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

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proof of Markov chain Monte Carlo

This is the first step of proof of MCMC in my notes I have a question, how come $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$? Is it true for any markov chains which are ergodic and aperiodic? The ...
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21 views

confused about the proof of Markov chain Monte Carlo

This is the proof from notes I'm confused about the $\pi(x_p|x)$ and $\pi(x|x_p)$ Let's say $X\sim $Bin$(10,0.3)$, so $\pi(x)=\binom{10}{x}0.3^x0.7^{(10-x)}$, so what does $\pi(x_p|x)$ or $\pi(x|...
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16 views

Using Monte Carlo simulations with subsequent element removal

I'm attempting to build an evaluation set for a logistic regression classifier and I've run into a statistical problem. The study involves a very large population (G) that has two properties of ...
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14 views

Sampling from Hierarchical Distribution

Let's say I know $$ X \mid \mu, \sigma \sim \mathcal{N}(\mu, \sigma^2), \qquad \mu \sim Unif(a, b), \quad \sigma \sim Unif(c, d), $$ and I want to sample from $X \mid \mu, \sigma$. Is there a go-to ...
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Can Machine Learning or Deep Learning algorithms be utilised to “improve” the sampling process of a MCMC technique?

Based on the little knowledge that I have on MCMC (Markov chain Monte Carlo) methods, I understand that sampling is a crucial part of the aforementioned technique. The most commonly used sampling ...
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2answers
63 views

Rao-Blackwellization of Gibbs Sampler

I am currently estimating a stochastic volatility model with Markov Chain Monte Carlo methods. Thereby, I am implementing Gibbs and Metropolis sampling methods.Assuming I take the mean of the ...
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1answer
24 views

Confidence interval for the p-value estimate when doing Monte Carlo testing

The second paragraph of the Monte Carlo testing section of the Wikipedia article on resampling statistics, the values of a confidence interval for a p-value from a MC sampling is given: After $N$ ...
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58 views

Is correlation between parameters a problem when fitting a Bayesian model using MCMC?

Assuming some Bayesian model, for example: $$y \sim N(X\beta, \sigma)$$ where this model has: Response vector: $$ y = \pmatrix{y_{1} \\ y_{2} \\ \vdots \\ y_{n}} $$ Predictor matrix: $$ \...
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26 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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11 views

Statistical comparison between two stochastic algorithms

I am working on a mechanics problem with variability in material properties. For that I need to analyze the efficiency of two methods for same accuracy (confidence level I guess?), the first one being ...
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702 views

Monte Carlo method cannot be used

This is an example from notes I don't understand why we should think about the $E(x)$ and $Var(x)$ first?
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15 views

Simulating probability through random sampling: Is this valid?

What is the name of a simulation that randomly takes two samples from a population and then compares it to another random sample with a predefined condition? I'm writing an assignment and I need some ...
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34 views

proof independency of two tests, how to go about?

I am trying to prove that two tests are investigating different hypotheses. So for example, test A tests for differences in variance while test B test for differences in means between two groups ...
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23 views

Random forest classfication: the Monte Carlo approach to train/test split

I'm trying to build a classification random forest. The problem is, that the output (dependend) variable is skewed (the more important class, let's call it < true >, happens ~20% of time) and the ...
4
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2answers
53 views

Parameter Estimation for intractable Likelihoods / Alternatives to approximate Bayesian computation

Suppose that I have a stochastic model with some parameters that I want to fit to some observed data. Let's assume the Likelihood intractable, i.e. for some reason I cannot work with the analytical ...
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1answer
49 views

What distribution describes this process?

I have three clinics that are testing for Zika virus. We know the proportion of positives aggregated across all clinics, i.e., the first clinic has identified 20% of all positives across the three ...
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14 views

Picking tuples from a list with low discrepancy

Given a list of m items, I am looking for a way to repeatedly pick a tuple of n distinct items from this list with low discrepancy. For example, suppose I have a list of 3d points, and I want to ...
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23 views

Sample dependency in Neural Net Training cross-validation

I've created a Monte Carlo simulation that randomly divides my data into "test" and "training"-Samples and then trains a neural network. The ratio of 0 and 1 (19.62%) Category is stabilized on ...
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59 views

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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18 views

Order of generation in the VEGAS algorithm

The VEGAS algorithm is a way of efficiently generating Monte Carlo events. In the first iteration, it generates an n-dimensional (n can be configured) set of uniformly distributed random numbers ...
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26 views

Optimizing a function available only through (monte-carlo) stochastic approximation

I am working on a problem where I want to estimate the maximum of a density that I can, in practice, evaluate (pointwisely) using a Monte-Carlo approach (because of intractable integrals). Obviously, ...
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28 views

Bayesian MCMC Fitting

I am doing a Bayesian MCMC fit using emcee in python. I first maximize the log of the likelihood and use the results as initial parameter starting points in my MCMC. I am using a uniform prior and ...
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33 views

Nuisance Parameter in Bayesian MCMC

I am doing a Bayesian MCMC fit to some data using a simple model and I want to understand how to handle nuisance parameters. I am looking at this tutorial. The model is a line: $$y = m x + b$$. The ...
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63 views

Show estimate converges to percentile through order statistics

Let $X_1, X_2, \ldots, X_{3n}$ be a sequence of iid random variables sampled from an alpha stable distribution, with parameters $\alpha = 1.5, \; \beta = 0, \; c = 1.0, \; \mu = 1.0$. Now consider ...
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10 views

MCMC diagnostics for EMC or SMC

Are diagnostics developed for MCMC (e.g. Gelman-Rubin, Geweke) suitable for output from Evolutionary Monte Carlo (EMC) or Sequential Monte Carlo (SMC)?
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31 views

Monte Carlo conditional pdf

I have a question which as been bothering me. It's best explained by way of a dumb example: Suppose one wants to compute the value of pi by sampling within the unit square (i.e. https://en.wikipedia....
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What is the difference between the Monte Carlo Method in R package 'DMwR' and a normal Monte Carlo Method?

