Questions on Monte Carlo methods, methods that require the repeated generation of (pseudo-, quasi-)random numbers for computing their results.
1
vote
1answer
198 views
Monte Carlo simulation on sphere: unbiased random steps
Im doing a Metropolis Monte Carlo simulation with particles on a sphere and have a question concerning the random movement in a given time step.
I understand that to obtain a uniform distribution of ...
4
votes
0answers
213 views
Simulating from a Multivariate Gaussian without Cholesky
I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
1
vote
1answer
135 views
Exponential Probability Monte Carlo simulation
I need to write a Matlab program to estimate the quantity $\theta = \mathrm{Pr}(X < 1)$, where $X$ is an exponential random variable with mean $1$. I am doing this for multiple monte carlo ...
2
votes
1answer
262 views
Monte Carlo - Control Variates & Antithetic method
Supposing $g(x)=\sqrt[3]{x}$, I want to calculate the expected value of g, $E(\sqrt[3]{x})$, using Monte Carlo method, by generating $x_i$ from a Weibull distribution with parameters $(1,5)$.
After ...
0
votes
2answers
32 views
Numerical estimation of simple integral
Considering the problem of numerical evaluation of the integral of a 'good' function $f(x)$ over a unit interval
$I = \int_0^1f(x)dx$
Why can we say $I = E[f(U)]$, where $U\sim Uniformly[0, 1]$?
2
votes
2answers
34 views
Maximum of $3x^2e^{-x^3}$
I have a PDF which looks like:
$f(x) = 3x^2e^{-x^3}, \quad x \geq 0 $
I need to find it's maximum (to sample from it using the rejection method), so I differentiate and set the result to $0$:
...
0
votes
0answers
19 views
Monte Carlo Technique and Convergence
I am solving $I = \int_{0}^{1}f(x)dx$ by Monte Carlo, e.g $I = E[f(U)]$ where $U\sim unif([0,1])$ so have
$I = E[f(U)]\approx \hat I_M\colon= \frac{1}{M}\sum_{m=1}^Mf(u_m)$
I am interested in how ...
3
votes
2answers
145 views
Markov chain stationary probability simulation
Having a defined markov chain with a known transition matrix, rather than to calculate the steady state probabilities, I would like to simulate and estimate them.
Firstly, from my understanding there ...
1
vote
1answer
34 views
Bound of the variance of a random Variable
I am having trouble trying to prove that given a random variable $Y$ where $0 \lt m_1 \lt Y \lt m_2 < \infty$, where $m_1$ and $m_2$ are constants the
$\displaystyle Var(Y) \le \frac{(m_2 - ...
1
vote
1answer
23 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 ...
0
votes
1answer
32 views
Question on Joint Posterior
Likelihood: $f(x^T, n^T|\theta^T) = \prod_{i=1}^{30} \binom{n^T_i}{x^T_i}{\theta^T}^{x^T_i}{(1-\theta^T)}^{n^T_i-x^T_i}$
Prior: $ log(\frac{\theta^T}{1-\theta^T})\sim N(\mu_T,\sigma_T^2) $
I am ...
2
votes
1answer
30 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 ...
0
votes
0answers
25 views
optimum weights for minimum variance
If $X_i$ is an estimator for $x_i$, and an estimate of the sum i.e. $ Y = x_1 + x_2 + ... + x_n $ is made by the ration $X_i / p_i$ where $p_i$ is the probability that $X_i$ is chosen , then what are ...
1
vote
3answers
93 views
transformation of integral from 0 to infinity to 0 to 1
How do I transform the integral $$\int_0^\infty e^{-x^2} dx$$ from 0 to $\infty$ to o to 1 and. I have to devise a monte carlo algorithm to solve this further, so any advise would be of great help
0
votes
1answer
65 views
Will I have learned the prerequisites for self learning stochastic calculus and monte carlo method?
I'm an undergraduate econ major, and my main focus is in actuarial sciences, which as you may or may not know it's pretty mathematical. Some of the topics I will have to learn at some point on my own ...
0
votes
0answers
17 views
selection for variance reduction in monte-carlo
I need to prove that the following variance reduction operations lead to the optimum value. A related question i asked here about variance reduction gives the optimal proportions are according to ...
1
vote
1answer
33 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 ...
0
votes
1answer
42 views
Metropolis Hastings definition - Proving $\pi(x)$ is the invariant density of our transition matrix
I'm currently working through the proof of the Metropolis-Hastings algorithm, and using two sources:
page 328, section 3
page 1704-1705
I have a good understanding of most of the proof until ...
1
vote
0answers
29 views
Monte-Carlo for SDE with square root diffusion term
I've recently got a question from a Master student about a numerical simulation/integration of the SDE of the following shape
$$
\mathrm dX_i(t) = \left(\sum_{j=1}^M \nu_{ji} a_jX_j(t)\right)\mathrm ...
1
vote
0answers
24 views
Importance sampling of finite path of stochastic difference equation
Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
2
votes
0answers
25 views
Determining Distributions for Monte Carlo
I'm trying to run a Monte Carlo to determine a set of given weights.
I have 5 weights (w1 to w5) that add up to 100%. Many people have different opinions on what these weights should be. We have ...
0
votes
0answers
19 views
Location based Simulations
I wasn't sure if I should ask this on CrossValidated stack, but since this is a math/programming problem for me I figured I'd ask here first. I am data mining user locations via schedules - so I know ...
3
votes
2answers
248 views
Reaching all possible simple directed graphs with a given degree sequence with 2-edge swaps
Starting with a given simple, directed Graph G, I define a two-edge swap as:
select two edges u->v and x->y such that (u!=x) and (v!=y) and (u!=y) and (x!=v)
delete the two edges u->v and x->y
add ...
