Using (pseudo-)random numbers and the Law of Large Numbers to simulate the random behavior of a real system.

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4
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0answers
99 views

How to generate two groups of $n$ random numbers in $U(0,1)$ such that sum of these two groups equal? [on hold]

I want to have two groups of $n$ random numbers $u_i$ and $v_i$ in $U(0,1)$, such that $\sum u_i = \sum v_i$ What I tried is: I can firstly get $u_i$ by $U\sim U(0,1)$, make $s=\sum u_i$. Then I ...
18
votes
5answers
356 views

Approximate $e$ using Monte Carlo Simulation

I've been looking at Monte Carlo simulation recently, and have been using it to approximate constants such as $\pi$ (circle inside a rectangle, proportionate area). However, I'm unable to think of a ...
0
votes
0answers
15 views

Estimate number of unique items by number of duplicates in a sample

We have a v. large (1e6) population with unknown number of types of items. We draw a small sample (~100) of a certain size, and find that exactly one item was duplicated. The question is to estimate ...
0
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0answers
18 views

Absolute error or RMSE when you know the exact value?

I'm testing the $k$ neighbour correlation of a uniform random sequence $x_i$ in $[0,1)$. I know its exact expected value to be $1/4$ and I want to show that the error decreases with $\sqrt{N}$ where ...
2
votes
2answers
54 views

Monte Carlo Simulation of Complex Dynamical System

Assume that $\vec{z}(t)$, the state at time $t$ of a particle in a two-dimensional space, can be fully described by its position and velocity: $\vec{z}(t) = [r_x(t)\ r_y(t)\ v_x(t)\ v_y(t)]$. ...
1
vote
1answer
22 views

How to explain simply that the set of runs for Non Intrusive Polynomial Chaos cannot be used as a Monte Carlo sample

I had quite an annoying problem at work, a few days ago. I was doing a forward Uncertainty Quantification analysis using Non Intrusive Polynomial Chaos (NISP) (see for example here). Basically, you ...
1
vote
1answer
82 views

Does this Monte Carlo Technique Have a Name?

I sketched this algorithm out the other night. I am sure it has a name, I just do not know what it is yet. It would be helpful if someone could point me in the right direction for research. I ...
1
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1answer
15 views

Must cross-entropy method's true distribution be in-family with the proposal distribution?

The cross entropy method is used in rare event estimation, such as when estimating $$\mathbb{E}_u\left[H(X)\right] = \int H(x) f(x; u) dx\text{,}$$ where $H$ is some performance function and $f(x; u)$ ...
1
vote
1answer
37 views

justification of Monte Carlo integration

I came across the following justification of Monte Carlo integration, where $p(x)=\frac{1}{b-a}$ $E[F_N]=E\bigg[ \frac{b-a}{N} \sum \limits_{i=1}^{N}f(X_i) \bigg]$ $= \frac{b-a}{N} \sum ...
0
votes
1answer
42 views

Monte Carlo rolling forecast of time series - details needed

I know I'm doing a short term forecast of a volatile time series using Monte Carlo, but I'm unsure as to the details - for example, I'm sure I had a very good reason for naming a term 'drift', but I ...
3
votes
2answers
58 views

Can I use bootstrapping to estimate the uncertainty in a maximum value of a GAM?

I have data from an experiment where I look at the development of algal biomass as a function of the concentration of a nutrient. The relationship between biomass (the response variable) and the ...
1
vote
1answer
47 views

Consistent estimator of the expectation of a conditional probability

I'm stuck in a problem where I have distribution distribution $P(\boldsymbol{x})$, from which I know how to sample from (i.i.d.) and two functions of the random variable $\boldsymbol{x}$: ...
1
vote
0answers
47 views

Monte Carlo error propagation

Consider a set $X$ of $N$ iid random variables, each one with its own standard deviation: $$X: \{x_1\pm\sigma_1, x_2\pm\sigma_2, ..., x_N\pm\sigma_N\}$$ Say I have a "black box" numerical function ...
0
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0answers
33 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\infty}_{\infty} f(x)p(x) dx$ (with p(x) ...
1
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0answers
14 views

using markov matrices for projections

Top of the morning folks .. Business context i am doing an experiment with an index time series (NSE, India's national stock exchange ..NIFTY to be precise) and i am using markov matrices to ...
0
votes
0answers
35 views

how to do MCMC simulation

I've been diving through the internet, trying to get the best possible understand on how to do this type of simulation, most of the information is theory or MCMC in simple english, but its been hard ...
0
votes
1answer
22 views

