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

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Variance of Importance Weights in Importance Sampling

What is the variance of importance weights in Importance Sampling? Is it good to reduce the variance of importance weights and why? And what is the optimum proposal in importance sampling? I assume ...
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How to determine the transition probability in Sequential Importance Sampling (SIS) for Particle Filter

Given a state-space model \begin{align} x_k &= f_k(x_{k-1}, v_{k-1}),\\ z_k &= h_k(x_k, w_k), \end{align} where $x_k \in {\mathbb R}^{n}$ and $y_k \in {\mathbb R}^{m}$ are the system state ...
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1answer
15 views

How to generate “independent” quasi random numbers

I am studying Monte-Carlo simulations using quasi random numbers and encounter the following problem: I am given a set of 1D quasi-random numbers $(X_i)$ over $[0,1)$, and would like to generate ...
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Exercises on the following topics on Markov Chains

We are being taught the following topics in Markov Chains: 1) Markov Chain Monte Carlo: Hard Core model, Counting random q-colourings of a graph 2) Total variation distance for a Simple Symmetric ...
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122 views

Monte Carlo Simulation- Simulating Sum of a DICE. Matlab CODE.

Hello everyone, I try to solve the following problem: Use Monte Carlo simulation to approximate the sum of the 100 consecutive rolls of a fair die. My work in math lab is: ...
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$\theta=\int_{0}^{2}3e^{-3x}dx$ Compute theta using Monte-Carlo Method

$\theta=\int_{0}^{2}3e^{-3x}dx$ Using Monte-Carlo method estimate the confidence intervals for the integral above. Use a distribution different from the uniform distribution to minimize the ...
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1answer
46 views

Quasi Monte Carlo or Gaussian Hermite quadrature for a statistical model with random effect

In my likelihood function, I need to integrate a random effect out as follows $$\int g(x,c)\exp(-c^2/2)dc .$$ Since the likelihood function is really complicated, I need the approximation to be fast ...
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38 views

Adequacy of Monte-Carlo simulations

Suppose we have a number of independent random variables of the form $X_1 \sim U[a_1,b_1], X_2 \sim U[a_2,b_2], X_3 \sim U[a_3,b_3]$. Now, suppose we generate a random variable $Y$ as follows: $$Y = \...
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1answer
23 views

Measurement, lognormal distribution, Monte-Carlo

I do have problems to understand the lognormal distribution. So, I do have one measurement M, measured with a sensor having a std S. As the sensor is not too accurate I want to build up a list of ...
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1answer
21 views

Why does this MCMC algorithm to estimate parameters of a linear equation not converge to the posterior distribution?

As a kind of proof of principle I'm trying to estimate the parameters of a linear equation (before moving on to ODEs) using Markov Chain Monte Carlo sampling. The post that I am following can be found ...
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2answers
66 views

Montecarlo estimate of a integrand from 0 to $\infty$

I have a question about monte carlo estimation of integrals. Suppose I am told to estimate using monte carlo, the integral: $$f(y) = \int_{0}^{y}\frac{4}{1+x^{2}}dx$$ I want to estimate $f(\infty)$. I ...
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1answer
54 views

Monte Carlo estimation of a constant?

I am currently learning monte carlo and I dont quite understand it. In the question I am given, I am asked to estimate $\pi$. So I am to write a Matlab code that computes a numerical estimate of $\...
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27 views

Integrating a function containing a conditional Gaussian mixture

I wonder if the following integral has an analytical solution. $$ \int_{-\infty}^{\infty}\frac{w_1 N_1(x)}{\sum_{i=1}^{n} w_i N_i(x)} N_0(x) dx $$ where $w_1, \ldots, w_n$ are positive constants, $N_0,...
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1answer
55 views

Sampling uniform equilibrium distribution with Markov Chain Monte Carlo

I'm wanting to sample the discrete uniform distribution over $n = 10$ integers using MCMC. My question concerns the transition probability matrix, $P$. As I understand it, any symmetric, irreducible ...
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14 views

Finding marginal posterior distributions (Gibbs Sampling)?

When using Gibbs sampling I need to find the conditional distributions of the parameters. In all textbooks and examples they seem to unanimously suggest that "it's obvious". Take for example page 56 ...
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48 views

why we use uniform distribution on accept reject method?

the accept-reject method have the following algorithm: Given known random number generators $U \sim Unif(0,1)$ and $X \sim g$, we can generate $Y \sim f$ by the following algorithm. Let $c$ be a ...
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Monte Carlo integration to solve coefficients of an orthogonal series - reusing the set of random points

I'm trying to approximate a function by summing a series of orthogonal functions. $f(x) \approx \sum_i a_i \phi_i(x)$ Since the set of functions $\phi_i(x)$ are orthogonal with respect to each other,...
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35 views

How to derive the conditional given the following joint probability

I encountered this question while reading about MCMC methods to solve image reconstruction problems. Consider a black and white image where $-1$ corresponds to white and $+1$ to black. $X_{i,j}$ ...
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20 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function $f(a,...
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22 views

Cdf of truncated distribution

Let $X$ be a random variable with density $f_x$ and distribution function $F_x$. Define the interval $I = (a,b)$. Given that we know these and the inverse distribution function $F^{-1}_x$, how can we ...
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Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
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What is the benefit of stochastic models over deterministic models? [duplicate]

I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask: Consider I have a simple and deterministic model $M$, with a number of input ...
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1answer
71 views

Monte-Carlo simulation with sampling from uniform distribution

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for ...
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12 views

The exact usage of Sequential Monte Carlo for distributions over time?

I have wondered the usage of Sequential Monte Carlos and it is used as an alternative to Kalman filter for example. However I wonder if this can be also used for simulating a distribution over time? ...
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1answer
48 views

Average of Monte Carlo simulations of continuous functions again continuous?

