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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Estimate at which point a linear model hits a certain value (including probabilities)
I have a simple 1D set of datapoints with a trend, I want to estimate at which point $X_t$ (i.e., at which point in the future) the model will hit a certain threshold $Y_t$:
I can fit a trendline to ...
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Monte Carlo Gradient Estimation in Auto-encoding Variational Bayes
I am currently reading paper Auto-encoding Variational Bayes and I am not being able to understand the highlighted part in the screenshot below:
I am not understanding why there is f(z) and what is ...
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Use of Monte Carlo Tree Search
I was talking with someone much more experienced in stats than I am and they suggested the use of Monte Carlo Tree Search for a problem I am facing.
Problem Statement: I am collecting jitter ...
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Figure of merit for multiple simulations of point patterns
I am having problems understanding how I can evaluate a set of (Monte Carlo) simulations based on randomly distributed points.
Assume you simulate a random point pattern in a square and you plot the ...
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Variance of samples drawn from different known distributions
I have a discrete random variable $Z$. Every possible outcome of $Z$ has a given probability $p(z)$ and a value given by some normal distribution with unknown mean, but known variance $z_i \sim \...
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Why greedy leads to best among all epsilon-soft Monte Carlo
In the RL book of Barto and Sutton, the authors give the definition of epsilon-soft and the pseudocode. I understand this step proves that we can keep improving a epsilon-soft policy. But I don't ...
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Monte Carlo integration for a random integrand (or a set of integrand)
I understand that the Monte Carlo integration works for a fixed function, say $$\int f(x)p(x)\mathrm{d}x\approx\frac{1}{N}\sum_{i=1}^Nf(X_i),$$ where $p$ is a probability density function on sample ...
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How to average several posteriors distributions from a Monte Carlo Simulation
Say you produce several posteriors distributions from different runs of the same model under different seeds. That is to say you have something like the following:
...
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Modeling voting paradoxes through Monte Carlo Simulations
Recently, I have been reading about voting paradoxes, more specifically the Condorcet paradox which states that there is no majority will. With other words, it may be difficult to aggregate individual ...
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Can Thomson sampling be used for better results in a 1 player-MCTS
I made a Monte Carlo tree search (MCTS) algorithm for the travelling salesman problem inspired by this paper which uses UCB1.
When I was digging to see where does the UCB1 formula comes from, I read ...
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Comparing two types of confidence intervals in R using Monte Carlo: trouble understanding what's going on
In a course I'm taking, my professor includes the following code in his slides.
I'm trying to understand what this code does, but perhaps more importantly I'm also trying to understand the ...
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best loss function to fit model if observations contain montecarlo noise?
I have observations on the sphere and I'm trying to fit spherical-harmonic coefficients to best approximate and interpolate the observations.
I'm using a solver library for non-linear least squares ...
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How does Particle Filters work?
I'm trying to figure out how particle filter works.
Assume that I have selected propability function called $a \sim Gauss(\mu, \sigma)$. We call it proposial (Gaussian) Distribution.
Then we have ...
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What are some methods to choose a $n$ for Quasi Monte Carlo Integrations?
When studying "simple" Monte Carlo integration methods, such as Hit or Miss, Crude , Importance Sampling, etc. A common problem for first time learners is to choose a number $n$ of points ...
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Sampling marginal distribution from joint density
Suppose we know that random vectors $x, y$ have joint density $p(x, y) \propto \exp(-U(x_1, \ldots, x_m, y_1, \ldots, y_n))$, and we want to draw a random sample from the marginal $p(x)$ (i.e. we want ...
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Compute a Monte Carlo estimate. Which of the variances (of $\hat{\theta}$ and $\hat{\theta}^{*}$) is smaller, and why?
Compute a Monte Carlo estimate $\hat{\theta}$ of $$ \theta = \int_{0}^{0.5} e^{-x} dx $$ by sampling from Uniform$(0, 0.5)$, and estimate the variance of $\hat{\theta}$. Find another Monte Carlo ...
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Simulating exponential Vasicek/Ornstein-Uhlenbeck
I am trying to simulate commodity prices using the exponential Vasicek/Ornstein-Uhlenbeck model from Schwartz 1997 p. 926 Equation (1). I am using the closed form solution from Vega 2018 p. 5 Equation ...
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Monte-Carlo Estimation of conditional expectation term
I want to ask if my approach to estimation of the following quantity is correct:
I have $n$ i.i.d. draws $\{(X_i,Z_i) \}_{i=1}^n$ and I want to estimate for a fixed $(i,j)$ pair the quantity:
$$
\...
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Using Monte Carlo standard error to determine the ideal number of trials?
I am doing a simulation study that involves estimating the parameter $\theta$ under a specific experimental design. $\theta$ is the parameter that take on the value 1 if algorithm A is better than ...
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Metropolis Hastings for BART: Calculation of Tree Prior and Transition Kernel
I am trying to understand the details of BART (Bayesian Additive Regression Trees). In particular, I would like to know how the Metropolis Hastings acceptance probability is calculated for BART. My ...
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Sampling from a distribution which results in heteroskedasticity
I have a pretty basic model whereby using a monte carlo approach I am seeking to recreate the monthly stock market returns over the last 30 years. Using the actual monthly returns of an index, I have ...
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Correlated asset paths to derive VaR
So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...
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Number of MC Simulations in Multivariate Model with Copulas
I am working on a project with the following characteristics:
4,000 hydroelectricity generation time series
800 futures time series (each corresponding to at least one hydroelectricity time series)
...
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Can I use regime-switching when bootstrapping relatively few data (<50 samples)?
Consider a large set of data, each with a series of 25-50 independent, non-negative observations and the following characteristics: a) most series have positive values only, b) some series have <10 ...
