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

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2answers
788 views

Monte Carlo simulation on sphere: unbiased random steps

Im doing a Metropolis Monte Carlo simulation with particles on a sphere and have a question concerning the random movement in a given time step. I understand that to obtain a uniform distribution of ...
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1answer
74 views

Should I ignore $0$ when do inverse transform sampling?

Generic method Generate $U \sim \mathrm{Uniform}(0,1)$. Return $F^{-1}(U)$. So, in step 1, $U$ has domain/support as $[0,1]$, so it is possible that $U=0$ or $U=1$, but $F^{-1}(0)=-\infty$. ...
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1answer
176 views

Help me with Monte Carlo method : Importance Sampling

I have a little problem with applying a Monte Carlo method: Importance Sampling. I need to calculate: $$ \int_0^\infty \int_0^\infty \frac{dx\;dy}{2 \pi (1 + x^2 + y^2)^{3/2}} $$ Can somebody help ...
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1answer
169 views

A problem of inversion method for sampling

I am reading the book Stochastic Simulation, Algorithms and Analysis by Asmussen. On page 39, when talking about inversion method for sampling a distribution, he gave an example: an r.v. ...
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0answers
35 views

Monte Carlo Integration via Ray Casting

Let's suppose we have a 2D line segment $S$ at $y=0$ and extending from $x=-h$ to $x=+h$. We define a function $f(x)=1$ for every point $x$ of $S$. We wish to integrate $f$ over $S$ with Monte Carlo ...
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33 views

How do I solve a under-determined quadratic multi-variate system?

I have the following equation: $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{11} X_{1}^2 + \beta_{22} X_{2}^2 + \beta_{33} X_{3}^2 + \beta_{12} X_{1} X_{2} + \beta_{23} X_{2} ...
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0answers
23 views

How to compute transition probabilities?

I have a stationary process $(X_t)_{t \geq 0}$ with distribution $$\mathbb{P}[X_t \in A ] = \int_A f(x) \, dx$$ for any measurable set $A$ and any $t \geq 0$. I want to compute $$ \mathbb{P}[X_\tau ...
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17 views

Markov-Chain Monte-Carlo: Are transformations on the inputs valid?

The problem: I am trying to solve a high dimensional (up to ~50) class of data fitting & modelling problems. The user specifies the problem, so I would like to make the configuration as easy as ...
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2answers
28 views

Would like some help formulating an optimization problem

I have a function $f$ that takes $n \geq 1$ positive real-valued arguments $\mathbf{a} \in R^n_+$. This function is defined for all amounts of inputs (e.g. $f(1)$ and $f(3, \pi, 17)$ are both valid) ...
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0answers
15 views

Estimate the volume of a convex body given a uniform random sample of points inside it?

Let $K$ be a convex, full-dimensional, bounded region of $R^n$. More precisely, there exist two balls of radiuses $0<r<R$ such that the ball of radius $r$ is fully contained inside $K$ and the ...
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0answers
31 views

Gibbs sampling truncation for contrastive divergence

I am following Yoshua Bengio's Learning Deep Architectures for AI and at page 31 there is a phrase that confuses me. Starting by lemma 7.1 in the same page: Lemma 7.1. Consider the Gibbs chain ...
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3answers
73 views

Exact concept of Monte Carlo Method [closed]

I am a programmer and just came across the section where in Monte Carlo was discussed. I would like to know the exact concept of Monte Carlo simulation. In net i have read about it that it is the ...
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0answers
28 views

computing the area of a region using Monte Carlo integration

Suppose that I am interested in estimating the area of $\Gamma \in \mathbb{R}^2$. I do not know the exact shape of $\Gamma$ but I have a sufficiently large number of sample points $(X,Y) \in \Gamma$ ...
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0answers
20 views

Multi-armed bandit with infinitely-many arms

Has anyone studied variants of the multi-armed bandit algorithm with infinitely many arms? I have a collection of distributions parametrized by an integer $n$. Unfortunately, I can't analytically ...
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0answers
31 views

MCMC/E-M limitations?MCMC over E-M?

