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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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Optimal scaling of the Random Walk Metroplis-Hastings algorithm and the speed measure of the limiting diffusion

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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Assumptions on the target density in the RWM optimal scaling paper by Roberts, Gelman and Gilks

In the famous paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, at the bottom of page 116, the supremum of the third derivative of $\ln f$ ...
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41 views

Variance of Rao Blackwellization for Monte Carlo Estimate of Expectation

from https://arxiv.org/pdf/1401.0118.pdf If we have a function $J(X,Y)$ of two random variables $X$ and $Y$ and we want to compute the expectation $\mathbb E_{p(X,Y)}[J(X,Y)]$. We define $\hat J(X)= ...
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72 views

Bayesian approach to report simulation studies?

I am running a simulation study where we want to estimate a proportion $p$. We are reporting the coverage of credible intervals with a uniform prior, and we are doing $500$ Monte Carlo simulations. We ...
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30 views

Monte Carlo testing: number of required permutations

I want to perform a statistical hypothesis test, however I don't know the exact distribution of my test statistic under $H_0$. Therefore, I need to calculate a Monte Carlo estimate $\hat{p}$ of the ...
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12 views

Non-deterministic MCMC model with updates

0 down vote favorite I have historical data for three variables : Y, X1, X2, for example 1000 points. The distribution of future values of Y depends from X1 and X2 and can't be expressed in ...
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21 views

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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1answer
32 views

Sample from aggregate portfolio distribution versus individual asset distributions

Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=\begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} $, expected returns $R=\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix}$, ...
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1answer
45 views

Monte Carlo $\epsilon$ - greedy policy is better than $\epsilon$- soft policy

In the RL book of Barto and Sutton, the authors have proved that any $\epsilon$-greedy policy with respect to $q_{\pi}$ is an improvement over any $\epsilon$-soft policy $\pi$ is assured by the policy ...
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30 views

How to solve systems of linear equations with random variables? How to identify model parameters?

I want to learn know how to solve systems of linear equations with randomness. Example of a deterministic version of the sort of problem I want to solve: ...
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11 views

ensemble of an ensemble in Scikit Learn

I am trying to get my head around ensemble learning and need some advice. Basically, my database contains a deterministic target variable and the feature variables are all stored as probability ...
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24 views

Generate pseudodata by Monte Carlo

What does it mean for "Generate pseudodata by Monte Carlo"? For the text I was reading it says suppose that the real data $Z_n \sim N(\mu(\theta), \Sigma (\theta))$, then the pseudodata are generated ...
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Approximating the first moment of $h(x)$ where $x$ ~${\rm log\,normal}(\mu, \sigma)$

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$)? So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
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Bound for the bias of ergodic averages for non-stationary Markov chains

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\mathcal E)$ be a measurable space $X$ be a $(E,\...
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1answer
23 views

Why is the probability of one variable bigger than another differ for my Monte-Carlo simulations and theory?

Suppose: $A \sim \mathcal{N}(0,\,1)$ $B \sim \mathcal{N}(1,\,1)$ In theory: $P(A>B) = \Phi\left(\dfrac{\mu_A-\mu_B}{\sqrt{\sigma_A^2+\sigma_B^2}}\right) = \Phi(-1)\approx 0.1586$ In my Monte-...
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How are deltas chosen for the proposal distribution in multivariate metropolis hastings sampling?

Say I want to use Metropolis Hastings algorithm to get posterior draws of multivariate parameters. In the one variable case, you could manipulate delta until you found something that worked (gave 40% ...
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1answer
16 views

Sample whole number from distribution with average less than 1

I am trying to create some simulated data, where for each participant, the average number of events that occur in a given week can be any positive number, but usually in the range of 0 to 5. Trouble ...
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Bootstrapping in sampling distribution [duplicate]

What is the fundamental importance of bootstrapping? To generate a sampling distribution, what are the purposes of taking just one sample and resampling from it multiple times; as opposed to taking ...
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2answers
86 views

What test to use to find the probability of highest value?

