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

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1answer
28 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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1answer
76 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 ...
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0answers
44 views

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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0answers
25 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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0answers
18 views

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
80 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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0answers
50 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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1answer
18 views

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) = ...
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42 views

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, ...
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1answer
57 views

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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0answers
67 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, ...
2
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1answer
81 views

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 ...
2
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1answer
30 views

Concentration inequality without variance

Let $X$ be a positive random variable with $\Bbb E X \leq M$. I would like to compute the expectation using Monte-Carlo method, so I am looking for the bounds on $\Bbb P(|\bar X_n - \Bbb E X| \geq ...
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0answers
9 views

Generating a sample of Multivariate Epanechnikov Kernel

Is there an efficient algorithm to generate random deviates from a multivariate Epanechnikov kernel? I know it can be performed by rejection sampling but it is ineffective in high dimensions.
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0answers
22 views

Parameter Values From Asymmetric Probability Distributions

I am performing a Multipole Decomposition Analysis on some experimental data, essentially fitting a set of experimental data with a linear combination of functions. Annoyingly these functions are not ...
0
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1answer
16 views

Unbiasedness Importance Sampling

In the 2003 survey paper on MCMC methods by Andrieu et al, there is a section on importance sampling. More specifically, in the section included above it is claimed that $\sum_{i=1}^N ...
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0answers
23 views

Metropolis algorithm and formula

In the article by M. Creutz and B. Freedman "Statistical Approach to Quantum Mechanics" authors provide a formula for Metropolis algorithm: $$ W(x_j, x'_j) = \frac{1}{N_0} \left( \theta \left[ S(x_j) ...
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1answer
90 views

Generation of random variable from a complicated CDF

Suppose I am given a CDF of a distribution, given by $F(x) ∝ \int_0^1 x^y e^{-y} dy.$ Here,'x' ranges from 0 to 1. How do I generate a random variable from this distribution?
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46 views

What calculus material can prepare me for MCMC?

I am looking to revise calculus from scratch to move on to Monte Carlo Markov Chain Methods and Quasi Monte Carlo Methods. I studied calculus properly a couple of years ago however it was back in high ...
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0answers
36 views

Parametrization of the $Ax=b, x \geq 0$ domain for Monte-Carlo simulation

I have a linear system, $n=15$, with $6$ constraints. There's no problem finding a single solution or establishing the null space; so I can see the full solution space. But I'm only interested in ...
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0answers
54 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 ...
3
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1answer
108 views

Optimal algorithm for guessing random variable

Let's say you have some unknown quantity $$X\in [0,1]$$ We have N tries to guess the value of X - if you guess $$g_{i}\le X$$ then you capture value $$V_{i} = g_{i}$$ while if your guess is over the ...
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1answer
82 views

How can one read the order of convergence from a loglog-graph?

I am making a task which includes running a Monte Carlo simulation and calculating the order of convergence experimentally. I have to calculate (or approximate) the order of convergence using ...
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1answer
19 views

Inner product estimator - random variable

I'm curently working on the functional space $L^2(\mathbb{R}^n,B(\mathbb{R}^n),\mathbb{P}_X)$ where $\mathbb{P}_X$ is a probability measure. If I generate randomly $N$ realizations of $x_i$ following ...
2
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1answer
60 views

Approximation to a compounded Binomial distribution

I need to find an approximation, from which I can easily sample, to the following compounded Binomial distribution: $X \sim \mathrm{Binomial}(e^{-\epsilon}, \ n)$ where $\epsilon \sim ...
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37 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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0answers
29 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 ...
2
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1answer
60 views

Use of ergodic theory in numerical simulations

Is ergodic theory used in numerical simulations? The kind of application I have in mind is: for $\alpha$ irrational, $( n\alpha \mod 1)_{n \geq 0}$ is equi-distributed on $[0,1]$, and I imagine that ...
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0answers
12 views