I am currently learning hierarchical bayesian models using JAGS from R, and also pymc using ...
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1answer
54 views

Approximate an integral using Monte Carlo method

I have a question on an assignment Calculate the value of the integral I = $\int_0^\pi sin^2(x)dx$ using the Monte Carlo Method (by generating $ 10^4 $ uniform random numbers within domain [0, ...
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0answers
53 views

Random numbers generator

If I know how to generate random numbers from Gaussian distribution (using Box-Muller method), how can I generate random numbers from distribution with pdf ...
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2answers
289 views

Random directions on hemisphere oriented by an arbitrary vector

Hy, i'm writing a raytracer, and for that I need to generate n random vectors that are inside an hemisphere oriented by the surface normal. Ideally, I would also like being able to restrict the rays ...
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0answers
57 views

Standard deviation of the mean through bootstrap resampling of dependent samples

I'm trying to do a Monte Carlo approximation of an integral where the samples are not independent (how much so can be tuned by a parameter giving how often I sample). Therefore the regular expression ...
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0answers
92 views

Monte Carlo sampling and Kolmogorov–Smirnov test in practice

I have two deterministic algorithms, Algorithm 1 and Algorithm 2. The first has $m$ inputs and one output, and the second has $n$ inputs and one output. The distributions of the inputs of the ...
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0answers
32 views

Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
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0answers
50 views

Linear Filtering Problem (Keynman Fac/Particle Model)

$lienar Filtering Problem $$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$ above is $$\approx$$ $$dX_n^1 ...
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0answers
38 views

Normalisation of Monte Carlo overlap

Background: In quantum mechanics you are sometimes required to compute the overlap (inner product) of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ ...
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1answer
33 views

Find Monte Carlo Variance When Expected Value is not Known

I'm working on a problem that can be approached in two different ways. Both are Monte Carlo algorithms--but it's a hard problem, so I am unsure whether the expected values are indeed the same. I ...
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0answers
118 views

Convergence of Monte Carlo simulation

I am not sure if this is a valid question but here goes. For the monte carlo method I know that estimation of the mean is also a random quantity and follows a normal distribution. The standard error ...
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0answers
129 views

What does “sequence is equidistributed in [0, 2]” mean?

I was reading an article in which they are mentioning this sentence: "sequence is equidistributed in [0, 2]" where the sequence in question, is a sequence of real number (the article in question is ...
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0answers
374 views

Monte Carlo standard deviation of the mean estimate too small.

I'm doing a Monte Carlo calculation and use the standard deviation of the mean $\sigma_M$ as the error. To get an estimate of this from the regular standard deviation I use $$\sigma_M=\dfrac\sigma ...
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1answer
139 views

Sample uniform direction within cone

My question is pretty much the same as this question below, however I came up with a potential solution to this problem that I didn't see an answer to in the other question and I was wondering if it ...
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1answer
45 views

How do we calculate when to shout for the optimal payoff?

For example: A non-dividend paying stock is currently priced at $20, and you hold a put that allows early exercise in 2 months and in 4 months. The option expires in 6 months. Volatility is 30%, and r ...
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0answers
73 views

Monte Carlo Integration help.

I need to evaluate the following internal: $I=\int^{+\infty}_{-\infty}\dots\int^{+\infty}_{-\infty} f(x_,x_2,x_3\dots x_n) {1\over (\sqrt{2\pi})^N}e^{-{1\over2}x_1^2-{1\over2}x_2^2 \dots ...
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1answer
40 views

finding the optimal decision value for two dependent random events.