6
votes
1answer
72 views
Monte-Carlo for the Wasserstein metric
Let $(X,d)$ be some metric space and assume that $d\leq 1$. Further, let $\mu, $ $\nu$ be two Borel probability measures on $X$ and let
$$
\Gamma(\mu,\nu) = \{\gamma - \text{measure on }X\times ...
0
votes
0answers
34 views
Finding a closed-form formula for the variance of the absorption direct estimator
I need to solve a system of linear equations via monte carlo methods, i.e.
$$Ax=B$$
I need to derive the a formula for the variance of an estimator given that the estimator is equal to:
- let $Q$ be ...
2
votes
1answer
85 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 ...
4
votes
1answer
141 views
Expected Value and Variance of Monte Carlo Estimate of $\int_{0}^{1}e^{-x}dx$
Given the integral:
$I=\int_{0}^{1}e^{-x}dx$,
use standard Monte Carlo with 1000 random numbers and repeat the simulation 1000 times.
(a) What is the expected value and variance of the simple Monte ...
1
vote
0answers
151 views
Matlab: Monte Carlo Probability Problem
A student is looking for a date who is older than 23, 170cm < height < 180cm, weight < 160lb.
The probability distribution for age $a$, weight $w$, and height $h$ is given by:
...
0
votes
1answer
51 views
Efficient method of approximating a distribution with Gaussian
Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$),
how to find the best ...
0
votes
2answers
74 views
How to recognize ellipse/ellipsoid from random points? UN-weighted average?
Suppose we are getting random points in 2D (or 3D) which tend to be on ellipse (or ellipsoid). We can't guarantee points are uniformly distributed over ellipse (ellipsoid surface).
The task is to ...
3
votes
0answers
48 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 ...
1
vote
0answers
27 views
Simulating of GBM
I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem:
Given a GBM of the form
$dS(t) = \mu S(t) dt + ...
1
vote
0answers
55 views
Volume with MonteCarlo
I'm asked to find the volume of a given bounded solid in $\mathbb{R}^3$ by MonteCarlo means: since it is contained in a prism, I generate random points in the prism and see what proportion of them lie ...
1
vote
0answers
64 views
Approximation of SDE
I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I ...
0
votes
0answers
72 views
Computing Margin-of-Error using Monte Carlo simulations
I am interested in computing the margin-of-error for a metric computed on a random sample. The underlying distribution (finite) from which the random sampling is done is not normal (its extremely ...
1
vote
1answer
43 views
Quasi-random sequence on the unit $2$-sphere for Monte-Carlo based method
Please forgive my possible misuse of the appropriate definitions. I'm looking for a quasi-random sequence of directions in the unit $2$-sphere, to be used in a Monte-Carlo method to calculate an ...
1
vote
1answer
36 views
Use of this condition on the instrumental density in importance sampling?
From Rubinstein's Simulation Monte Carlo Method, assume r.v. $X$ has density function $f$, $H$ is a measurable function, and $g$ is another density function. If $g$ dominates $Hf$ in the sense that ...
0
votes
0answers
132 views
Use importance sampling to estimate the quantity [Matlab]
So I've been doing some importance sampling by hand for much easier problems but im troubled with this task and thus running it through MATLAB would be much easier I think. Use importance sampling to ...
2
votes
1answer
126 views
Acceptance probability of Metropolis-Hastings
I am an IT guy writing my masters thesis on MCMC methods for use in predicting the outcome of football(soccer) matches. Right now I am trying to wrap my head around MCMC and Metropolis-Hastings in ...
1
vote
0answers
58 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 ...
1
vote
1answer
64 views
Should I ignore $0$ when do inverse transform sampling?
Generic method
Generate $U \sim \mathrm{Uniform}(0,1)$.
Return $F^{-1}(U)$.
So, in step 1, $U$ has domain/support as $[0,1]$, so it is possible that $U=0$ or $U=1$,
but $F^{-1}(0)=-\infty$. ...
0
votes
1answer
69 views
Monte Carlo sampling a binomial expansion
I want to figure out the following question
$$ 1 = (10 - 9)^{100} = 10^{100}-100 \times 10^{99} \ 9 + \frac{100 \times 99}{2} 10^{98} \ 9^{2} - \frac{100 \times 99 \times 98 }{3}10^{97} 9^{3} ...
0
votes
1answer
30 views
Why cannot the Markov Chains used in MCMC simulations be null recurrent?
I am aware this question borderlines retardedness, but I am seeking an accurate explanation.
I understand in null-recurrent cases, the expected amount of time to explore states can be infinite. Is ...
2
votes
2answers
119 views
Confusion related to the calculation of value of $\pi$
I was referring to this lecture Markov Chain Monte Carlo. However, I didn't get how the $\pi$ value was calculated. Here is a screenshot
As far as I know the integral gives the area under the ...
0
votes
2answers
240 views
How to decide what is the probability distribution in a Monte-Carlo simulation?
For a Monte-Carlo integration of
$$\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
81 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 ...
2
votes
2answers
154 views
What does it mean for MCMC to converge?
I know that a Markov Chain is a discrete random process where the current state decides the next and in a random walk, the probability that we move from node u to v is 1/N(u). An MCMC sample will ...
2
votes
3answers
253 views
Buffon needle experiment for $\pi$ approximation
What are the "best" values for length of needle $(l)$ and distance between paralles $(d)$ for an accurate approximation of $\pi$? Does it have to be $l=d=1.0$ or $l<d$ or $l>d$? Thank you
5
votes
2answers
286 views
Is there a natural way to multiply measures?
Given two measures $\mu$ and $\nu$ on some measurable space $X$, is there a way to multiply them to get $\mu \cdot \nu$, another measure on $X$ (and not $X \times X$, as for the usual notion of ...
3
votes
1answer
87 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 ...
