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

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48 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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14 views

Monte Carlo study - Simple linear regression [on hold]

I generated Monte Carlo code for 100 random observations Y according to this model: Y = β0 + β1x + e, set.seed(123) n=100 b0=2 b1=1 eps=rnorm(100) for(i in 1:n){ x=seq(0,10, length.out = 100) ...
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21 views

Monte Carlo, Simple linear regression [on hold]

A Monte Carlo study of simple linear regression. Let $\beta_0 = 2, \beta_1 = 1$ and assume the following relation holds true $$Y = \beta_0 + \beta_1 x + \varepsilon$$ where $x \in [0,1,\ldots, 10]$, ...
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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
12 views

Beta-PERT with Confidence Interval? [on hold]

I'm working on something that does Monte Carlo simulation using PERT input values, and I put a Beta distribution on the values to model the random spread. However, the function I am using, which is ...
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22 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
14 views

Is there a good text book on serial tempering?

I've read that serial tempering is an approach for "MCMC sampling from a sum of parametrized distributions". I've only found two papers (Marinari and Parisi and Geyer and Thompson) introducing this ...
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0answers
23 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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38 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
36 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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17 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 ...
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1answer
35 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
20 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
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1answer
77 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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0answers
22 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
18 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
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1answer
48 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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0answers
38 views

Probability for a “sure lose” blind Nil hand in the game of Spades?

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
55 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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0answers
22 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,...
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1answer
32 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
98 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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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
19 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
27 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
16 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
38 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
36 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
94 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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0answers
15 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 ...
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1answer
23 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
28 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
71 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
45 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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0answers
15 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 ...
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2answers
144 views

Random process not so random after all (deterministic)

I would like to show (demonstrate by simulation) a random process that turns out after $i$ interactions to be deterministic, i.e. ends at predefined value (roughly) known at time $t=1$. Conditions ...
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0answers
17 views

Dau Genton test - in grandmothers terms

tl;dr Can you explain the Dau Genton test in terms a median grandmother could understand? Background: So I am looking for an "in.chull" for multivariate, concave hull, and I was going through "...
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1answer
24 views

Comparison of results of monte carlo simulations

I am doing monte carlo simulations. In the first run the experiment is repeated e.g. 10000 times. The result looks like x+/-y, y is the relative error. Next, I change a part of the experiment and run ...
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42 views

Unbiased sampling and subsampling

Suppose I have a distribution $\mathbb{F}$ with mean $M$. Also, assume we have a set of i.i.d samples of size $n$ denoted by $X=\{x_1, x_2,..., x_n\}$ from $\mathbb{F}$. As a result, all $x_1, ..., ...
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0answers
30 views

Covariance in an error propagation leading to negative variance

Does it ever make sense that, propagating the error from a set of data, the covariance term being very negative makes the variance go to a negative value (which by itself makes no sense)? Context: ...
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1answer
103 views

Why does Off-Policy Monte Carlo Control only learn from the “Tails of Episodes”?

I was reading through section 5.7 of the second edition of Sutton and Barto's "Reinforcement Learning: An Introduction" when I came across this passage: where the "method" that the author is ...
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1answer
22 views

Monte Carlo simulation--Have I applied Bernoulli distribution properly? [closed]

I am trying to run a monte carlo simulation and I just wanted to make sure that I set it up properly. A salesman visits 100 different homes. Someone answers the door 80% of the time. Of that ...
4
votes
1answer
174 views

Why temporal difference (TD) method has lower variance than Monte Carlo method?

This question might be a little trivial. However, I had a hard time understanding it or finding some formal proof for it. In many papers, it is being said that for estimating the value function, one ...
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0answers
21 views

Predict Next Purchase Order Value

Let's say we have historical customers order by value and we're trying to predict the next order value for each one of them. We don't have the date/time of past orders, and we don't care about the ...
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0answers
45 views

“Change of measure” for fast simulation?

$\newcommand{\var}{\operatorname{var}}$The problem is here: Consider a nonnegative random variable X whose PDF is close to being exponential, of the form $$f_X(x) = g(x)e^{−x},$$ where $g(x)$ is a ...
2
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1answer
86 views

Estimate correlation between data and data-fit model for variance reduction in Monte Carlo estimate

Say that I want to estimate the mean of a function $f$, $\mathbb{E}[f(X)]$, given some input distribution $x\sim P(x)$. I don't know anython about the form of $f$ except that it is smooth and ...
4
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1answer
169 views

How to choose best proposal distribution for importance sampling

From Robert & Casella p95, we know that the choice of proposal distribution $g(x)$ with minimal variance is the $g$ proportional to $|h(x)|f(x)$. If we restrict our proposal distribution to cetain ...
4
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1answer
38 views

Show, Attend, Tell why monte carlo sampling?

In the paper Show, Attend and Tell the authors derive the Loss function as $$ L_s = \sum_s p(s|a)\log(p(y|s,a)) , $$ with $s_i \sim \text{Multinoulli}(\alpha_i)$ so that $p(s_i|a) = \alpha_i$, which ...
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0answers
29 views

type I error using montecarlo method in R

i want to calculate the type i error rate and power for the correlation test for bivariate normal data using Monte Carlo simulation. But i am getting unexpected values for the type I error and for ...
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2answers
35 views

Efficiently drawing from non-parametrically estimated distribution

Suppose I can estimate a distribution $G(x)$ as $$ \hat G(x) = f(x, X)$$ where $X$ are my data points and $f$ is a known, but computationally heavy function. Eventually I'm interested in $$ h(\...