If I have a vector of around 40 values each with a normally distributed error, is there an easy way to figure out the probability of each element being the element with maximal true value? For context,...
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1answer
89 views

Monte Carlo maximum likelihood vs Bayesian inference

I recently heard about MCMLE (Monte Carlo maximum likelihood estimation) for finding $$ \hat\theta = \underset{\theta}{\text{argmax}} \frac{\exp\left(\theta^TT(y)\right)}{c(\theta)} $$ when the ...
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Determining the Relationship Between Monte Carlo Breaks and Model Volatility

I'm looking for a statistical test to understand the relationship (if any) between the model volatilities of a stochastic process, and the occurrence of a'break', defined as an instance when an ...
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24 views

How to do Monte Carlo with conditional input values?

Please apologise my likely ignorance of the correct terminology and notation. Any edits and suggestions to improve the question are very much appreciated. I want to perform a Monte Carlo simulation ...
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28 views

Using Monte Carlo approximation for posterior expectation

We have two groups of people, members of each group have a chance of getting a disease $\theta_i$. The groups have $n_i$ members and $Y_i$ diseased members. Group 1: $n_1=100, Y_1=21$, Group 2: $n_2=...
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44 views

Interpreting a Monte Carlo Simulation’s Results

I generally run 1000 iterations whenever I simulate things like stock prices. When interpreting the results, how do I find which simulation is the median, or highest probability of occurring in ...
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1answer
49 views

How to lower the standard deviation in a Monte Carlo Simulation [closed]

I am trying to simulate a stock's price with a Monte Carlo simulation. I am using this formula in excel: $S_{t+1}=S_t\cdot exp(d\Delta{t}+s\varepsilon \sqrt{\Delta{t}})$, where $d=\bar{x}-\frac{s^2}{2}...
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Construct a new confidence interval (CI) from a published asymmetric CI which was determined using Monte Carlo simulation

Background: NOAA publishes precipitation frequency estimates for the U.S. where the authors use a Monte Carlo simulation technique to generate 1,000 synthetic data sets to estimate a 90% confidence ...
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1answer
36 views

Find joint maximum of a sampled density

I used a sampling method to fit a model with three parameters to data, by supplying the likelihood function and priors. (I'm using JAGS but I think this applies to any method). I obtain triplets of ...
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1answer
27 views

Monte Carlo Metropolis: Standard Error and Acceptance

In a time series data generated by Monte Carlo Metropolis algorithm, when is the standard error (correlation between two data points is assumed to be negligible) is higher - when the change in the ...
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1answer
121 views

Proof that an epsilon greedy policy w.r.t. $q$ values is better than the original policy $\pi$?

I was trying to understand the proof why policy improvement theorem can be applied on epsilon-greedy policy. The proof starts with the mathematical definition - I am confused on the very first line ...
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23 views

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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1answer
24 views

Monte Carlo Sample input into OLS Regression

I am trying to find the best way to conduct a risk-based regression study. I have distribution data for both X and Y and used a monte-carlo sampling of the distributions to generate a data set. I then ...
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1answer
49 views

What's done with the expectations in this proof?

This is a proof of the per-decision importance sampling (theorem 1) from the appendix of: https://www.google.co.uk/url?sa=t&source=web&rct=j&url=http://scholarworks.umass.edu/cgi/...
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1answer
66 views

Spades - Probability for a “sure lose” blind Nil hand?

Spades is a trick-taking card game. The object is to take at least the number of tricks (also known as "books") that were bid before play of the hand began. Spades is a descendant of the Whist family ...
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1answer
76 views

What is intuition behind high variance of Monte Carlo method? [closed]

I'm studying Reinforcement learning from lectures of David silver, where he says that Monte Carlo method is not biased and has very high variance. But I don't understand in which sense the bias and ...
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Hamiltonian MC on Riemannian Manifolds Explanation

I'm new to manifolds; so looking for a more detailed explanation of Hamiltonian Monte Carlo on Manifolds I was reading the paper "Riemann Manifold Langevin and Hamiltonian Monte Carlo" In the paper,...
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1answer
38 views

Can I use Hamiltionian Monte Carlo when my likelihood is not a direct function of my parameters?

