Questions tagged [metropolis-hastings]

A special type of Markov Chain Monte Carlo (MCMC) algorithm used to simulate from complex probability distributions. It is validated by Markov chain theory and offers a wide range of possible implementations.

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Metropolis and Barker Algorithms: Irreducibility

A question from Brémaud, Markov Chains Gibbs Fields, Monte Carlo Simulation and Queues. Exercise 11.5.2. Metropolis and Barker Algorithms: Irreducibility how do you show explicitly that $$p_{i,j}= \...
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Can you give an example of Metropolis and Metropolis-Hastings algorithm?

I have studied many books and tried to understand both the Metropolis and Metropolis-Hastings algorithm. Everywhere it is written in the context of the Ising model or Lenard-Jones Energy. I am having ...
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Do the following normalizing constants cancel out in Reversible Jump ratio?

We know that a Strauss Point Process has density $$p(x_{1}, x_{2},..., x_{K})\propto \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}$$ where ...
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Hamiltonian Monte Carlo vs. "Metropolis-Hastings with a Hamiltonian step"

In Hamiltonian Monte Carlo the proposal is accepted with probability: $$ \alpha\left(\mathbf{x}_n(0),\mathbf{x}_n(L\Delta t)\right) = \min\left(1, \frac{\exp\left[-H\left(\mathbf{x}_n(L\Delta t),\...
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Calculating Metropolis-Hastings algorithm correction term

I am teaching myself about different samplers and currently going through a course. They write up an example of the Metropolis algorithm (MA), which I am pretty sure I understand. They next propose ...
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Is the Jacobian term needed if the prior is on the transformation parameter?

Suppose I have a strictly positive parameter $\sigma$ and I need to estimate it using the random walk Metropolis-Hasting algorithm. I know that I can do a parameter transform, i.e., $\beta=log(\sigma)$...
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Interpret and assess the output of MCMC in high dimension?

In one of my recent presentations I used MCMC to generate 50,000 samples from a 13-dimensional random variable. And the audience of this presentation is largely made up of layperson that has no ...
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How can I find the acceptance probability for a joint Metropolis-Hastings proposal?

Suppose that I want to generate a proposal $(x^*,y^*,z^*)$ with the following: $$z^*\sim p(z|\alpha,\beta)$$ $$x^*\sim p(x|z^*,\boldsymbol{\gamma}_x)$$ $$y^*\sim p(y|z^*,\boldsymbol{\gamma}_y),$$ ...
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Is this Categorical Distribution a valid proposal for Metropolis Hastings?

Is it possible to use a categorical distribution as proposal distribution for Metropolis-Hastings? For example, suppose that at the current iteration we are at the state $x_{i}=1$, from which there ...
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Beta-Binomial Gibbs Sampler

I am self-studying Bayesian statistics from the book Computational Bayesian Statistics by Turkman et al, but I am stuck on Problem 6.3 from the book: Suppose we want to consider a Binomial (unknown $\...
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Metropolis-Hastings algorithm for logarithmic probability density

Similar question to posted here: Metropolis-Hastings using log of the density however my question is around sampling a random number from a uniform distribution. I am following the steps outlined in ...
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multivariate potential scale reduction factor less than one

I am attempting to implement the multivariate potential scale reduction factor (PSRF) mentioned in this answer and originally described by Brooks and Gelman (1998). When I use a basic Metropolis ...
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Is the Markov property important in the Metropolis algorithm?

I’m taking a class in Bayesian statistics, and we’re learning about the Metropolis algorithm. Suppose for simplicity that we just have one parameter we’re trying to estimate: $\theta$. According to ...
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Proposed transition matrix for MCMC in two-state Markov Chain

Suppose we would like to model the weather (either sunny $S$ or cloudy $C$) using a two-state Markov Chain, given a set of data collected from 10000 days: $$CCCSSSSSSCCCSSSSSSCCCC...$$ We can use the ...
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With knowing the exact form of the prior density, how to improve the convergence/acceptance rate of an independent Metropolis Hasting algorithm?

I have a set of parameters $[\eta_1,\eta_2,...,\eta_k]$ (k can be a bit large, sometimes k=40) to estimate via the Bayesian MCMC method. I know that each component of $\eta_{k}$ is independently ...
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Help to Proof of Gibbs sampler acceptance rate

i was wondering if someone could explain how the acceptance rate in the Gibbs Sampling works. In the literature it says that in Gibbs case, the acceptance is always 1 because part of the "...
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Bayesian analysis example with convergence under Gibbs but not Metropolis-Hastings

Having a conceptual understanding of algorithms such as Metropolis-Hastings, Gibbs and Hamiltonian Monte Carlo can provide ideas of remediation to apply when models do not converge. This question ...
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In Bayesian models, can you use Uniform(-inf, inf) as a prior?

In Bayesian models, can you use Uniform(-inf, inf) as a prior? I ask because in an class, we looked at MH MCMC sampler, and showed that to sample from a distribution, we need not explicitly solve for ...
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Optimal way to calculate the sample variance of MCMC data?

