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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9 views
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^...
1
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0answers
16 views
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 $\ln f$ ...
1
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2answers
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)= ...
2
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2answers
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 ...
1
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0answers
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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0answers
12 views
Non-deterministic MCMC model with updates
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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 ...
1
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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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0answers
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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0answers
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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0answers
88 views
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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0answers
19 views
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,\...
0
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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-...
4
votes
1answer
46 views
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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0answers
12 views
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 ...
1
vote
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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0answers
9 views
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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0answers
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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0answers
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 ...
1
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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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0answers
18 views
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 ...
4
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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 ...
0
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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 ...
2
votes
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 ...
3
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0answers
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 ...
0
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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 ...
2
votes
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/...
3
votes
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 ...
2
votes
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 ...
1
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0answers
24 views
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,...
1
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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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0answers
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), ...
0
votes
0answers
8 views
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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0answers
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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0answers
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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0answers
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 ...
2
votes
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)})...
1
vote
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 ...
4
votes
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 ...
1
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0answers
17 views
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 ...
0
votes
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 ...
1
vote
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, ...
4
votes
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+...
2
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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) \\
& ...
3
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0answers
16 views
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 ...
