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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In sports modelling, are hot simulations better or cold simulations?

I'm thinking here largely of the context in which someone has an Elo rating model for a particular sport. To calculate things such as how often the team makes the Finals series, or wins the ...
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Expected value of softmax transformation of Gaussian random vector

Let $\mathbf w_1,\mathbf w_2,\ldots,\mathbf w_n \in \mathbb R^p$ and $\mathbf v \in \mathbb R^n$ be fixed vectors, and $\mathbf x \sim \mathcal N_p(\boldsymbol{\mu}, \mathbf{\Sigma})$ be an $p$-...
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Population Monte Carlo Algorithm

I am trying to wrap my head around the Population Monte Carlo Algorithm. I want to implement it for a mixture model, but I am uncertain on how to proceed. I am mostly looking for references or ...
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Keeping track of the variance of a Metropolis-Hastings estimator

Let $(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces, $p,q$ be probability densities on $(E,\mathcal E,\lambda)$, and $\varphi:E'\to E$ be bijective and $(\mathcal E',\...
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Expected value of a "logistic uniform" multivariate

Let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb R^d$ and $b_1,\ldots,b_n \in \mathbb R$ be fixed. For $\mathbf{x} \sim \mathcal U([0,1]^d)$ and $j \in \{1,\ldots,n\}$, consider the real variable ...
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Endogeneity in spatially lagged regression model

The standard convention in Spatial Statistic is that the spatial lag term in a regression model will be biased due to simultaneity. Looking at the following model, it would be difficult to argue with ...
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Entropy of a mixture of Gaussians

I need to estimate as fast and accurately as possible the differential entropy of a mixture of $K$ multivariate Gaussians: $$ \mathcal{H}[q] = -\sum_{k=1}^K w_k \int q_k(\textbf{x}) \log \left[\sum_{...
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Looking for a recursive formula for asymptotic variance of importance sampling estimator (self-normalized)

Looking for a recursive formula to approximate variance of importance sampling estimator $Var_q\big[\delta_{IS}\big]\approx\sum_{i=1}^n\tilde w(X_i)^2\big[h(X_i)-\delta_{IS}\big]^2$. This is an ...
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Sampling from mixture of *unnormalized* densities

Suppose I have $n$ unnormalized densities $g_1(\textbf{x}), \ldots, g_n (\textbf{x})$, for $\textbf{x} \in \mathbb{R}^d$, and $n \gg 1$, which largely overlap but in a nontrivial way. I need to sample ...
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Computational entropy and Monte Carlo simulation

Is there a point at which the statistical properties of the random number generator will start to influence the results of Monte Carlo simulation? I have a scenario where I need to calculate the ...
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269 views

MCMC for Maximum Entropy?

Is there a way to sample from a discrete probability distribution, whose distribution itself is the solution to a Maximum Entropy problem with known linear constraints, without needing to solve for ...
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Estimating parameters using Kullback-Leibler or Kolmogorov-Smirnoff via Nelder-Mead

I want to find the parameters of a model which specifies a set of classification probabilities, for say M classes. (I'll use the parameters in another model later.) Given a set of parameters $\theta$,...
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Deriving priors for MCMC implementation

I have been working on an assignment lately wherein the object is to implement an MCMC approach to simulate from a generated posterior distribution. The posterior distribution is generated from a ...
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Rao-Blackwellization in Black Box VI

In the paper, "Black Box Variational Inference," by Ranganath et al. (2013), the authors derive a Rao-Blackwellized estimator of the gradient of the evidence lower bound with respect to a ...
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Why is my quasibinomial GLM estimator biased - Monte Carlo simulation

I'm playing with some Monte Carlo simulations to get an idea of the properties of some linear and non-linear models. The linear OLS model in my case is specified as: $Y_t = \beta_0 + \beta_1x+ \...
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Any known approximations of summing quantiles from joint (bernoulli / lognormal) distributions

This is my first post to this site! For an insurance-like scenario, I have several independent risks which I want to sum together and find a 95% percentile. Currently I do this by Monte Carlo but I ...
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Detailed Balance for Hamiltonian Monte Carlo

I am trying to understand the detailed balance proof present in this paper: https://arxiv.org/abs/hep-lat/9208011v2 (page 4). My question: Why do we consider the volume of a neighborhood of points ...
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What is the best way to report the results and uncertainty from a Monte Carlo simulation?