I am trying to estimate the performance of a machine learning model on time series data. I saw the example of model evaluation using Monte Carlo Estimates from the book "Data Mining With R Learning ...
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31 views

Markov chain Monte Carlo sampling using CDFs instead of PDFs

I wonder if there is any MCMC sampling method which uses the definition of the target CDF instead of the target PDF; however, I may use a proposal PDF. I would like to use Metropolis-Hastings but it ...
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16 views

How to relate distributions?

I have 100 objects. Each object has 10 (highly correlated) attributes that I can measure. For each object, I obtain 10000 samples of that object's attributes. I now want to relate the attributes ...
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1answer
51 views

Sampling from marginal distribution using conditional distribution?

I want to sample from a univariate density $f_X$ but I only know the relationship: $$f_X(x) = \int f_{X\vert Y}(x\vert y)f_Y(y) dy.$$ I want to avoid the use of MCMC (directly on the integral ...
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28 views

How is the Fermiac machine (Monte Carlo trolley) working?

There is a cool website showing the Markov chain with a machine. But nobody is explaining how it's working or showing a video of it's functioning. This is explaining the Markov chain monte carlo ...
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42 views

Monte Carlo vs simulation in GARCH (package “rugarch” in R)

What is the difference between a GARCH simulation and a GARCH Monte Carlo simulation? I look in the vingette for the "rugarch" package in R, Introduction to Rugarch. In section 6 Simulation on page ...
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144 views

How to extract distribution information from descriptive statistics?

In my thesis, I am trying to perform a Monte Carlo simulation with a set of parameters, where I take a random value from a known distribution to calculate a singe run of the simulation. However, for ...
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29 views

Indirect solution for maximum entropy through sampling?

Is there a way to sample from a finite set $\{A,B,C,D\}$ such that the limiting empirical proportions converges to the maximum entropy solution of their probabilities consistent with known constraints?...
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176 views

Why is the intercept of linear regression biased?

Out of curiosity, I conducted the following simulation (code below). Why is it that when the variance of the error term is large coefficient associated with the intercept is biased? Can you recommend ...
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13 views

Generating correlated uniform random variables [duplicate]

How can I generate two Uniform $(0,1)$ variables $U, V$ with correlation approximately .25?
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65 views

Convergence of the distribution of 0.05 quantiles through Monte-Carlo simulation

I am trying to get admitted to a masters in quantitative finance (I come from a computer science background), so next week I will have 3h to solve an exam in statistical computing using my favourite ...
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1answer
37 views

Effective sample size for MCMC with multimodal target

I am trying to evaluate an adaptive MCMC algorithm on a multimodal target density. Among other performance measures, I would like to evaluate the sampler in terms of Effective Sample Size (ESS). The ...
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1answer
40 views

Evaluating two sets of random samples

Let $p$ be a probability distribution that can be computed tractably for any given point. I use two MCMC methods to generate samples from the distributions. For each MCMC method, I run 1000 Markov ...
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40 views

Is there a survey that explores all the available Markov chain Monte Carlo methods?

I am interested in exploring the efficacy of various Monte Carlo methods. I am aware of the Metropolis acceptance criterion, Hamiltonian Markov chains, Gibbs sampling, importance sampling, slice ...
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24 views

Friedman's test or Monte Carlo?

I have two time-series data sets of the same five experiments. That makes two 5 X 7 matrices where the row is the experiment and the column is the day, and each matrix comes from a different treatment....
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1answer
20 views

Models for nonnegative (incl. zero) positively skewed multivariate time series (trade volumes)

I want to build a Monte Carlo simulation that is based in part on share amounts that are traded in the market for a set of stocks. I need to be able to take into account the co-dependence of trade ...
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51 views

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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31 views

Monte Carlo integration with density unknown

If I want to find the integral $\int f(x)dx$, I want to use the Monte Carlo method to calculate it. What I have is the data $x_1, \cdots, x_n$ follows $p(x)$. (In my application, $f$ is some function ...
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1answer
37 views

Estimating normalization constant with Monte Carlo integration

Be $f(x)$ a function. Suppose that $f(x)$ integrates to a finite value $k$: $$\int_{-\infty}^{\infty}f(x)dx=k$$ The normalization constant of $f(x)$ is $1/k$. Monte Carlo integration can give an ...
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16 views

Use Monte Carlo to find monthly premium on a Credit Default Swap

You are holding a 10-year 100 million bond newly issued by Risky Corp (A rated). You wish to insure against the possibility of default by entering into a credit default swap with me. Our contract ...
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21 views

Efficiently sampling from Markov Chain with low-probability transitions

I need to sample a large number of paths from a Markov Chain with known state transition matrix $T$, where some of the state transitions are low probability (~0.01%). For example, I might have a large ...
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2answers
38 views

Can I use the Bhattacharyya distance as an acceptance criterion for Approximate Bayesian Computation?

I am researching the spread of a disease through a population and want to capture the behavior of this disease with a model. I already have a model and patient data. The data is a value per patient ...
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52 views

Monte Carlo Methods

_I've tried using sqrt p(1-p)/n to get the standard error and then calculate the t test but for all parts I get a very large number of t so this means ...