Questions tagged [accept-reject]
Use this tag for accept-reject sampling methods. These are also known as rejection sampling methods. These methods sample a random variable from a dominating measure (h) and accepts the draw if an auxiliary random variable (a uniform) is less than the desired measure (g), so accept the draw if u<g. Otherwise draw another pair. You can think of this as sampling on a space with 1 additional dimension where the additional dimension is uniform under g.
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Acceptance-rejection problem confusion
My professor has given me a random variable $X$ with a probability density function:
$$f(x) = cos(x), 0 ≤ x ≤ π/2.$$
I have to write an acceptance-rejection algorithm to simulate values of $X$ based ...
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Estimating the integral $\int_{0}^\infty x^4 e^{-2x}\,dx$
We have been given a random variable having a Gamma distribution as shown below:
Using the accept-reject algorithm, we are supposed to sample from the Gamma distribution using exponential as the ...
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How to found the bound $M$ in rejection sampling when the differentiation of the ratio of target and proposal density is not possible?
Suppose we have the following PDF of X:
$f(x)=\frac{1}{4}(2-x)\;;-1\leq x\leq 1$
We want to use $g(x)\sim\mathrm{Unif}(-1,1)$ as a proposal density to generate samples from $f(x)$ using a rejection ...
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Perfect sampling and inverse transform sampling
Firstly, looking at the discussion https://math.stackexchange.com/questions/241315/three-ideas-of-perfect-sampling, the term "perfect sampling" does not seem adequate, since there are ...
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Rejection Sampling Kurtosis and Number of Iterations
I tried to implement a rejection sampling method in python based on something explained during class. The target distribution is the normal distribution and the proposal is the exponential ...
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How to validate rejection sampling?
What is a principled approach to validating samples generated from rejection sampling actually follow the target function?
I am looking for some thing more than a simple histogram + density plot.
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Choosing a proposal density g(x) for $ f(x)= {\Large \frac{e^{x}}{(e-1)} }$ [closed]
In finding an proposal distribution function $g(x)$ for the following function:
$ f(x)= {\Large \frac{e^{x}}{(e-1)} }$ where $0 \leq x \leq 1$
Tested with $$x^2+1, 1/x+1$$ and other variations, but ...
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Why does this algorithm generate a standard normal distribution?
I have this algorithm which I encountered:
(1) Generate $U_1$, $U_2$ independently from Uniform(0,1)
(2) Set $Y_1 = -\log{U_1}, Y_2 = -\log{U_2}$. If $Y_2 > \frac{(1-Y_1)^2}{2}$, accept $(Y_1, Y_2)$...
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Sampling according to a product of a known density and a probability function
Given a known density $p(x)$, I'd like to generate samples according to $q(x) \propto p(x) f(x)$, where $f(x)$ is some probability function, $\forall x f(x) \in [0, 1]$, e.g., a sigmoid function.
One ...
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Approximate Bayesian Computation with some information about the posterior distribution
ABC method works as follows.
At iteration $i \geq 1$ :
Draw $\theta_i \sim p(\theta)$.
Generate a sample $X^{(i)} \sim p(X|\theta = \theta_i)$.
Accept $\theta_i$ if $\rho(S(X^{(i)}),S_{obs}) < \...
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Sampling order of rejection sampling
Consider a target density $f(y) \leq C \, g(y)$, where it is easy to make samples from density $g$. We can draw samples from $f$ by repeating the following routine:
Draw $Y \sim g$
Draw $U \sim \...
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How to identify the distribution being sampled by this algorithm?
I've been studying a piece of code lately that uses the Monte Carlo acceptance-rejection method to draw random samples from a distribution I'm struggling to identify. Here's a Python implementation:
<...
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Capital Y vs small y
I want to use the acceptance-rejection method to generate a sample from the following target probability density function:
$$f\left(x\right)=1.25x^4+2x^3+0.25,\:0<x<1$$
Let the trial probability ...
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Can I get "fail to reject the null" when testing with test statistic, and "reject the null" when testing with P-value at the same time? [duplicate]
Can I get "fail to reject the null" when testing with test statistic, and "reject the null" when testing with P-value at the same time in a same nonlinear regression model?
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Rejection sampling ineffectiveness in high dimensions
I guess I really have two questions.
First, iv'e seen quoted in a couple of places that the probability of accepting a given sample in a rejection sampling algorithm (sampling from a density $f$ with ...
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Finding a proposal distribution in acceptance-rejection method
I'm learning the Acceptance-Rejection method but I am having a hard time finding a g(x) except using uniform distribution to simulate the f(x).
How could we find a g(x) that has a simple pdf and is ...
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Monte Carlo sampling ( accept/reject) for geographic dataset
I have a dataset consisting of latitude and longitude and I'm confused on which approach to use to determine the distribution of points so I can apply the monte Carlo accept/reject for sampling. this ...
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how to generate data from a distribution whose cdf is not in closed form? [duplicate]
I am working on a distribution whose pdf and cdf is
$$f(x,\alpha,\beta)=\frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta}}{(1+(\frac{x}{\alpha})^{\beta})^{2}}\frac{\sin(\frac{\pi}{\beta})}{(\frac{...
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Rejection region and two sided test. Absolute value of mean
Given following problem:
I've solved this problem assuming two sided test and rejection region $R=\{|\bar{X_n}| > c\}$ but it seems to be incorrect because correct answer assumed (I've checked it)...
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Escape unsuccessful accept-reject step in MCMC
I have an MCMC procedure that samples latent variables $h_1, \dots, h_T$. It is based on Shephard and Pitt (1997), https://doi.org/10.1093/biomet/84.3.653. Let $f$ be the true conditional posterior ...