I am fitting data to a model that has ~30 input parameters, each with their own uncertainty levels, and which can interact with each other in the model. I therefore decided the best way to fit the ...
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  • 141
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Modeling a Correlated Bivariate Beta Distributions in PyMC3

My goal is to perform a bayesian A/B test of probabilities of success in two groups considering a hypothesis about non-zero covariance between those probabilities. Bivariate beta distribution I am ...
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4 votes
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Convergence Proof of First Visit Monte Carlo Control

I am currently trying to find a formal proof of convergence for the Monte Carlo Reinforcement Learning Methods described in Sutton,Barto's Book "Reinforcement Learning - An Introduction" , Section 5. ...
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4 votes
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368 views

Convergence of gradient descent Monte Carlo Control with function approximation

Can anyone point me in the direction of a formal proof of convergence for a (on/off policy) Monte Carlo control algorithm with (non-)linear function approximation? In http://incompleteideas.net/book/...
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Sampling from distribution defined variationally

Suppose I define a probability distribution $\mu\in\mathrm{Prob}(\Sigma)$ over some compact $\Sigma\subseteq\mathbb{R}^n$ using a variational problem: $$ \mu:=\arg\min_{\mu\in\mathrm{Prob}(\Sigma)} F[\...
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How to Validate a Monte Carlo Simulation

I have historical data of a production process, and I've being asked to build a simulation model to predict its performance in the future. Using the historical data, I've being able to obtain the ...
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4 votes
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Uniform convergence of Monte Carlo approximation

Usually Monte Carlo method is used to compute integration. For example, let $g(x,\theta)$ be a continuous function about $x$ and $\theta$, $f(x \mid \theta)$ is a continuous pdf with parameter $\theta$...
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How to show that the variance of Sequential Importance Sampling estimates increase with the dimension?

I am trying to understand the Particle Filter and the motivation to use it over the regular Sequential Importance Sampling. As far as I understand until now: 1- We try to estimate the expectation of ...
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4 votes
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157 views

How to estimate the given function using Rao Blackwellization approach?

I have a function $X$ which is lognormal (0,1) and then another function $\log Y = 4 + 2 \log(X) + \epsilon$ where $\epsilon \sim \mathcal N(0,1)$ I want to estimate $E(Y|X)$ as a Rao Blackwellized ...
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4 votes
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323 views

Gibbs sampling from full conditionals

I have the following joint density: $p(x_1,x_2,y_1,y_2) \propto \exp\left(−\left(x_1^2+x_2^2+c_1(y_2-y_1)^2+c_2(y_2-y_1)^4\right)\right)$ Can I use Gibbs sampling to sample from that? How can I get ...
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4 votes
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72 views

Random walk with restricted graph knowledge

I have a very large graph and a function of its vertices, and want to estimate mean value of this function. It's not possible to sample vertices uniformly in this problem, so a reasonable choice for ...
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4 votes
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111 views

Averaged estimators in stochastic versions of EM

Recently I've been working EM algorithms for MAP estimation in a problem where the expectation is intractable, but the maximization is easy. Further, draws from the distribution in the E-step are ...
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How to calculate the uncertainty of a parameter when using a cost function other than the least squares?

Context: Given some measurements $y_i$ associated with the independent variable $x$ and uncertainties $\sigma_i$ and, on the other hand, a model $f(x;\theta)$ where $\theta$ is a free parameter (which ...
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A generalized randomized mean estimate based on the Chebyshev inequality

Let $X$ be a random variable that does not take on a single value with probability 1. Let “black-box" sample access to the random variable $X$ be given. Let $M_k$ be an upper bound on the $k$th ...
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Computationally + Statistically Efficient Unbiased Estimation of Chebyshev Polynomials of Expectations

Let $T_n$ denote the $n^\text{th}$ Chebyshev polynomial, defined by the recursion \begin{align} T_0(x) &= 1,\\ T_1(x) &= x,\\ T_n(x) &= 2x \cdot T_{n-1} (x) - T_{n-2} (x). \end{align} Now, ...
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3 votes
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Deriving the bias of the self-normalizing importance sampling estimator

Suppose we are interested in the expectation of a test function $f(X)$ with respect to target distribution $\pi(X) \propto \gamma(X)$ using importance sampling with proposal distribution $q(X)$ with $...
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How are the variance of an estimate to $\int_Bf\:{\rm }d\mu$ and the deviation of $f$ from the mean $\frac1{\mu(B)}\int_Bf\:{\rm d}\mu$ related?

