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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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Monte Carlo computational cost

Paper Hello. I'm reading the above paper and I do not understand how they managed to solve eq (17.35) -- i've seen many papers skip through this as trivial and didn't bother to show the method to get ...
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Implementing Gibbs Sampler on joint distribution $X$ and $N$ where $X$ is continuous and $N$ is discrete

Q Random variables X and N have joint distribution, defined up to a constant of proportionality, $$f(x,n) \propto \frac{e^{-3x} x^n}{n!} ~,\quad n=0,1,2, \ldots , x>0$$ Implement a Gibbs sampler ...
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Chose the correct inputs for a Monte Carlo simulation

Motivation If have found my self answering to a SO question about Monte Carlo simulation. The model to design is stated as this: Let 20 people, including exactly 3 women, seat themselves randomly ...
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Simple Monte Carlo uncertainty quantification: why are more samples required with additional uncertain inputs?

Having read a little about simple Monte Carlo methods, I understand that, given a random variable $X$, the difference between its true mean $\mu$ and its sample mean $\hat{\mu}_n$ is of order $\sigma /...
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Proposed 'exact' solution for the Asian Option

https://www.researchgate.net/profile/Moawia_Alghalith/publication/331075967_Exact_Pricing_of_the_Arithmetic_Asian_Options_A_Simple_Formula/links/5c704a07a6fdcc47159419a0/Exact-Pricing-of-the-...
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An “Edgeworth Series-esque” approximation of ratio distribution using Monte Carlo methods. What is this method called?

I am hoping someone can provide me with the name of the following technique that appears to estimate the density of the ratio of independent random variables (although it could work for other ...
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Minimise computational cost for given level of MSE

I am trying to understand how to minimise cost of a Monte Carlo implementation for a given value of MSE/RMSE. Please see the notes attached...I do not follow the second line. I would be grateful if ...
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Derivation Contrastive Divergence

I am trying to follow the original paper of GE Hinton: Training Products of Experts by Minimizing Contrastive Divergence However I can't verify equation (5) where he says: $$ -\frac{\partial}{\...
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Identifying a conditional distribution for Gibbs sampling

I have $N$ samples with $M$ features with class labels $T\in\{-1, 1\}$ which were generated by drawing each feature $m$ from a normal distribution $N\sim N(0, \sigma_m)$. Class labels were assigned ...
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Weighted random sample over continuous data

I'm attempting to write an algorithm which gives a random value x in the domain [0-1) and is weighted according to a function. I don't seem to be able to determine how to do this with continuous data, ...
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Show that the larger $c$ is the faster ${\rm d}U_t^c=\frac c2h'(U_t^c){\rm d}t+\sqrt c{\rm d}W_t$ converges to its stationary distribution

Given two Markov chains $\left(X^{(1)}_n\right)_{n\in\mathbb N_0}$ and $\left(X^{(2)}_n\right)_{n\in\mathbb N_0}$ with transition kernel $\kappa_1$ and $\kappa_2$, respectively, and a common ...
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Efficient simulation for distribution of time until coin toss pattern

Suppose we flip a biased coin (with probability $p$ being Heads) repeatedly until a certain pattern (e.g., HHHTT) appears. We are interested in the number of flips $N$ required. It is well-known that $...
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Stochastic approximation of a Cauchy principle value integral.

Suppose I have a random variable $X\sim f_X(x|\boldsymbol\theta)$ with a well-defined expected value. The usual integral for an analytic solution of this expected value is $$\operatorname EX=\int_{\...
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Monte Carlo Markov Chain Line fitting

I am working on creating a Metro hastings MCMC simulation to fit a line so that I can learn more about MCMC by building one and learn more about statistics. My confusion: I am having trouble wrapping ...
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Prove Rate of Convergence of Monte Carlo

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. How does \begin{equation} \mathbb E\left[\,\left|\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right|\,\right] \to O\...
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How do I find the expectation of a non-stationary auto-regressive time series, with absorbing states?

Apologies for not knowing Latex! Consider the following recursive function: $$ y(t+1) = y(t) (1+r) + R - e(t) $$ Where $r,R$ are known constants, $r>0$, and $e(t)$ is distributed as a truncated ...
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Acceptance-Rejection ahas a Measure Theoretic Interpretation?

Question Does there exist a measure $\mathbb{Q}\ll\mathbb{P}$ justifying the "acceptance-rejection" sampling $$ \mathbb{E}_{\mathbb{P}}\left[X \mid f(X)\leq C\right] = \mathbb{E}_{\mathbb{Q}}\left[X \...
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Drawing random realisation from quantity with poisson error

How would I draw a random realisation of a variable with an upper and lower error determined from Poisson statistics using the Gehrels 1986 formula? See: http://adsabs.harvard.edu/abs/1986ApJ...303.....
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Integrating with a non-analytical solution (random effects)

I would like to integrate a function with two random effects, implying three successive integrations. My problem is that after the first integration, it is not possible to obtain an analytical ...
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33 views

Integrating a function over a random effect (Normal distribution)

I would like to integrate a function with a random effect. The function is : $G(t; \beta) = exp(- \lambda t^\gamma \exp(\beta Z))$, $\beta$ being the random effect taken from a normal distribution ...
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Monte Carlo method to find minimum value of a function

As far as I understand the Monte Carlo methods from a non-rigourous point of view because unfortunately I didn't study mathematics formally. For example to find a minimum value of a function $f(x)$ ...
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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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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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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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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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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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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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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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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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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 ...