By "not a direct function of my parameters" I mean the following. I have some observed K-dimensional data and a model that can generate synthetic data based on 6 free parameters. I use this model to ...
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150 views

How to simulate data for generalized estimating equations (GEE) with logistic link function?

I am working with a pre/post test structure, measuring dichotomous outcomes. I am using GEE to estimate the coefficient for time (also a binary variable, 0 representing pre and 1 representing post), ...
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How to generate a point set with predefined discrepancy?

I just learned the low discrepancy sequence and find it will be helpful in my research. However, my goal is not to get a sequence with a low discrepancy, but generate different 2D point sets with ...
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45 views

Autocorrelation, Autocovariance and Large lag Standard Error

I have time series generated data from Monte Carl-Metropolis Simulation. I have estimated correlation coefficients using: $r_k = \frac{c_k}{c_0}$ where $c_0$ is the varaiance and $c_k = \frac{1}{N}\...
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41 views

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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21 views

Standard Error in Auto correlated Data

I have time-series data generated via Metropolis algorithm - Monte Carlo simulations. Since these data must have some correlation between them, the formula of the standard error for IIDs variable must ...
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2answers
41 views

A doubt on the formula for updating the weights in Sequential Importance Sampling in a State-Space model

Let $x_{0:t}^{(i)}$ be the states from time $0$ to $t$ from sample $i$. Similarly for the observations $y_{1:t}$. The normalized weights are updated according to Where does the term $p(y_t|x_t^{(i)})...
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1answer
43 views

MCMC samples for constructing a histogram

I am interested in generating samples from a density $\pi(\theta)$ to construct a histogram for $\pi(\theta)$ and to use these samples to generate samples of $f(\theta)$ for some function $f$. I may ...
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1answer
120 views

In exactly what sense do MCMC draws approximate the target?

Background We want to sample from some intractable density $\pi(\theta)$. Using an MCMC algorithm, we generate a sample of draws $\{\theta_i\}_{i=1}^N$ from a Markov chain that has $\pi(\theta)$ as ...
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Choosing a custom proposal distribution in Metropolis-Hastings Monte Carlo

I have many states and have calculated a good custom proposal distribution for my Monte Carlo simulation. The system reaches a good solution faster than if it were to just use a randomly selected ...
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1answer
30 views

Monte Carlo / bootstrapping to generate a Kaplan Meier curve

I have 4 survival datasets from 4 different trials examining 2 different drug classes independently. I would like to model the likely survival curve resulting from a pooled selection of either drug ...
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1answer
29 views

How can the support of proposal distribution impact convergence of RH-MH algorithm?

In the book Introducing Monte Carlo Methods by Casella and Robert, there's a sentence with which I'm having some trouble to understand. «If the domain explored in $q$ [proposal] is too small, ...
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3answers
72 views

Conditional expectation on an estimator for defensive sampling

In Introducing Monte Carlo Methods, by Robert and Casella, we have How do we derive the second equality? Shouldn't it be $$E\left[\frac{f(X_i)}{g_{Y_i}(X_i)}|X_i\right]=\frac{f(X_i)}{g_1(X_i)}\rho+...
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1answer
48 views

SIR explanation in Robert and Casella Intro to Monte Carlo Methods - How to do this derivation?

Why is it an exact simulation from $f$, and not only an approximation? I get $\begin{split} P(X^*\in A) & = \sum_i^n P(X^*\in A , X^* = X_i)=\sum_i^n P(X^*\in A | X^* = X_i)P(X^* = X_i) \\ & ...
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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 ...