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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0answers
13 views
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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1answer
47 views
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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1answer
23 views
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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1answer
25 views
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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0answers
15 views
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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1answer
19 views
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 ...
2
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0answers
15 views
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:
$$
-\frac{\partial}{\...
2
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0answers
49 views
Identifying a conditional distribution for Gibbs sampling
I have $N$ samples with $M$ features with class labels $T\in\{-1, 1\}$ which were generated by drawing each feature $m$ from a normal distribution $N\sim N(0, \sigma_m)$.
Class labels were assigned ...
0
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1answer
17 views
Weighted random sample over continuous data
I'm attempting to write an algorithm which gives a random value x in the domain [0-1) and is weighted according to a function. I don't seem to be able to determine how to do this with continuous data, ...
1
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1answer
102 views
Show that the larger $c$ is the faster ${\rm d}U_t^c=\frac c2h'(U_t^c){\rm d}t+\sqrt c{\rm d}W_t$ converges to its stationary distribution
Given two Markov chains $\left(X^{(1)}_n\right)_{n\in\mathbb N_0}$ and $\left(X^{(2)}_n\right)_{n\in\mathbb N_0}$ with transition kernel $\kappa_1$ and $\kappa_2$, respectively, and a common ...
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9 views
Efficient simulation for distribution of time until coin toss pattern
Suppose we flip a biased coin (with probability $p$ being Heads) repeatedly until a certain pattern (e.g., HHHTT) appears. We are interested in the number of flips $N$ required. It is well-known that $...
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0answers
29 views
Stochastic approximation of a Cauchy principle value integral.
Suppose I have a random variable $X\sim f_X(x|\boldsymbol\theta)$ with a well-defined expected value. The usual integral for an analytic solution of this expected value is
$$\operatorname EX=\int_{\...
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0answers
18 views
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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2answers
41 views
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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0answers
11 views
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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0answers
13 views
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 \...
1
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1answer
55 views
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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1answer
17 views
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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1answer
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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1answer
73 views
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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1answer
29 views
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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31 views
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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0answers
16 views
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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0answers
13 views
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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0answers
17 views
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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0answers
43 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$
$...
2
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0answers
47 views
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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0answers
16 views
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 ...
3
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0answers
56 views
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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23 views
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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59 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
45 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
75 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 ...
0
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1answer
25 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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0answers
17 views
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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1answer
685 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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0answers
17 views
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}$$
and the ...
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0answers
62 views
Why is the Monte Carlo integration dimensionally independent?
Suppose that the random variables $x_1, x_2, ... x_N$ are drawn independently from the probability density function $p(x)$.
Now the convergence rate of a deterministic numerical integration method ...
1
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1answer
76 views
Why does the Monte-Carlo Method Work?
I've been reading about the Monte-Carlo Method and how it is much simpler for computers to use the Monte-Carlo Method to guesstimate solutions to complex problems like the Standard Model. It is ...
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71 views
Random numbers associated in Monte Carlo Simulation
An Outcome has a probability of 35% of occurring. To Simulate this outcome, which integer random numbers on a scale of 0-99 should be associated with it for Monte Carlo Simulation?
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1answer
31 views
Approximating the first moment of h(x) where $x$~Lognomal($\mu, \sigma$)
What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$).
So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below:
\begin{align}
E(h(X)) &= ...
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0answers
31 views
How can we measure how good a Metropolis-Hastings estimator of an integral is?
Let
$(E,\mathcal E,\mu)$ be a $\sigma$-finite measure space
$Q$ be a Markov kernel on $(E,\mathcal E)$ with $$Q(x,\;\cdot\;)=q(x,\;\cdot\;)\mu\;\;\;\text{for all }x\in E\tag1$$ for some $\mathcal E\...
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0answers
43 views
Markov kernel for the simulated tempering algorithm
Let
$(E,\mathcal E,\mu)$ be a measure space
$I$ be a set with $|I|\in\mathbb N$ and $\zeta$ denote the counting measure on $(I,2^I)$
$w_t:E\to[0,1]$ be $\mathcal E$-measurable for $t\in I$ with $$\...
3
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0answers
82 views
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 ...
2
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1answer
74 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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0answers
14 views
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 ...
0
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0answers
21 views
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 ...