Why standard normal samples multiplied by sd are samples from a normal dist with that sd

This answer notes that if a programming language/libraries provide a procedure that returns random samples from a standard normal distribution, we can generate samples from another normal distribution ...
0
votes
1answer
24 views

Likelihood that a given outcome was generated by a Markov model

I am new to the concept of Markov Models and Markov Chain Monte Carlo simulations. I would like to take a piece of data and determine the likelihood that it was generated by a known MCMC ...
1
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0answers
16 views

How to choose a constant for reject sampling

When using a non-Markov Monte Carlo sampling method, for example acceptance-rejection sampling, we choose a density $\ h(x) $ and a known constant $\ c $, such that $\ ch(x) $ acts as a blanketing ...
1
vote
1answer
24 views

Difference between monte carlo based analysis and a hypothesis test

Is it reasonable to use Monte Carlo methods to resample a dataset of weekly rainfall amounts to statistically test for difference between two timeseries? That is, randomly pull ~30 paired observations ...
3
votes
1answer
25 views

Generating random samples from a marginal distribution

I have a joint distribution $p(a,b)$ that I obtained through numerical integration- that is, I don't have a formula for $p(a,b)$ but a bunch of samples drawn from this joint distribution. I would ...
6
votes
1answer
438 views

What is the difference between Metropolis Hastings, Gibbs, Importance, and Rejection sampling?

I have been trying to learn MCMC methods and have come across Metropolis Hastings, Gibbs, Importance, and Rejection sampling. While some of these differences are obvious, i.e., how Gibbs is a special ...
3
votes
1answer
22 views

Is there a technique where we keep the proposal in Adaptive Rejection Sampling?

As I understand, the proposal distribution, which I'll call $h(x)$, in adaptive rejection sampling is a linear piece-wise function which converges to the true distribution as the number of iterations ...
7
votes
2answers
134 views

How to generate samples uniformly at random from multiple discrete variables subject to constraints?

I would like to generate a Monte Carlo process to fill an urn with N balls of I colors, C[i]. Each color C[i] has a minimum and maximum number of balls which should be placed in the urn. For ...
6
votes
2answers
627 views

meaning of 'Monte Carlo' in this sentence

This is from a paper 'Algorithms for Inverse Reinforcement Learning' by Ng, Russell (2001) We assume that we have the ability to simulate trajectories in the MDP (from the initial state $s_0$) ...
5
votes
1answer
128 views

Issues with Importance sampling for flat prior

I am trying to draw Bayesian inference via importance sampling for a parameter $\xi$ attached with an (unbounded) flat prior. This seems problematic as this is clearly not a probability measure but ...
0
votes
0answers
15 views

Sensitivity Analysis -Variance Decomposition

Hi ı would like make Sensitivity analysis to my model.First of all I made uncertainty analysis with Monte Carlo method and now ı would like to find the most influential parameter in my model. My ...
0
votes
2answers
94 views

Calculating integral with antithetic variables

Use simulation with antithetic variables to find $$\int_{-\infty}^{\infty} \int_0^\infty\, \sin(x+y)e^{-x^2+4x-y}\,dx\,dy.$$ My question is, how struggle with the infinite limit? It is easy for me ...
1
vote
1answer
35 views

How to generate series of pseudorandom autocorrelated numbers

Say I am Ok with the numbers getting drawn from a standard normal distribution, but I also want the autocorrelation of the series at lag 1 to be a specific number. How can I generate such a series of ...
2
votes
1answer
36 views

Using MCMC to sample from a posterior, are our posterior beliefs on parameters independent?

I've been given a classification problem in which MCMC (slice-sampling) is used to sample from a hierarchical posterior. After getting $n$ samples, the Monte Carlo method can be used to give an ...
0
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0answers
15 views

Conditional expectation via monte carlo [duplicate]

Suppose we want to evaluate $E[\theta|X]$, $X=(X_1,...,X_n)$ iid such that $X_i|\theta \sim f(x|\theta)$, $\theta \sim f(\theta)$. Suppose that we don't know how to simulate from $f(\theta | x)$, but ...
1
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1answer
40 views

Determine the best sample size for minimum expected loss

Let $\theta \sim Gamma(1,2)$ and $X_1,...,X_n$ iid such that $X_i|\theta \sim Poisson(\theta)$. It is asked to determine the best sample size $n^*$ such that the posteriori risk $$L(\theta, d) = ...
1
vote
2answers
92 views

Algorithm for `rmarkovchain` in R

What method or algorithm is used in the function rmarkovchain from the R-package markovchain to generate samples and how does it work? Edit: I was interested in ...
0
votes
1answer
31 views

