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

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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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44 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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39 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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1answer
64 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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24 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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18 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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43 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
125 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
56 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
42 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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1answer
36 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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48 views

Monte Carlo Integration

I was reading this document (I will reproduce the equation): ...
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1answer
131 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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30 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
89 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
287 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
45 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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1answer
41 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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1answer
114 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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1answer
119 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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85 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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1answer
46 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) ...
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49 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
97 views

What is the typical method for sampling uniformly in a convex polytope

The polytope in my case is the intersection of the k-plane $Ax=b$ and $\{x>0\}$ where $A$ is the constraint matrix and $b$ is some solution. I'd like to find a method that is fast and efficient for ...
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2answers
147 views

Monte Carlo Integration - determining if a random x,y coordinate falls within the circle or square

My textbook says you can take any random (x,y )coordinate between -1 and 1 like (-.3, .5) or (.4, -.7) and determine if the given coordinate falls within the circle if you calculate $\sqrt(x^2+y^2)$ ...
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2answers
561 views

Why does Monte-Carlo integration work better than naive numerical integration in high dimensions?

Can anyone explain simply why Monte-Carlo works better than naive Riemann integration in high dimensions? I do not understand how chosing randomly the points on which you evaluate the function can ...
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1answer
124 views

Conditional probability of the sum of r.v.

I have $n$ independent random variables $X_i$ with known PDF and CDF (say, Normal, but not necessarily with the same parameters). Given $U_1, U_2 \subseteq \{1,...,n\} $ such that $U_1 \cup U_2 = ...
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2answers
41 views

Numerical estimation of simple integral

Considering the problem of numerical evaluation of the integral of a 'good' function $f(x)$ over a unit interval $I = \int_0^1f(x)dx$ Why can we say $I = E[f(U)]$, where $U\sim Uniformly[0, 1]$?
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2answers
50 views

Maximum of $3x^2e^{-x^3}$

I have a PDF which looks like: $f(x) = 3x^2e^{-x^3}, \quad x \geq 0 $ I need to find it's maximum (to sample from it using the rejection method), so I differentiate and set the result to $0$: ...
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36 views

Monte Carlo Technique and Convergence

I am solving $I = \int_{0}^{1}f(x)dx$ by Monte Carlo, e.g $I = E[f(U)]$ where $U\sim unif([0,1])$ so have $I = E[f(U)]\approx \hat I_M\colon= \frac{1}{M}\sum_{m=1}^Mf(u_m)$ I am interested in how ...
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1answer
67 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
30 views

Condition for Law of Large Numbers, Monte Carlo

In some lecture notes I am reading, there is the following; Consider $X_{1},...,X_{n}$, each with pdf $g$ (the instrumental distribution). Our aim is to estimate $E_{f}[h(X)]$ where $h(X)$ is some ...
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1answer
104 views

Monte Carlo Rejection Sampling Method

I have the following passage from a set of lecture notes I am working on that I would like to understand a little better. $\underline{\text{Algorithm for Rejection Sampling}}$: Given two densities ...
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1answer
58 views

Question on Joint Posterior

Likelihood: $f(x^T, n^T|\theta^T) = \prod_{i=1}^{30} \binom{n^T_i}{x^T_i}{\theta^T}^{x^T_i}{(1-\theta^T)}^{n^T_i-x^T_i}$ Prior: $ log(\frac{\theta^T}{1-\theta^T})\sim N(\mu_T,\sigma_T^2) $ I am ...
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3answers
2k views

transformation of integral from 0 to infinity to 0 to 1

How do I transform the integral $$\int_0^\infty e^{-x^2} dx$$ from 0 to $\infty$ to o to 1 and. I have to devise a monte carlo algorithm to solve this further, so any advise would be of great help
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1answer
187 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
42 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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1answer
95 views

Metropolis Hastings definition - Proving $\pi(x)$ is the invariant density of our transition matrix

I'm currently working through the proof of the Metropolis-Hastings algorithm, and using two sources: page 328, section 3 page 1704-1705 I have a good understanding of most of the proof until ...
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80 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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75 views

Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
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36 views

Determining Distributions for Monte Carlo

I'm trying to run a Monte Carlo to determine a set of given weights. I have 5 weights (w1 to w5) that add up to 100%. Many people have different opinions on what these weights should be. We have ...
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1answer
113 views

Monte-Carlo for the Wasserstein metric

Let $(X,d)$ be some metric space and assume that $d\leq 1$. Further, let $\mu, $ $\nu$ be two Borel probability measures on $X$ and let $$ \Gamma(\mu,\nu) = \{\gamma - \text{measure on }X\times ...
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1answer
307 views

Stratified Monte Carlo

Consider the integral $I=\int_{0}^{1}e^{-x}dx$. Now consider the stratifed Monte Carlo estimate $\hat{I^{s}}$, that has $N_{st}=8$ strata. What is the variance of $\hat{I^{s}}$? What is the percent ...
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1answer
1k views

Expected Value and Variance of Monte Carlo Estimate of $\int_{0}^{1}e^{-x}dx$

Given the integral: $I=\int_{0}^{1}e^{-x}dx$, use standard Monte Carlo with 1000 random numbers and repeat the simulation 1000 times. (a) What is the expected value and variance of the simple Monte ...
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1answer
98 views

Efficient method of approximating a distribution with Gaussian

Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$), how to find the best ...
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70 views

Does this count as a Monte Carlo simulation?

Let's say I have a group of robots that walk on a 11x11 grid of tiles in four directions, N, S, E, W, and each robot has different probability distribution functions that assign different ...
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2answers
363 views

How to recognize ellipse/ellipsoid from random points? UN-weighted average?

Suppose we are getting random points in 2D (or 3D) which tend to be on ellipse (or ellipsoid). We can't guarantee points are uniformly distributed over ellipse (ellipsoid surface). The task is to ...
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43 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 + ...