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Does the accepted sample based on the acceptance-rejection algorithm has the same distribution as X~$f(x)$? [duplicate]
In Bayes Statistics, does the accepted sample based on the acceptance-rejection algorithm has the same distribution as X~$f(x)$?
Intuitionally it does, but how to prove it under the discrete and ...
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Accept-Reject algorithm from binomial or other non-uniform distribution
I'm currently researching monte carlo simulations and the different methods. What I'm finding is that methods such as accept-reject typically sample from a uniform distribution and then compare that ...
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Rejection sampling - total probability of acceptance [closed]
I am given the following pdf $$f(x)=3 x^{2}, \quad 0 \leq x \leq 1$$ which i need to simulate by using rejection sampling.
I have used the following code below in R.
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Rejection Sampling when the proposal is a mixture of two distributions
Given a density $f(x; \alpha, 1) = x^{\alpha-1}\exp(-x)/\Gamma(\alpha)$ and a proposal density $q(x) = \alpha_1.q_1(x) + \alpha_2.q_2(x)$ I want to generate random samples using Rejection Sampling ...
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Integration with accept reject sampling Monte Carlo
I've got a quick question with regards to accept-reject Monte Carlo integration that I can't solve. Suppose I want to integrate some function, $f(x,y)$, with samples of $x, y$ from $p(x,y)$.
Now, ...
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generating process of acceptance-rejection algorithm
The acceptance-rejection algorithm is described as follows:
suppose you have RVs $X$ and $Y$ with densities $f_X$ and $f_Y$, respectively, and there exists a constant $c$ such that $\frac{f_X(t)}{f_Y(...
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Understanding the Rejection Sampling for a vector
I am thinking about this classical problem in statistical simulations. We want to apply the Rejection sampling to simulate a random vector $Y=(Y_1, Y_2)$ of a uniform distribution from a unit disc $D =...
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Confused about the sampling method
I want to sample from a density using the rejection method. The density is defined as $$ f(x) = \frac{e^x}{e-1},~~~ 0\leq x \leq 1.$$
Following the definition of the rejection method, we can find ...
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Sampling from Gamma Distribution using the Rejection Method
I'm having some issues working through this practice problem. I have worked through the first portion of it, and I have the solution, but I don't understand how/why the solution does two things at the ...
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Devising an acceptance sampling plan for False Negative Rate
I need to evaluate a binary classifier that classifies inputs in positives and negatives. Since all predicted positives (PP) are assessed, I have complete data on the true positives (TP) and the false ...
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How the conditional probability is being calculated in Rejection sampling
In a class lecture, the "Acceptance-rejection algorithm" was presented as follows:
To generate $𝑋 \sim 𝑓(𝑥)$, Find density $g$ satisfying
$\frac{f(t)}{g(t)}<=c$ for some constant $c$ for ...
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Metropolis-Hastings - interpreting the transition kernel: alpha*proposal
I thought I had great intuition and mathematical understanding of the Metropolis-Hastings algorithm, until closer inspection... as I started compiling my notes, I realized I do not understand the ...
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Interpretation of the region of rejection in hypothesis testing in binomial distribution
The pharmacy company Life Co. has developed a new drug against insomnia. To check the effectiveness, this drug was tested with n = 10 patients. At present, the standard medication can cure 30% of the ...
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How does the Metropolis Algorithm "get off the ground"?
I'm thoroughly confused by the Metropolis Algorithm as defined in Casella and Berger's Statistical Inference. Namely, here's the definition (p.254):
Let $Y \sim f_Y(y)$ and $V \sim f_V(v)$, where $...
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Acceptance-Rejection using Functional
Setup
Let $X\in L^1(\Omega,\mathcal{F},\mathbb{P})$.
As far as I've seen, Monte-Carlo methods generate $x_1,\dots,x_n$ from the distribution of $X$ and uses the Glivenko-Cantelli theorem to conclude ...
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Posterior of $\text{Normal}(\theta,1)$ with a Cauchy prior distribution
If $X \sim N(\theta,1)$ with Cauchy as robust prior
$$\pi(\theta) = \frac{1}{\pi(1+\theta^2)} \qquad -\infty < \theta < \infty$$
What will be the posterior distribution when Cauchy is $(-2 <...
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Understanding the Delayed Rejection Metropolis algorithm (Mira, 2001a)
I'm having trouble understanding the algorithm as briefly described here, and I can't find the original paper by Mira since it seems to be from some obscure print journal (Metron Volume 59).
The ...
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Metropolis-Hastings acceptance ratio for truncated proposal
I have a proposal distribution for one parameter theta_guess
theta_guess = guessleft(theta_accept(1,r-1), 0.01,0)
which is a ...
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Exact Sampling from Improper Mixtures
Suppose I want to sample from a continuous distribution $p(x)$. If I have an expression of $p$ in the form
$$p(x) = \sum_{i=1}^\infty a_i f_i(x)$$
where $a_i \geqslant 0, \sum_i a_i= 1$, and $f_i$ ...
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Using a Random number Generator to draw samples from a Cumulative Distribution function
I am given a Rayleigh, distribution function:$$f(x)=\frac{1}{5}x\exp\left(\frac{-x^2}{10}\right)$$ with $x>0$ and asked to:
Use an appropriate random number generator algorithm to draw 500 samples ...
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Proof of Rejection Sampling
I'm trying to go through the proof of rejection sampling and I found a paper ACCEPTANCE-REJECTION SAMPLING MADE EASY which provides several helpful explanations. For Lemma 2, the paper claims that if $...
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Sampling from Skew Normal Distribution
I want to draw samples from a skew normal distribution as part of a matlab project of mine. I already implemented the CDF and PDF of the distribution, but sampling from it still bothers me.
Sadly, the ...
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