Using (pseudo-)random numbers and the Law of Large Numbers to simulate the random behavior of a real system.
1
vote
1answer
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 ...
0
votes
0answers
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)
...
0
votes
0answers
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 $x \in [0,1,\ldots, 10]$, ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
1
vote
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=...
0
votes
0answers
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 ...
1
vote
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}...
0
votes
0answers
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 ...
4
votes
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 ...
0
votes
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
votes
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 ...
3
votes
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 ...
0
votes
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
votes
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/...
2
votes
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 ...
2
votes
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 ...
1
vote
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,...
1
vote
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 ...
0
votes
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), ...
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 ...
0
votes
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}\...
1
vote
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_{...
0
votes
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 ...
2
votes
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)})...
1
vote
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 ...
3
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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, ...
4
votes
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+...
2
votes
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) \\
& ...
3
votes
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 ...
4
votes
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 ...
2
votes
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 "...
0
votes
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 ...
0
votes
0answers
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, ..., ...
1
vote
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:
...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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
votes
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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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(\...