Sobol variance based decomposition

I have 6 input variables, each of which is normally distributed. Can I use Sobol variance-based sensitivity analysis? I have read some articles where they said that input variables must have uniform ...
1
vote
1answer
42 views

Combine multiple Monte-Carlo estimates

I use a Monte Carlo simulation (say 100.000 runs) to estimate parameter in R. I have memory problems and my first thought is to run multiples times my estimation program (say 500 times) . My ...
0
votes
0answers
22 views

Standard error of function of sample mean without using Delta Method

I am interested in estimating $SE(g(\bar{x}_m))$, where $\bar{x}_m$ is a Monte Carlo mean with M.C. sample size $m$, and $g(\cdot)$ is a nonlinear continuous function. $plim(\bar{x}_m)=\mu$ by law of ...
2
votes
0answers
27 views

Do Monte Carlo perturbations capture all the uncertainty in prediction?

I have a model $M$ that I use to predict a value $y = M(\vec x)$. I have known one-$\sigma$ error bars on each input $x_i \in \vec x$. I want to know the one-$\sigma$ error bar on my prediction $y$. ...
0
votes
0answers
29 views

What is Bayesian and Monte Carlo Simulation? [duplicate]

Can someone explain in plain language for a layperson what are Bayesian and Monte Carlo simulations and the relationship between the two? I thought Bayesian was the same as Monte Carlo Simulation...
2
votes
1answer
102 views

How to sample from a rectified Gaussian distribution?

Is the rectified Gaussian distribution in the following case the same as the truncated Gaussian distribution within the interval $[0,\infty)$? Here is the link to the paper. In this paper, ...
2
votes
1answer
50 views

Computation of average number of hops for an atom to reach a certain distance

I apologize if this has been asked before. I tried searching all previous posts to look at different forms of this problem (such as random 2d walks on a lattice, relation to Isling's work, etc.). I ...
2
votes
1answer
39 views

What should be the underlying distribution behind Monte Carlo simulation?

When we are trying to use Monte Carlo simulation to solve a problem that does not have analytical solution, how do we decide what should be the underlying distribution from which we draw these random ...
0
votes
1answer
72 views

use of monte carlo simulation within regression in R or any avaliable programme

I have a data set; sample size is 16, the number of independent variable is 18 and one dependent variable . there are correlations between independent variables. I want to conduct Monte Carlo ...
4
votes
1answer
406 views

Is this alternative method to Metropolis-Hastings salvageable? What is it called?

For my application, I need to calculate an integral over a specific distribution. This distribution is obtained by Bayesian inference - the density at $\Theta$ is proportional to $P(\Theta)f(\Theta)$, ...
0
votes
0answers
15 views

Monte Carlo VaR assuming logistic distribution

I have a Monte Carlo model which measures the Value at Risk (VaR) for given portfolio. I use the geometric brownian motion to model the prices. But let's say I assumed the returns of prices follow the ...
3
votes
2answers
83 views

Simulation for the random vector (X,Y) with density f(x,y,a)

I need to generate data from a random vector with joint density $(X,Y)$ with density: $$ \frac{x^\alpha(x+y+2)}{x+y+1}e^{-x-y}~~~~~,\alpha,x,y> 0 $$ Do you have any hints on how to start?
0
votes
0answers
28 views

Is a Monte Carlo approximation to a consistent estimate itself a consistent estimate?

Let $A(x)$ be a consistent estimate of some population quantity $A_0$, where $x$ is the data and there are $N$ observations. However, $A(x)$ is difficult to calculate directly, but can be ...
3
votes
1answer
51 views

Are the mean of samples taken from Metropolis-Hasting MCMC normally distributed?

I've come across the following theorem while studying MCMC. It seems to suggest that the sample mean taken from the MCMC – the posterior marginal expectation – should be normally distributed, using ...
0
votes
1answer
66 views

Does principal components analysis lose any information regarding the interdependence of the variables?

I have often heard that a copula describes in full the interdependence of a set a random variables. Lets say I want to generate a set of random variables that conform to an observed joint probability ...
0
votes
0answers
13 views

I-MR Charts when there is only two observation

I have a data having only two data points (specifically project A and project B). I need to draw an I-MR chart for quality control. Since I have only two points, my graph is not meaningful. How can I ...
1
vote
1answer
30 views

Combining prediction intervals

I'm using ML regressors (neural networks and random forests) to predict some numbers. I can put in my inputs and get out a value and its prediction interval. The inputs to my regressors are noisy, ...