/36659034/trying-to-create-a-pi-estimator-in-r Briefly, the steps are: ...
• 63
1 vote
47 views

• 11
1 vote
39 views

### Probability of sum of random variables exceed a certain theshold

I have a minor technical issue. Let's say $Y = \sum^{n}_{i=1} X_{i}$. Now I want to find $P(Y > \gamma)$ by Monte Carlo. Let's assume the $X_{i}$ are i.i.d. Gamma distributed. How I see the ...
• 47
126 views

### Generate random variables from some PDF

My problem is the following. I am given a PDF, say $p(x) = x \cdot \mathrm{e}^{-x}$ for $x > 0$. I want to generate $n$, say $n = 1000$, random variables given this density, such that I can ...
• 47
9 views

### Monte Carlo - Why is the density of $\hat\beta$ the same regardless of interval size?

I am trying to obtain the estimated density $\hat\beta$ given a large number of sample size from intervals [0, 1] of a normal distribution. I was able to get the correct $\hat\beta$ using the interval....
1 vote
111 views

### Expected value of a norm

I am currently studying the Monte Carlo methods for solving PDEs with random coefficients. My problem here is basically just doing with some algebraic properties of the expected value function which I ...
1 vote
71 views

### Scaling and translating function so that we may assume that it is defined on the unit interval [0,1] with values in [0,1].

For a introductory course in statistics with python, i'm supposed to approximate the integral $\int\limits_{-1}^{1}f(x)dx$, where $f(x) = x^4$. But before I can start with the practical work in python,...
• 13
11 views

### The gradient vector in Hamiltonian Monte Carlo (leapfrog method)

Let $x_{t}, \omega_{t} \in \mathbb{R^{d}}$ The Hamiltonian Monte Carlo says this: Deterministic: it relies on the Hamiltonian dynamics so given an initial state, at any time $t$, specified by the ...
• 5,957