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

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3
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3answers
29 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 ...
0
votes
1answer
33 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 ...
1
vote
0answers
12 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 ...
1
vote
0answers
26 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
votes
2answers
40 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
votes
2answers
95 views

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

We have a sequence of pairs of discrete, real-valued RVs $X_n$ and $Y_n$. Each pair is characterized by a discrete probability measure on $\mathbf{R}^2$, which we will just denote $\mu_{X_n,Y_n},$ ...
1
vote
3answers
60 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 ...
1
vote
0answers
26 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 $(X,Y) \in \Gamma$ ...
0
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0answers
26 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 ...
2
votes
0answers
14 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 ...
0
votes
0answers
10 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
vote
0answers
17 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 ...
1
vote
1answer
55 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 ...
0
votes
0answers
19 views

Finding normal curve given the minimum and maximum - is it possible?

I have a quick question regarding the normal distribution, or really any kind of distribution as it can also be skewed if need be. I was wondering if it were possible to let the curve have a minimum ...
0
votes
0answers
6 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
votes
1answer
26 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 ...
0
votes
0answers
25 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
votes
1answer
36 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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0answers
6 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). ...
-2
votes
1answer
15 views

Variance Reduction Using Antithetic Variates

I found this online: http://en.wikipedia.org/wiki/Antithetic_variates For example #2, can someone please provide step by step procedure on how to answer the integral using antithetic variates? I ...
3
votes
0answers
39 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
0
votes
0answers
10 views

Volume of a region given by a CSP

I have a Linear Constraint Satisfaction Problem i.e. I have variables $ x_1, x_2,...,x_m$, with corresponding domains $D_1,D_2,...,D_m $ satisfying linear constraints $C_1, C_2,...,C_n$ with $n ...
2
votes
2answers
32 views

What's the average length of a random line segment in a $1 \times 1$ field?

What is the average length of a line segment in a $1 \times 1$ field? Given $$x_1, y_1, x_2, y_2 \in [0,1]$$ $$S = (x_1,y_1,x_2,y_2)$$ $$dist(x_1,y_1,x_2,y_2) = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$ ...
0
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0answers
15 views

Forecasting disputed transaction frequencies

Problem I would like to forecast credit card chargeback/dispute frequencies using historical dispute data I have recorded over time. The data I currently store includes: Disputed transaction date ...
0
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0answers
15 views

Estimating the Square of a Mean

Suppose I want to estimate $\theta = (\mathbb{E}[f(X)])^2$, where $f: \mathbb{R} \to \mathbb{R}$ and is Borel-measureable, and $X$ is a random variable. I'll use Monte Carlo, for which one ...
3
votes
1answer
42 views

Metropolis Hastings

So I have seen the Metropolis Hastings algorithm written 2 ways, and I don't quite understand how they can be equivalent: The first way is by defining the 'acceptance probability' as: ...
2
votes
0answers
76 views

Question about Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies - Kannan Lovasz Simonovits 97

I've been trying to understand this paper: "Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies", Ravi Kannan, Laszlo Lovasz, Miklos Simonovits. Motivation: The paper is about ...
1
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0answers
30 views

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

I am currently learning hierarchical bayesian models using JAGS from R, and also pymc using ...
0
votes
2answers
48 views

Approximating an integral using Monte Carlo Method

I wrote a solution for 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, π] × [0, ...
1
vote
1answer
49 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, ...
1
vote
3answers
115 views

Evaluating Difficult Monte Carlo Integration in R

I recently posted a simple version here (Simple Monte Carlo Integration). I was able to verify that the answer was indeed close to 1/3 when I wrote the following R code, and got a mean of X of ~1/3: ...
1
vote
1answer
65 views

Monte Carlo gamma function

This question was asked before but I'd like to ask something more precise given the answer that was given. [ Estimate gamma function using monte carlo ] What is the criterion for a random point to ...
2
votes
2answers
96 views

Simple Monte Carlo Integration

I am trying to use Monte Carlo Integration, which is nicely described in the answer here (Confusion about Monte Carlo integration). I am using Monte Carlo Integration to evaluate $\int_0^1x^2\,dx$. ...
0
votes
2answers
76 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?
0
votes
0answers
27 views

Sequential Importance Sampling

What are the pros and cons of the the basic SIS-algorithm? I know there is some drawback considering the weight degeneration, but not so much about the pros. Also, is there a proof that the extra ...
0
votes
0answers
18 views

Importance Sampling Distribution

I have an infinite set of events and these event are either ture or false. I perform a monte carlo simulations to find the probability of an event being true. Now I have the knowledge that $A$ % of ...
0
votes
0answers
5 views

Why is importance sampling always framed as calculating E(h(x))?

In all of the tutorials I've seen, importance sampling is always framed as a way to calculate: $$E(h(x)) = \int h(x) f(x) dx \qquad x \sim f$$ I don't understand why it is not framed as a more ...
1
vote
0answers
49 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 ...
0
votes
1answer
21 views

What is the average minimum distance between two Sobol points?

Having the first n points of a d-dimensional Sobol sequence, what is the average Euclidean distance from one arbitrarily point to its nearest neighbour?
1
vote
2answers
235 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 ...
1
vote
2answers
61 views

Probabilities seem to be growing exponentially

We have instituted random drug testing at our company. I was charged with writing the code to generate the weekly random list of employees. We've gotten some complaints because of people getting ...
0
votes
1answer
81 views

Histogram normalization

I need to generate random numbers from Gaussian distribution and to draw an equalized histogram. I've generated them in Matlab using Box-Muller transformation. Since I wasn't sure how to equalize the ...
1
vote
0answers
41 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 ...
1
vote
0answers
79 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 ...
0
votes
0answers
91 views

Importance Sampling of 2D constant piecewise function convertible to 1D?

So I have a constant piecewise 2D function (luminance values of pixels of an image) that I am writing an importance sampling algorithm for. I was going to write my algorithm by first sampling the 1D ...
0
votes
0answers
13 views

LDA with fixed topics?

Suppose I have a collection of "topic" probability distributions $\{\phi^{z}\}$ for LDA (Latent Dirichlet Allocation) that I have found via alternate methods; is there a closed form MLE for the ...
0
votes
1answer
86 views

Double integral estimation with Monte Carlo method by importance

I have this integral $\int_{0}^{\infty}\int_{0}^{\infty} e^{-x-y-xy}dxdy$ I have to estimate the value of it using the Monte Carlo method, using as importance function the exponential PDF. Does ...
0
votes
0answers
51 views

Monte Carlo simulations, accuracy of mean vs variance of answers

I am working on a Monte Carlo simulation where two inputs are being used, $N$ is the amount of simulations I use, and $M$ controls the detail of each simulation (specifically the amount of time step ...
0
votes
1answer
37 views

Looking for Method to evaluate the optimal node rate vs number of simulation rate in a Monte Carlo simulation

I am currently working on evaluating an American Option using a Monte Carlo simulation, and I am getting answers but they vary quite a bit. The two variables that I can alter are number of simulations ...
2
votes
0answers
48 views

Mixing time for metropolis chain on graph coloring

I'm reading the Markov Chains and Mixing Times by David Levin et al.. In section 5.4 page 71 a proof is given for a bound of mixing time for the Metropolis Chain on graph coloring. In the proof, such ...