The acceptance rate (acceptance ratio, acceptance fraction) for a Markov Chain Monte Carlo sampler indicates the fraction of accepted over proposed moves.
Questions tagged [mcmc-acceptance-rate]
33 questions
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How does the celebrated result about the diffusion limit of the Random Walk Metroplis-Hastings algorithm help us to find the optimal scaling
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 $(\mathbb R^d,\mathcal B(\mathbb R^d))$ with…
0xbadf00d
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Mean acceptance rate for Metropolis-Hastings algorithm
My question relates to the result stated on page 4 of: http://stat.columbia.edu/~gelman/research/published/baystat5.pdf
which claims that the mean acceptance probability when performing the Metropolis-Hastings algorithm with a standard normal target…
Matt Barrett
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Do I need to evaluate acceptance rates in Metropolis within Gibbs algorithm?
Consider the Gibbs sampler
Sample $\theta' \sim p(\theta|\tau, D)$
Sample $\tau' \sim p(\tau|\theta', D)$
where $\theta,\tau$ parameters of the data $D$. Now assume that we can only sample from $p(\tau|\theta', D)$ using a Metropolis step, for…
tomka
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Erroneous expression for Metropolis-Hastings acceptance ratio in a paper
Let
$(E,\mathcal E)$ be a measure space;
$\rho:E\to[0,\infty)$ be $\mathcal E$-measurable, $p:E^2\to[0,\infty)$ be $\mathcal E^{\otimes2}$-measurable, $$r(x,y):=\left.\begin{cases}\displaystyle\frac{\rho(y)p(y,x)}{\rho(x)p(x,y)}&\text{, if…
0xbadf00d
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3
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How to tune MCMC with unwieldy posterior
Let's say I have $n$ observations of a random variable, $X_1, \dotsm, X_n \sim \mathcal{N}(0, \sigma^2)$. I also assume $\sigma^2$ has a Gamma(1,1) prior distribution, $\pi(x) = \exp(-x)$.
I'm now attempting to use Metropolis-Hastings to sample…
CLL
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Expression for the mean acceptance rate of the Metropolis-Hastings algorithm
Let
$(E,\mathcal E,\lambda)$ be a measure space
$p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$
$q:E^2\to[0,\infty)$ be ${\mathcal…
0xbadf00d
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Is the MC produced by HMC reversible?
I know that the deterministic dynamics in Hamiltonian Monte Carlo/Hybrid Monte Carlo are reversible and the numerical integrators one uses to approximate them are reversible too. But HMC consists of 2 steps:
Sample velocity
Run Hamiltonian Dynamics…
Lochend
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How to apply MCMC to bayes when likelihood is not easy to compute
Let $z$ be observations and $w$ be the parameter that we want to infer. Assuming that we know the prior $p(x)$, by using Bayes law, we have $p(x|z) = p(z|x)p(x)/p(z)$ where $Z$ is the marginal likelihood. For the purpose of sampling from $p(x|z)$,…
orematasaburo
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Need to understand a statement for Random Walk Metropolis algorithm's proposal distribution?
I was told that the proposal distribution of Random Walk Metropolis needs to be symmetric. But today I was reading a book about Bayesian Analysis which contains the following statement:
"The proposal distribution does not have to be a rotationally…
StoryMay
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Metropolis Hastings - Acceptance ratio, proposal and lkelihood
From a previous post :
First to explain the MH algorithm, it's used to approximate
numerically a target distribution, in this case $p(\theta|D)$. At
each stage of the algorithm:
A value $\theta_{proposed}$ is proposed using the jumping or…
youpilat13
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How worried should I be about low acceptance rate in cold chain (parallel tempering MCMC sampler)
I have a very noisy/multimodal likelihood function for a 6-parameter model. The popular emcee sampler fails miserably (no matter how many chains I use and for how long I run them, they always get stuck in local minima), but its parallel-tempered…
Gabriel
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How to understand the scaling in Metropolis Hastings MCMC
We know the Metropolis Hastings (MH) in MCMC:
target distribution: $\pi(x)$
proposal distribution: $p(y|x)$
acceptance: $\alpha(x,y) = \min \Big(1, \dfrac{\pi(y)p(y|x)}{\pi(x)p(x|y)}\Big)$
Here are some examples of a proposal…
user6703592
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Metropolis Hastings for Poisson Distribution
Studying Bayesian Inference and Markov chain Monte Carlo (MCMC) algorithms, I am facing a self study question on a MCMC approach to a Poisson distribution with parameter $\lambda$.
Using R, my code is:
### Parameters
# observation
x <- 60
# alpha…
Arduin
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How can I tune my Random Walk Metropolis Hastings algorithm on the fly?
I just have a very general question. I've recently started to use Random Walk Metropolis-Hastings (RWMH) to sample from a distribution in order to calculate integrals. But I've noticed that the efficiency of integral calculations is highly dependent…
AlphaBetaGamma96
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what is the optimal step size for metropolis-hastings algorithm to have independent state
In the PRML chapter 11, The Metropolis-Hasting algorithm,
For a sampler with Gaussian distribution as proposal distribution.
The original distribution is correlated multivariate Gaussian distribution, with standard deviations $\sigma_{min}$ and…
Kelvin_neverKnow
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