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

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Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...
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10 views

Gibbs sampler for MCMC is equilibrium distribution

Consider a Gibbs sampler, as in https://en.wikipedia.org/wiki/Gibbs_sampling, which we use to generate a Markov chain $x^0, x^1, x^2,...$ which samples from a distribution $\pi$. If we call ...
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15 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
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9 views

control variates - estimating definite integrals

What are the good techniques to find $g(x)$ so that $f (x) - g(x)$ is minimal, in order to evaluate $\int_a^b [(f(x) - g(x)) + g(x)]dx$?
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29 views

Analytic approach to find probability and total value of a set of independent events

I have a forecasting worksheet which describes a set (worksheet) of independent events, all of which have a likelihood of happening given as a probability (e.g. 0.7). Every event also has a yield ...
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56 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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28 views

Determining Errors in Monte Carlo Simulation

I was wondering if anyone could throw light on possible errors associated with Monte Carlo sampling. I seem to be getting values that are slightly different each time despite running my model for ...
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79 views

Algorithm to find best in class of groups with weighting?

I have widgets and a single widget will have attributes of: Name Weight (decimal from 0-1) Group (letter A-F) Price (an integer from 1 - 100) I must pick one ...
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16 views

Antithetic pair of non-independent normal random variables

Suppose that I have two non-independent normal random variables, X and Y such that $(X,Y)$ has mean 0 and the following variance covariance matrix: \begin{bmatrix} 1 & \rho ...
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21 views

Understanding graph obtained from Monte-Carlo simulations

I am running a Monte Carlo Simulation where I sample from about 65 Normal Distributions. I also keep track of the probability associated with each sample by approximating a thin area in the Normal ...
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18 views

What is the difference between Monte Carlo Simulation and Law of Lig Numbers

As far as I understood, Monte Carlo simulation uses several randomly chosen candidates to see where the result converges. On the other hand, Law of Large Numbers theorem describes the result of ...
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14 views

low discrepancy of halton sequences

I want to proof, that the Halton sequence is low discrepancy. I have to show that $$D_N^*(\mathcal{S})\le C\frac{\ln(N)^s}{N}$$ where $D_N^*$ ist the star discrepancy, $\mathcal{S}$ is the Halton ...
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197 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
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79 views

Obtaining useful information from graph obtained via Monte-Carlo Simulations

I've been running Monte Carlo Simulations on some Matlab code and then plot the graph shown below. I was just wondering what useful information I could collect from this graph? Edit: fit ...
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1answer
28 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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1answer
60 views

Buffon's experiment with squares

Say, we'd like to make the Buffon's experiment but with squares instead of needles. Notation: $d$ is the distance between lines $b$ is the square side length $y$ is the distance from the center of ...
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2answers
40 views

How many simulations for a game?

Lately I was interested by Monte-Carlo simulations. I found many papers about this approach in the Internet but for now they are too hard for me. I just want to start understanding this method with ...
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25 views

Average over all positive functions on the unit interval whose Lebesgue integral is one

I want to average over all positive functions on the unit interval whose Lebesgue integral is one. Formally, I want to compute the mean of the following probability distribution defined over function ...
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1answer
24 views

Bias Method in monte carlo integration

This is from a proof in my monte carlo course. let $h$ be a smooth function, $T_n = h(\bar{X})$ $\mu = E(X)$ then by taylor expansion $E(T_n -\tau) = E[h(\bar{X} -h(\mu)] = E[\bar{X} - \mu]h'(\mu)+ ...
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1answer
82 views

Monte carlo estimation of maximum likelihood estimators

I'm interested in numerically finding the maximum likelihood estimator of a parameter $\theta$, as well as the confidence interval of this estimator. First I'll describe the method I've been trying, ...
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1answer
49 views

Numerical integration of innocent-looking singular integrand

Consider the rather innocent integral: $$I=\int_{0}^{1}a x^{a-1}dx=1,\quad 0<a<1$$ Numerically, this integral converges awfully slowly, and one must use a recursive method to get anywhere near ...
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14 views

Cholesky decomposition correlated varibles.

Hi all I'm looking to generate correlated random variables. So I know that you can use the Cholesky decomposition of the correlation matrix to obtain the correlated values, however I keep being told ...
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1answer
62 views

Monte Carlo estimations of e

I need to estimate $e$ with a monte carlo method. We only learned the crude monte carlo integration, so I can't use any robust monte carlo simulations. I know that $\displaystyle \int\limits_1^x ...
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39 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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43 views

$\pi$ Monte-Carlo - Probability that O-Lock hit a Spoke?