Let $(E,\mathcal E,\mu)$ be a probability space, $(X_n)_{n\in\mathbb N_0}$ be an $(E,\mathcal E)$-valued ergodic time-homogeneous Markov chain with stationary distribution $\mu$ and $$A_nf:=\frac1n\...
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  • 539
3 votes
1 answer
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Erroneous expression for Metropolis-Hastings acceptance ratio in a paper

Let $(E,\mathcal E)$ be a measure space; $\rho:E\to[0,\infty)$ be $\mathcal E$-measurable, $p:E^2\to[0,\infty)$ be $\mathcal E^{\otimes2}$-measurable, $$r(x,y):=\left.\begin{cases}\displaystyle\frac{\...
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  • 539
3 votes
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Probability that one Gaussian RV exceeds all others

Imagine we have $k$ Gaussian RVs $$ X_i \sim N(\mu_i, \sigma_i^2) \text{ for } i=1, \ldots, k $$ and we sample from each of them independently to produce a vector, $\vec{x} = (x_1, \ldots, x_k)$. For ...
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Why do we use the log-derivative trick before Monte Carlo?

I still don't understand how we can approximate the gradient of an expected value... Indeed it's impossible to sample points and then to average the gradients of them as we have only samples... (How ...
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  • 357
3 votes
1 answer
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Numerical examples proving and disproving the optimal scaling heuristic by Roberts et al

Let $d\in\mathbb N$ with $d>1$ $\ell>0$ $\sigma_d^2:=\frac{\ell^2}{d-1}$ $f\in C^2(\mathbb R)$ be positive with $$\int f(x)\:{\rm d}x=1$$ and $g:=\ln f$ $Q_d$ be a Markov kernel on $(\mathbb R^...
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  • 539
3 votes
0 answers
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Visualization of unbiasedness of high dimensional paramter estimates

Assume a statistical model $f_{\theta}(X)$ that allows to estimate a parameter vector $\hat{\theta}\in \mathbb{R}^p$ from data $X$ and assume that $p$ is high dimensional (you may assume something ...
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3 votes
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References for Texas Holdem

Does anybody know any good papers or software that use Monte Carlo techniques to estimate the probability of certain hands or winning/losing a hand in Texas Holdem? Ideally I'd like to have some ...
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3 votes
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Hamiltonian Monte-Carlo in Reinforcement Learning

I was wondering if someone can point me out a reinforcement learning algorithm/paper (continuous variable), in which the Hamiltonian Monte-Carlo has been directly used. Thanks!
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1 answer
194 views

Gibbs Sampling vs. Using Raw Probability in Contrastive Divergence

In Hinton's Practical Guide to Training Restricted Boltzmann Machines, Section 3, he discusses different situations in which one should take a sample from the Gibbs sampling process, and other ...
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Estimating standard error of Monte Carlo integration, non-MCMC version

Let us suppose that we're to evaluate the expectation of a random variable $h$ with respect to some distribution $\pi$, $\text{E}_{\pi}[h]$. The standard Monte Carlo estimate, using a sample of $X_1, ...
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3 votes
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163 views

Monte carlo method and Convergence in Distribution

Monte Carlo method, from what I could gather, allows one to obtain observations/draws from a possibly unknown statistical distribution. Let's say $T(X)\sim F$, where $T$ is a statistic, $X$ is a ...
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3 votes
0 answers
548 views

Importance sampling: what is this bias?

I am experiencing what seems to be a bias in importance sampling, which, given that it's an unbiased procedure, should not be there. Consider linear regression $$ y = X\beta+\epsilon $$ where there ...
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3 votes
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154 views

Error bars for sum of Monte Carlo integrations

I want to calculate a quantity $I$ which is the sum of integrals $I_k$, $$ I = \sum_k I_k\;,\quad I_k = \int_{\Omega} f_k(x)\,\textrm d x\;, $$ Every integral $I_k$ is evaluated independently by a ...
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  • 251
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sampling from a posterior derived from hierarchical Bayesian using HMC

I have a complex pdf based on hierarchical Bayesian formalism where x depends on the priors w'and w'', and I consider hyper-prior for the latter's as w=Php(zeta,beta) where Php stands for the hyper ...
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  • 55
3 votes
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Confusion on three “types” of Markov Chain Monte Carlo for the same inference

This is a long question but I would be very grateful if someone can help or provide some reference! And I believe this is a common confusion among beginners of MCMC. Background Given a directed graph $...
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  • 1,613
3 votes
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924 views

Monte Carlo simulation for fitting distributions (Weibull and log-normal)

I am working on a computer project which needs statistical analysis and I am not much of a statistic person. I have a Bluetooth (BT) detector device which detects passing Bluetooth devices (i.e one ...
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3 votes
0 answers
108 views

Inconsistent results with Monte Carlo solutions to similar problems in probability

I am presently going through the book Fifty Challenging Problems in Probability with Solutions and implementing Monte Carlo solutions to most of the problems in R to get familiar with the language, ...
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