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What do the specific values of the Sobol' indices mean?
I understand that first order and total effect Sobol' indices demonstrate the relative importance of the input parameters on the output of a given model. My question is, do the specific values of each ...
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SVM and Monte Carlo simulation to compute misclassification error rate
I am trying to solve the following problem with R:
use simulation to evaluate (by Monte Carlo) the expected misclassification error rate given a particular generating model. Let $y_i$ be equally ...
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LRT and Manually Finding Significance when Wilks Theorem isn't Valid
Hello and thank you for taking the time.
I'm performing an LRT for a likelihood distribution which violates the regularity conditions for Wilks theorem and wald intervals. I'm running a monte-carlo ...
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What is the inference behind the momentum variable and the Kinetic energy for a weakly non-linear inverse problems in the HMC method?
We generate an auxiliary momentum variable in the HMC method to provide gradient for the propagation of trajectory (m, p) (model or position, momentum) in the phase space.
If we look into Newton's ...
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Local sensitivity analysis with exponential and uniform distributions as input
For my thesis I need to run a sensitivity analysis on the input factors for a supply chain model. I am supposed to change the mean and the standard deviation (sd) of all input factors respectively by ...
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Surprising nonlinear variance-based scale est (bias adj) for Laplace Distribution competes with MLE?
Background:
Using the quantile function (inverse cumulative distribution) for the Laplace distribution supplied with uniform random deviates (per the RAND() spreadsheet function), I examined an ...
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Impact of correlation bounds for Monte Carlo simulations
As the lognormal distribution imposes bounds of attainable correlations as discussed in Attainable correlations for lognormal random variables my question would be what happens if say we want to do a ...
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Calculate fifth and sixth polynomials for Headrick (2002) method for non-normal multivariate distribution
I am trying to perform a 3-variable correlated multivariate Monte Carlo simulation. As the asset class returns are non-normal, I found the following function ...
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Desirable properties of datasets and models for benchmarking Bayesian posterior inference algorithms
Are there canonical datasets for benchmarking the performance of posterior inference algorithms? For example in machine learning literature, the MNIST dataset (and others) is often used in ...
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Training for multiple epochs vs. Monte Carlo cross-validation to evaluate performance of non-linear algorithms (models)
I am using Shao's 1993 article "Linear Model Selection by Cross-Validation" as a starting point for the following cross-validation strategy for a machine learning algorithm:
NOTE: I am ...
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Compute which of a finite number of integrals is minimal (not interested in the actual value of the integral)
Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...
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Insufficient floating point precision for the correct computation of a density
I'm using the Metorpolis-Hastings algorithm in a setting where the acceptance function is essentially of the form $$\alpha(x,y)=1\wedge\frac{u(x,y)}{v(x,y)},$$ where $$u(x,y)=p+(1-p)\prod_{i=1}^mu_i(x,...
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Metropolis-Hastings with a "Dirac transition"
I'm running the Metropolis-Hastings algorithm on a product space $\tilde E:=I\times E'$, where $I$ is a finite nonempty set and $E'=\bigcup_{i\in I}E'_i$ with $E'_i:=[0,1)^{d_i}$ for $i\in I$.
Given ...
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How to tell if an estimator is good by reading results of Monte Carlo simulations?
I'm new to parameter estimation world and I'm studying a model with two parameters $\mu$ and $\sigma$:
$$
dX_t = \mu X_t dt + \sigma X_t dB_t^H
$$
where $B_t^H$ is a fractional Brownian motion (fBm) ...
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Proposal Function For Variables That Sum To 1 (Dirichlet Prior)
Recently I've been trying to use MCMC to infer a set of 50 random variables (species frequencies) that sum to 1 with the Metropolis-Hastings algorithm. However, the algorithm is not working well ...
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Sampling states of an "unnatural" Hamiltonian System
I would like to sample from a Gibbs distribution given by
$$f(p, q) = \frac{1}{\mathcal{Z}}e^{-H(p, q; \omega, J)}$$
where $H$ is the Hamiltonian on generalized coordinates $(p,q)\in \mathbb{R}^{2n}...
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Can we find an asymptotically consistent Metropolis-Hastings estimator based on this proposal scheme?
I'm running the Metropolis-Hastings algorithm for a target distribution $\hat\mu$ (see below for the formal setup including the definition of $\hat\mu$) on a product space $I\times E'$. I'm using the ...
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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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Why the Monte Carlo Control algorithm is written this way?
I am having trouble to understand this algorithm, since this is not how I would have written it.
To me, we should first start to fix a policy. Then, we evaluate the Q values associated with this ...
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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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Robust sum of non-independent random variables
What approach could be used to sum non-independent variables?
I have probability distributions of stock prices and want to calculate the probability distribution of the portfolio price (sum of some ...
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ABC SMC: How do weights scale proportionally with number of parameters
Having some problems with the ABC SMC algorithm. I'm trying to implement the methods taken from here: Simulation-based model selection for dynamical systems in systems and population biology
How do ...
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How to use Hamiltonian Monte Carlo when some parameters result in ill-defined likelihoods?
I want to use Hamiltonian Monte Carlo for an estimation problem where, for some parameters, the solution does not "make sense," so I cannot compute the log likelihood or its gradient. In addition, I ...
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Probability that output is result of the same process?
I've developed a simple Monte-Carlo simulation. Output of this simulation is a histogram. This histogram is possibly a log-normal distribution, but I don't want to assume that. But I do know that the ...
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Predictions after SMC
I have a statistical model given by
$$
y_t\sim p(y_t|x_t, \theta)\\
x_t\sim p(x_t|x_{t-1},\theta)\\
\theta\sim p(\theta)
$$
where $y$ is the only observed component. Using a sequential Monte Carlo ...
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