I hope the following question is clear: Suppose, we have a continuous functions $f:\mathbb{N}^2 \rightarrow \mathbb{N}$. Now, suppose we run Monte Carlo simulations on the function, where the input ...
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32 views

How to estimate a distribution from samples in a histogram

Given a r.v. $\tau$ , I've computed $\Bbb{P}(\tau >a)=e^{-Nx}(e^{Nxe^{-a}}-1) $ , where $N\in\Bbb{N}_{>1} $ and $ x\in \Bbb{R}_{>0} $ are just fixed parameters; say $N = 2 $ and $ x = 1$, ...
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34 views

Calculating success probability variance using Monte Carlo simulation

For a Monte Carlo simulation where each sample can produce a success value (1) or a failure value (0), what is the variance of the probability for success? Given n samples and r successes, the ...
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Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
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31 views

Biggest rectangle inside a random geometric shape

I'm looking for the most efficient algorithm to find the rectangle with the greatest area inside a random geometric shape. The rectangle can be also rotated of course. I am sure that there exists a ...
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Opposite of Monte Carlo

In this lecture, at 1:08:35, the lecturer goes from $$\text{argmin}\frac{1}{N}\sum\limits_{i=1}^{N}\text{log}\frac{p(x_i|\theta_0)}{p(x_i|\theta)}$$ to $$\text{argmin}\int\log\frac{p(x|\theta_0)}{...
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26 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\inf}_{inf} f(x)p(x) dx$ (with p(x) some ...
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17 views

Numerical method to fit arbitrary 3D curve by distributing perturbing elements on a 2D grid

I am looking for help in choosing and possibly implementing an appropriate algorithm or method to solve the following problem: I have a surface that has a property $A(r)$ that I want minimized. I ...
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58 views

Intuition behind rejection sampling proof

I have a quick question about the proof of rejection sampling. Suppose we know how to sample from a distribution with $Y$ pdf $q$, and want to sample from a distribution $X$ with (known) pdf $\pi$. ...
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1answer
55 views

Comparing Monte Carlo estmated PI and the real value PI

A famous example of Monte Carlo integration is the Monte Carlo estimate of PI. The unit disk { (x, y) : x2 +y2 <= 1 } is inscribed in the square [ 1, 1] x [ 1, 1], which has area 4. If we ...
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Monte Carlo Importance Sampling - Finding the new distribution

I'm currently working on a project which requires the implementation of importance sampling to reduce the variance when pricing an option. I think I understand the theory behind what should be going ...
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26 views

Calculating integral with antithetic variables

Use simulation with antithetic variables and find $$\int_{-\infty}^\infty \int_0^\infty \sin(x+y)e^{-x^2+4x-y} \, dx \, dy.$$ so, my question and doubt is how struggle with the infinite limit ? It ...
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35 views

Mistake in generating random numbers - no irrational ones

Hi I just wondered if the probability densities have to be corrected when using them on a PC since the number representation is not at all continuous. So we cant simulate any irrational numbers and ...
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82 views

Matlab code, approximate an integral using Monte-Carlo method.

so i have to program the approximation of these two integrals using Monte-Carlo method: $$\int\int_D e^{x^2+y^2} \, dy \, dx $$ $$D=\{(x,y) \in \Bbb R \mid x^2+y^2\le9\}$$ and: $$\int_0^2 \int_{-1}^1\...
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Ammunition Depot: Monte Carlo Method

I was given the following question from a friend of mine and I can't seem to understand it to well: A squadron of 10 bombers attempts to destroy an ammunition depot. The fighter jet flies in the ...
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30 views

Optimization of Inputs to Monte Carlo Simulation Based on Outputs

I have an optimization process that seems to work, but I want to better understand why it works and whether there's a better way to do what I'm trying to achieve. Basically I am optimizing two (or ...
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Do Monte Carlo perturbations capture all the uncertainty in prediction?

I have a model $M$ that I use to predict a value $y = M(\vec x)$. I have known one-$\sigma$ error bars on each input $x_i \in \vec x$. I want to know the one-$\sigma$ error bar on my prediction $y$. ...
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1answer
103 views

monte carlo simulation - confidence intervals construction

I am starting with Monte Carlo Simulation. I have run simulation to estimate the mean and the variance of the exponential distribution. Simulation: I have generated random sample from uniform ...
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54 views

Validity of Monte Carlo

My question regards the fundamental validity of the concept of Monte Carlo. In the text where I learned about Monte Carlo some time ago and also on all resources I found on the internet, all authors ...
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Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = [1+(...
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How to use the Monte Carlo method with curve fitting

I have an assignment in which I need to choose statistical data from the Australian Bureau of Statistics and fit a curve to it. I've chosen the slaughtering of bulls in NSW (first Excel file, C:143-C:...
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How does the Metropolis Algorithm work? (for idiots)

I have the mathematical skills of a house brick and I am desperately trying to learn this algorithm from a computer science perspective. Below is my knowledge of the algorithm. Can someone please ...
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76 views

Integral over space filling curve

Generalizations of the Dirac delta ($\delta$) function ([1]) seemingly enable the expression of $d-1$ dimensional (surface) integrals as $d$ dimensional (volume) integrals in the following form: $$\...
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1answer
35 views

Equivalence of Probability spaces. Monte carlo integration

Pondering about the independence of dimension of Monte Carlo Integration, I came up with the following explanation: An integral over a square is not harder, thus has the same rate of convergence, ...
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1answer
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How Can ı validate my Monte Carlo Simulation?

Now I am writing my thesis The topic is about error analysis on turbocharger test bench. In order to estimate uncertainties at test bench first of all I used Monte Carlo Method. Now I have to ...