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

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2
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1answer
38 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 ...
2
votes
2answers
154 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
1
vote
0answers
31 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} ...
1
vote
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 ...
0
votes
1answer
15 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). ...
1
vote
0answers
17 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 ...
4
votes
1answer
59 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 ...
2
votes
1answer
26 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 ...
0
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0answers
8 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 ...
1
vote
2answers
28 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) ...
3
votes
3answers
36 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 ...
3
votes
2answers
100 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 $\mu_{X_n,Y_n},$ ...
0
votes
1answer
35 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
13 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
29 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
50 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, ...
1
vote
3answers
67 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
28 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
votes
0answers
29 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 ...
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
19 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
59 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
20 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 ...
3
votes
1answer
37 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
26 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 ...
2
votes
2answers
102 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 $\int_0^1x^2\,dx$. ...
-2
votes
1answer
21 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
41 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
2answers
216 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 ...
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
17 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
votes
0answers
17 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 ...
1
vote
3answers
136 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: ...
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 ...
3
votes
1answer
47 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: ...
1
vote
0answers
31 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
53 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
52 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
1answer
67 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 ...
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
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0answers
28 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
7 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
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0answers
52 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
22 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
259 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
1answer
149 views

Three ideas of perfect sampling

From David J.C. MacKay's Information Theory, Inference, and Learning Algorithms 32.2 Exact sampling concepts Propp and Wilson's exact sampling method (also known as "perfect simulation" or ...