# Questions tagged [monte-carlo]

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

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### Expected value with dependent samples

It is well known that the expected value of a function can be approximated with i.i.d. samples: $$E_X[f(X)] = \frac{1}{n}\sum_{i=1}^n f(x_i),\quad x_i\sim_{i.i.d.} X$$ What methods exist to ...
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### Standard deviation in Direct-Sampling

In a computational physics course, I was asked to do direct-sampling for the numerical value of $\pi$ and then I estimated the standard deviation of $\pi$ , ...
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### How to generate data with given correlation, one distribution counting intergers, the other normal?

I would like to do a Monte Carlo simulation related to this post: How to predict the degree to which an extraneous variable will attenuate a correlation? I need to generate a dataset with a Pearson r ...
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### Confusion in Sampling using the IP algorithm (Bishop PRML)

I'm reading Bishop's PRML p. 537 and I don't understand one piece of the IP (data augmentation) algorithm. Namely, the part that says "we use the samples $\{\mathbf{Z}^{(l)}\}$ obtained in the I step ...
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### 2-dimensional inverse transform sampling [closed]

Let $M\subseteq\mathbb R^3$ be a disjoint union of orientable surfaces. In Monte Carlo ray tracing, we sample a surface point $p'\in M$ by drawing a random direction $\omega_{\rm i}\in S^2$ on the ...
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### Sampling from unknown probability distribution [duplicate]

I'm reading about Monte Carlo methods. Suppose that $X_1,...,X_n$ are i.i.d $p(x_i|\theta)$, where $\theta$ is an unknown parameter of interest. My textbook states: Suppose we could sample some number ...
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### Simulating Laplace's needle to estimate $\pi$

A recent question sought assistance with computer simulation of Buffon's needle problem in R, with the goal of obtaining a Monte Carlo estimate of $\pi$. This is ...
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### Validity of Monte-Carlo method to estimate a probability distribution which follows a power law

I am using a Monte-Carlo method to estimate a probability distribution function (pdf). Basically, I have several input parameters following known distributions, from which I can draw samples, that I ...
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### Stratified sampling to generate random numbers (eg. for Monte-Carlo applications)

I am using a Monte-Carlo method to compute a value of interest $y$ from some input parameters $x_{i}$, that I use to draw statistical sets from simple distribution laws. In my case, for a single Monte-...
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### Minimizer of $\int\mu({\rm d}x)\int\kappa(x,{\rm d}y)|g(x)-g(y)|^2$ for a jump kernel $\kappa$ of the Metropolis-Hastings algorithm

Let $\kappa$ be a sub-Markov kernel on a measurable space $(E,\mathcal E)$ and $\mu$ be a probability measure on $(E,\mathcal E)$ reversible with respect to $\kappa$. Assume $\kappa$ and $\mu$ admit a ...
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### What's the advantage of importance sampling? [closed]

Here is a nice example of importance sampling: ...
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### Monte Carlo Sampling is performing better then Bayesian Optimization

So I am testing the Bayesian optimization library for determining where to sample next by quering a test function such as 2d Rosenbrock to better reconstruct that function using Gaussian Process ...
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### assessing the stability of importance (sampling) weights

I have read that when importance weights are used, the stability (variability) of the weights should be assessed (Levine and Casella, 2001) -- however, I wonder how this might be accomplished. For ...
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### Monte Carlo simulation to generate random numbers from Pareto distribution

I am trying to generate random numbers using Monte Carlo simulation from Pareto distribution using R. But I am not able to find the codes for the same. It would be very helpful if someone could share ...
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### Can we use a Dirac kernel in the proposal of the Metropolis-Hastings algorithm?

I'm running the Metropolis-Hastings algorithm on a product space $(I\times E,2^I\otimes\mathcal E,\zeta\otimes\lambda)$, where $I$ is a finite nonempty set and $\zeta$ denotes the counting measure. ...
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### How to Simulate Pure-Jump Process associated Marked Point Process in R

Fix $M>0$ and let $(\tau_i,\zeta_i)_{i =1}^{\infty}$ be a marked point process associated to the poisson-random measure $\mu$ on $[0,1]\times [-M,M]^2\times [0,M]\times [-M,M]$ with uniform finite-...
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### Use both historical prices and fundamentals data for predicting portfolio profit?

Get highly accurate probability distribution for the future price of portfolio using all the data available. Each stock has two historical data - daily prices (scalar time series updated daily) and ...
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### Do I want pnorm or one sample t-test?

I have an algorithm that calculates a metric from my real data, I have a simulation that generates random data of the same type and dimensions, runs the algorithm, and produces the same metric once ...
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### Kalman Filter and Monte Carlo

What is the consequence on the uncertainty of our estimate when applying the standard kalman filter to nonlinear systems? If we are unaware of the functional form of these non linear systems how do ...
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### What is better in Monte Carlo integration: product of means or mean of products?

Let $X$ and $Y$ be two independent continuous random variables with pdfs $f_X$ and $f_Y$, respectively. Let $\varphi_1$ and $\varphi_2$ be two continuous functions from ${\mathbb R}$ to ${\mathbb R}$. ...
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### Propogating Standard Error through monte carlo? [closed]

I have a model that has a bunch of parameters that I experimentally determined the mean and accompanying standard error/STD. Because the model has some tricky terms in it I'd like to propagate the ...