(Edit: can someone please help me migrate this to physics stack? I think they would be more interested in helping me out with this problem. Thanks.) I have a bicycle with one of those O-locks on it ...
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24 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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23 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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42 views

Monte-Carlo calculation of a pdf

I need to evaluate the two first moments of pdf using mcmc methods. I'm already familiar with Metropolis algorithms in simulating statistical physics problems but I never had to approach a conditional ...
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38 views

Product Involving Sines

I'm studying the following product: $$p(a,\omega)=\prod_{k=1}^{\infty}a\sin (k\omega\pi),\quad \omega \in \Bbb R,\quad a\in \Bbb R_+.$$ It's easy to see that for $a\in (0,1]$ this product diverges to ...
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49 views

Monte Carlo method error in Bernoulli random variables

Assume I am flipping an unfair coin. Flipping the coin will be heads with probability $p$ and tails with $1-p$. I have no idea what $p$ is (it could even be $.5$!) Let's say I decide to use the Monte ...
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20 views

Weak convergence of discretization scheme with correction

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback ...
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12 views

Logit Nomal Prior Distribution

$$\mu \sim N(\mu_0,\sigma_0)$$ $$ X_i \sim LN(\mu,\sigma_x)$$ Does anyone know any method for finding the posterior distribution $P(\mu|X)$ or at least any idea of how to estimate it numerically. I ...
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16 views

Inner product of Gaussians times another multivariate Gaussian derivation help

Assume that: $$p(R|U,V,\alpha) \sim \prod N(R_{ij}|U_i^{T}V_j,\alpha^{-1})$$ $$p(U|\mu_u,\Sigma_u) \sim N(U_{i}|\mu_u,\Sigma_u)$$ $$p(V|\mu_v,\Sigma_v) \sim N(V_{i}|\mu_v,\Sigma_v)$$ Here $R$ is an ...
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37 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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3answers
116 views

How do I generate $100$ numbers in $[0,1]$ which are more dense at $0$ and $1$?

I just need to generate random numbers in $[0,1]$ which are more dense at the end points. I first thought of generating two sets of numbers from $N(0,1)$ and $N(1,1)$, and then using those. But that ...
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1answer
53 views

Monte Carlo estimator

I have hopefully a short/simple question regarding monte carlo estimators. The expected value of a function of a random variable can be defined as: $$E[f(x)] = \int_{-\infty}^{\infty} f(x) p(x) dx$$ ...
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2answers
37 views

Analytic methods vs Monte Carlo (terminology)

What's the correct terminology to say "We can calculate the probability exactly using pure math, as opposed to Monte Carlo simulation"? Analytically sounds like we need Calculus, which we may not ...
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35 views

Question about the Monte Carlo Algortihm

I was reading the Monte Carlo algorithm for finding the area under a curve, say $y=f(x)$. The algorithm considers, $0\le f(x)\le M$ over the closed interval $a\le x\le b$. My question is,that why is ...
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44 views

Monte Carlo Integration

I was reading this document (I will reproduce the equation): ...
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118 views

Alternatives to Monte-Carlo simulation

Imagine I have a model of economy of a region, which consists of several companies, importers and population. Let's assume that all local companies in question produce food and agricultural ...
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27 views

Calculating a weighted mean / product of likelihoods from samples only

Suppose for argument's sake I have two distributions of arbitrary form (P(A) and P(B)), but they have well-defined first and second moments. Calculating the weighted mean of these two distributions ...
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1answer
83 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
183 views

Please explain Monte Carlo method

Generally I understand the idea of the Monte Carlo method. However, when I read articles about it, there is always shown an example of calculating pi using a square, into which we insert 1/4th of a ...
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17 views

Monte Carlo by point or by interval

Say I compute monte carlo output from input scenarios. Input are discrete time series. I choose time series as an example to make the problem more obvious - this could be also any curve. Computation ...
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2answers
40 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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37 views

Confusion about Monte Carlo integration

I find I can not really understand the Monte Carlo integration, even I use it for many applications, like stochastic ray tracing. Let us take circle-area-calculation for an example, First, we think ...
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101 views

Monte Carlo integration, expected value of the sample mean and expected value of f(x)

I am still progressing in my learning of probability and monte carlo method. I understand a basic MC estimator can be written as: $$\bar x = { 1 \over N } \sum_{i=1}^N f(x_i) \approx E[f(x)]$$ I ...
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107 views

Markov Chain Monte Carlo in plain English

I barely know what a markov chain is (I had a terrible teacher) and I probably have an idea of what a stationary distribution is... but I don't know how a Monte Carlo method works and I don't know how ...
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76 views

Monte Carlo Simulation

If we have a random variable such that $$ H=B_1G_2\min(1,G_1)+B_2\frac{\min(2,G_1 G_2+G_2)}{n-B_1-B_2}, $$ where $n$ is constant, $G_1$ and $G_2$ are independent lognormal with different parameters, ...
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42 views

expected value involving probability of inequality random variables.

I have a question, not sure if this can be solved by calculation or Monte Carlo method For random variable G2+G2*min(2/G2-1,G1) where G1, G2 are indenpendt, G1~Lognormal(mu1,cigma1) ...