1
vote
0answers
58 views

What does “sequence is equidistributed in [0, 2]” mean?

I was reading an article in which they are mentioning this sentence: "sequence is equidistributed in [0, 2]" where the sequence in question, is a sequence of real number (the article in question is ...
0
votes
0answers
20 views

Monte carlo error: Combining “experimental” and statistical errors

I'm doing a slightly involved Monte Carlo approximation of a quantity $E$ where I end up with the following formula: $E=\frac{\sum_{i=1}^np_ie_iG_i}{\sum_{i=1}^np_iG_i}\ .\ \ \ \ \ \ \ \ \ $ (1) ...
0
votes
0answers
10 views

control variates - estimating definite integrals

What are the good techniques to find $g(x)$ so that $f (x) - g(x)$ is minimal, in order to evaluate $\int_a^b [(f(x) - g(x)) + g(x)]dx$?
3
votes
2answers
111 views

Algorithm to find best in class of groups with weighting?

I have widgets and a single widget will have attributes of: Name Weight (decimal from 0-1) Group (letter A-F) Price (an integer from 1 - 100) I must pick one ...
4
votes
2answers
211 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
1
vote
1answer
28 views

Bias Method in monte carlo integration

This is from a proof in my monte carlo course. let $h$ be a smooth function, $T_n = h(\bar{X})$ $\mu = E(X)$ then by taylor expansion $E(T_n -\tau) = E[h(\bar{X} -h(\mu)] = E[\bar{X} - \mu]h'(\mu)+ ...
1
vote
1answer
100 views

Monte carlo estimation of maximum likelihood estimators

I'm interested in numerically finding the maximum likelihood estimator of a parameter $\theta$, as well as the confidence interval of this estimator. First I'll describe the method I've been trying, ...
1
vote
1answer
31 views

finding the optimal decision value for two dependent random events.

I have been struggling with this problem regarding options (bermuda) for some time now. You can exercise this option on two seperate occasions namely at $T_1$ or $T_2$ with a strike price $E$. The ...
0
votes
0answers
19 views

Logit Nomal Prior Distribution

$$\mu \sim N(\mu_0,\sigma_0)$$ $$ X_i \sim LN(\mu,\sigma_x)$$ Does anyone know any method for finding the posterior distribution $P(\mu|X)$ or at least any idea of how to estimate it numerically. I ...
0
votes
0answers
32 views

Calculating a weighted mean / product of likelihoods from samples only

Suppose for argument's sake I have two distributions of arbitrary form (P(A) and P(B)), but they have well-defined first and second moments. Calculating the weighted mean of these two distributions ...
1
vote
1answer
91 views

Why Gibbs sampling needn't “remixing”

I am generating $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)}$ using Gibbs sampling methods. So I want $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)} \sim$ some ...
1
vote
0answers
51 views

Gibbs / MCMC sampling for sum of parameters - how to improve slow mixing?

Suppose I have a hierarchical Bayesian model, where my observational prediction, $y'$, is calculated as the sum of other parameters, ${\alpha_i}$. My observation equation (the likelihood) is: $P(y | ...
1
vote
1answer
30 views

Condition for Law of Large Numbers, Monte Carlo

In some lecture notes I am reading, there is the following; Consider $X_{1},...,X_{n}$, each with pdf $g$ (the instrumental distribution). Our aim is to estimate $E_{f}[h(X)]$ where $h(X)$ is some ...
2
votes
1answer
109 views

Monte Carlo Rejection Sampling Method

I have the following passage from a set of lecture notes I am working on that I would like to understand a little better. $\underline{\text{Algorithm for Rejection Sampling}}$: Given two densities ...
1
vote
1answer
47 views

variance reduction

Say i have $n$ variables with variances $V_1,V_2,...V_n$. The sum of the variables will have a variance of $V=V_1+V_2+..V_n$ .Now if i am given N total simulations to reduce the variance V, how do i ...
6
votes
1answer
119 views

Monte-Carlo for the Wasserstein metric

Let $(X,d)$ be some metric space and assume that $d\leq 1$. Further, let $\mu, $ $\nu$ be two Borel probability measures on $X$ and let $$ \Gamma(\mu,\nu) = \{\gamma - \text{measure on }X\times ...
2
votes
1answer
350 views

Stratified Monte Carlo

Consider the integral $I=\int_{0}^{1}e^{-x}dx$. Now consider the stratifed Monte Carlo estimate $\hat{I^{s}}$, that has $N_{st}=8$ strata. What is the variance of $\hat{I^{s}}$? What is the percent ...
3
votes
0answers
73 views

Does this count as a Monte Carlo simulation?

Let's say I have a group of robots that walk on a 11x11 grid of tiles in four directions, N, S, E, W, and each robot has different probability distribution functions that assign different ...
3
votes
1answer
111 views

non-intuitive results in random sampling

I play a card game (Magic The Gathering) that involves creating a deck of cards that can loosely be split into two categories: land and spells. You are permitted to play only one land per turn, and ...
0
votes
2answers
412 views

How to decide what is the probability distribution in a Monte-Carlo simulation?

For a Monte-Carlo integration of $$\int_\Omega P(x)f(x)\ \text d x,$$ there seems to be no apriori distinction if $f$ or $P$ is the probability function. So does it matter if I consider $$P, f, P ...
2
votes
1answer
112 views

Designing an efficient sampling strategy

In a Monte Carlo simulation, my goal is to compute an estimate of the mean of a distribution via sampling. Traditional, straightforward statistics generates samples (via simulation) and computes the ...
5
votes
2answers
563 views

What is the relationship between the Poisson Distribution and the Monte Carlo Fallacy?

Gravity's Rainbow has this long passage about the Poisson distribution. Since Pynchon's education included a serious dose of mathematics, and his novels include many references to mathematics, I ...
1
vote
0answers
39 views

Integrating with respect to an MCMC Kernel

I'm currently trying to evaluate an integral with respect to an MCMC kernel, and I wanted someone else to read what I `proved' to see if it's really correct. It just doesn't feel right. Suppose I ...