I have been struggling with this problem regarding options (bermuda) for some time now. You can exercise this option on two seperate occasions namely at $T_1$ or $T_2$ with a strike price $E$. The ...
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0answers
74 views

Density estimation using conditional Monte Carlo simulation

In Stochastic Simulation: Algorithms and Analysis by Glynn and Asmussen on p 146 they provide the following example. Let $f(x)=a/(1+x)^{a+1}$ be the density of a pareto distribution and let $a=3/2$ ...
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0answers
62 views

Monte-Carlo tree search convergence proof

I have been doing some reading about Monte-Carlo tree search for games, recently. The Wikipedia article mentions that the algorithm converges to the minimax evaluation for finite zero-sum two-player ...
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1answer
108 views

Why Gibbs sampling needn't “remixing”

I am generating $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)}$ using Gibbs sampling methods. So I want $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)} \sim$ some ...
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2answers
51 views

Advantage gained by Blackjack rule variation

There are many tables, charts and simulations for standard Blackjack variations and the % change in house edge that each rule introduces, like this one, for example. I have come across a Blackjack ...
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0answers
64 views

Gibbs / MCMC sampling for sum of parameters - how to improve slow mixing?

Suppose I have a hierarchical Bayesian model, where my observational prediction, $y'$, is calculated as the sum of other parameters, ${\alpha_i}$. My observation equation (the likelihood) is: $P(y | ...
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1answer
84 views

Bound of the variance of a random Variable

I am having trouble trying to prove that given a random variable $Y$ where $0 \lt m_1 \lt Y \lt m_2 < \infty$, where $m_1$ and $m_2$ are constants the $\displaystyle Var(Y) \le \frac{(m_2 - ...
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1answer
225 views

Will I have learned the prerequisites for self learning stochastic calculus and monte carlo method?

I'm an undergraduate econ major, and my main focus is in actuarial sciences, which as you may or may not know it's pretty mathematical. Some of the topics I will have to learn at some point on my own ...
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1answer
53 views

variance reduction

Say i have $n$ variables with variances $V_1,V_2,...V_n$. The sum of the variables will have a variance of $V=V_1+V_2+..V_n$ .Now if i am given N total simulations to reduce the variance V, how do i ...
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0answers
122 views

Monte-Carlo for SDE with square root diffusion term

I've recently got a question from a Master student about a numerical simulation/integration of the SDE of the following shape $$ \mathrm dX_i(t) = \left(\sum_{j=1}^M \nu_{ji} a_jX_j(t)\right)\mathrm ...
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0answers
47 views

Simulating of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + ...
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0answers
165 views

Volume with MonteCarlo

I'm asked to find the volume of a given bounded solid in $\mathbb{R}^3$ by MonteCarlo means: since it is contained in a prism, I generate random points in the prism and see what proportion of them lie ...
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0answers
117 views

Approximation of SDE

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I ...
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0answers
313 views

Computing Margin-of-Error using Monte Carlo simulations

I am interested in computing the margin-of-error for a metric computed on a random sample. The underlying distribution (finite) from which the random sampling is done is not normal (its extremely ...
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0answers
47 views

How to sample the walk which visits each vertex of a graph specific number of times?

Is there any MCMC mathod that allow me to uniformly sample from all feasible walks where the following restrictions apply: ...
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0answers
62 views

Gibbs Sampling versus General Cases

Suppose we are given a prior distribution about an unknown parameter $\pi(\theta)$. Also we are given $f(x_{1}, \dots, x_n|\theta)$. We want to find $\pi(\theta|x_1, \dots, x_n)$. Now ...
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0answers
170 views

Markov Chains - Using Gibbs & Metropolis algorithm.

Suppose $f_x,_y$ is bivariate normal distribution. I was given the parameters $(μ_1, μ_2, σ_1^2, σ_2^2)$ and $ρ=0.95$ the correlation coefficient. I want to generate $(x_1,y_1), ...
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0answers
42 views

Integrating with respect to an MCMC Kernel

I'm currently trying to evaluate an integral with respect to an MCMC kernel, and I wanted someone else to read what I `proved' to see if it's really correct. It just doesn't feel right. Suppose I ...
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1answer
381 views

Why are random numbers necessary for a Monte Carlo simulation?

This may be somewhat of a question with an obvious answer, but I can not seem to understand the necessity of "truly" random numbers to make a Monte Carlo simulation a good one. I understand that not ...
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2answers
78 views

Estimate gamma function using monte carlo

Let $\Gamma(\beta) = \int_0^\infty x^{\beta - 1} e^{-x} dx$ how to estimate the above gamma function using monte carlo? Any idea?