Correct way to implement event rejection in Gillespie method

I am using the Gillespie method for use when generating snowflakes from copy-rotate-translate of basic geometric shapes. The model consists of a matrix, describing coalescence probabilities, and ...
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1answer
32 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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3answers
48 views

Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
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1answer
51 views

Simulating Random Vectors Based on Conditioning

I'm working on a project where I need to simulate random vectors $(Y, X_1,\dots,X_n)$ in order to understand the joint distribution $f(y,x_1,\dots,x_n)$. I wish to simulate enough random vectors so ...
3
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0answers
56 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 $\mathbb{R}^n$. More precisely, there exist two balls of radiuses $0<r<R$ such that the ball of radius $r$ is fully contained inside $K$ ...
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0answers
38 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 ...
3
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2answers
159 views

Lagrange multiplier and minimum variance

Looking into a control variate technique of Monte Carlo simulation I have run into a cost-optimization problem that I'm not quite sure I understand. It seems it has to do with Lagrangian multipliers, ...
3
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2answers
119 views

Mutual information of discrete RVs which converge in distribution to a continuous RV

$\mu_{X_n,Y_n}$ is a sequence of discrete joint-distributions on $\mathbf{R}^2$ that converge weakly to a continuous measure $\mu_{X,Y}$. That is, for any continuous function ...
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3answers
150 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
39 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
56 views

Why use rejection sampling in Monte Carlo simulations?

I've noticed that a lot of physics Monte Carlo simulations make extensive use of rejection sampling, rather than inverse transform sampling. In my research, I'm sampling random energy transfers from ...
4
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1answer
73 views

Bootstrap method failing where blocking works

I'm computing an average of individual samples that are not entirely independent and need an estimate for the true standard deviation. According to Newman and Barkema's book the most reliable method ...
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0answers
14 views

Performing inference on a further area of study, Bayesian model.

Consider the following model: $y_i \sim \text{Poisson}(n_i \theta_i)$ $\theta_i \sim \text{Gamma}(\alpha, \beta)$ $\theta_i \sim \text{Gamma}(\gamma, \delta)$ All other variables are constant. $ i ...
1
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1answer
46 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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1answer
104 views

How to Find a Probability with Monte Carlo Simulation [closed]

$$ f(x) = \begin{cases} C\exp(-\frac{1}{2}x^3), & \quad x >-1,\\ 0, & \text{othewise}. \\ \end{cases} $$ Here, $C=1/2.2702.$ I want to find the probability ...
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0answers
9 views

Deriving conditional distributions for a normally distributed change point problem

Considering the change point problem of $y_i \left\{ \begin{array}{ll} y_i \tilde{~} N(u_1, \sigma) & i=1,..,t \\ y_i \tilde{~} N(u_2,\sigma) & i= t+1,...,n \\ \end{array} ...
2
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1answer
29 views

Estimating quantities of a posterior distribution.

Consider the following model: $$ \alpha \sim N(0,1)$$ $$ \beta \sim N(0,1)$$ $$ d_i \mid \alpha, \beta \sim \mathrm{Bernoulli}(\Phi(\alpha + \beta x_i))$$ $d_i$ is $1$ if person $i$ has some ...
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0answers
34 views

Automatic differentiation for finance

we're estimating sensitivities with automatic differentiation. What we have read about it the adjoint (reverse) should perform more efficiently than the forward mode when there are more input ...
3
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1answer
41 views

On random rotational fluctuations in $\mathbb{R}^n$

Imagine first a disk that is mostly stationary, except for random ("thermal" if you like) "rotational fluctuations" around its axis (which is fixed). Something a bit like what's shown in the figure ...
0
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1answer
36 views

proposal distribution for metropolis algorithm

All, I'm wondering whether it is possible to use an asymmetric distribution, eg the exponential distribution as the proposal dist'n for a metropolis algorithm (wiki) (not the metropolis-hastings). ...