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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0answers
32 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 ...
0
votes
0answers
32 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 ...
1
vote
0answers
35 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
votes
1answer
52 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 ...
0
votes
1answer
16 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 ...
0
votes
1answer
16 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
votes
1answer
51 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 ...
1
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0answers
33 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 ...
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 ...
2
votes
1answer
33 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) ...
4
votes
3answers
40 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
36 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
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0answers
15 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
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0answers
31 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
56 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
104 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
74 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
32 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
votes
1answer
61 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
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0answers
11 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
20 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
67 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 ...
-1
votes
0answers
21 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
8 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
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0answers
27 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
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
votes
1answer
18 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
26 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
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
33 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
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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 ...
3
votes
1answer
49 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
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
54 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
54 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
165 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
74 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
111 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
78 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
32 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
19 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
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0answers
8 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 ...