I assume that the calculation of the average and variance of a sample generated by a Metropolis-Hastings MCMC does not depend on the existence of correlation. Recently, I was told that in the presence ...
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Metropolis Hastings on hierarchical bayes update question:

[I have this somewhat complicated hierarchical bayesian model]1 Here the $y$ on $\theta$ are Poisson, $\theta$ are deterministically generated from the $att, def$ (and $home$). Then the last ...
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Bayesian multivariate regression with common coefficients

In a hierarchical model I'm working on, I have $K$ different $N\times P$ predictor matrices, each denoted $X_k$ and $K$ length $N$ outcome vectors each denoted $y_k$. Essentially, I have a ...
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Dynamically adjusting parameters of Markov chain

I am using a Metropolis algorithm to generate samples from a complicated (high-dimensional) probability distribution. As is common, the proposed updates depend on some "step size" parameter $...
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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 ...
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Measure to capture within-chain fluctuations in MCMC?

I am using two kinds of updates for a particular parameter in MCMC estimation of my model. First update gives the following trace plot: Second update gives the following trace plot: Note that the ...
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Metropolis-Hastings Reversibility: What is the measure used in the definition of reversibility?

Let $\pi$ be a target probability distribution on a measurable space $(E, \mathcal{E})$. MCMC obtains dependent samples from $\pi$ by using a Markov Chain with transition kernel $\mathrm{K}:E\times \...
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What is the interpretation of this modified Metropolis algorithm?

Modified Metropolis-Hastings Consider a model with parameters $\theta = (\alpha, \gamma)$ and consider a modified Metropolis-Hastings algorithm which can be summarized (with brevity) as follows. ...
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Does the MCMC algorithm use the likelihood function to move to a new proposal value?

I'm trying to understand how MCMC and related algorithms work for Bayesian inference. In this paper the authors use an example of a normal distribution in explaining MCMC. They state, "If the ...
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Metropolis Algorithm for a high dimensional Bimodal distribution

I am using metropolis mcmc for an $n=8$ dimensional system on an (n-1)-sphere. I was considering the 2d case, as it can be visualized. For this case, the probability density,up to a normalization, is \...
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When is metropolis Hastings algorithm useful

I need to do a data analysis project and am considering the Metropolis Hastings algorithm to estimate the parameters of a logistic regression model. I would draw from the complete data log likelihood ...
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Is it possible to improve Markov Chain Monte Carlo performance by decomposing a Binomial Likelihood?

Suppose that we have sampled $y_{1},y_{2},...,y_{n}$ from a Binomial distribution $Bin(N,p)$. Also, let's assume that $p$ is known and our goal is to infer the unknown parameter $N$, with the use of ...
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MCMC algorithm for Hierarchical Bayes model with variable number of mixture components

I am trying to develop an MCMC algorithm for clustering $n$ data-points $y_{1},y_{2},\dots,y_{n}$ using a Gaussian mixture model, but with a prior defined on the number of components K. The ...
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Drift and Minorization for Metropolis Hastings algorithm

Can some one point me to articles or literature with an example of the drift and minorization condition proof. At the moment i have come across the Gibbs Sampler and the Random Walk Metropolis ...
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Question about MCMC independent proposals

I've a question re MCMC proposals, I was hoping you could help me with that. I need to implement for work an Independent Metropolis-Hastings algorithm to sample from a 10-dimensional posterior. I am ...
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Best Proposal Distribution for asymmetric target in Metropolis Hastings (Exponential Target)

Suppose I know my target distribution is asymmetric, for instance, suppose my target distribution is an exponential $$ p(x) = e^{-x} \qquad\qquad x\in [0, +\infty) $$ Is there any theory regarding ...
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Rao–Blackwellization of Metropolis–Hastings

I am trying to achieve a Rao–Blackwellization of Metropolis–Hastings algorithm. In the paper by Robert et al. 2018, the following is given. \begin{align} ℑ=&\frac{1}{T}\sum_{t=1}^Th(\theta^{(t)})=\...
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High Acceptance rate with Uniform Proposal Distribution

I'm using a metropolis algorithm to model a physical system. There are N sets of 8 real parameters, each set of 8 real numbers is normalized. The target is a Boltzmann distribution with the energy for ...
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Stratification of Random Walk Jumps in Metropolis-Hastings

I aim to achieve variance reduction in Random Walk Metropolis Hastings algorithm by introducing stratification to the random walk jumps. What I have tried is to make use of Latin Hypercube Sampling in ...
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Gibbs sampling step for variables that have a complex offline prior in an MCMC hybrid

I have a question about how to use an offline function as a prior when performing a Gibbs/hybrid analysis. Let's say I have data $y$ and some parameters which I'll simplify to $\theta_1, \theta_2$. ...
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How to sample from a joint probability distribution of two variables? [duplicate]

I want to randomly sample from a joint probability distribution characterized by two variables x and y. So, basically, I have the information of the joint PDF which has the dimension 100x50. As per my ...
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4 votes
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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-...
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Metropolis Hastings with Gamma Proposal Density

I am trying to use Metropolis Hastings to sample from a shifted gamma distribution. Since it is shifted, it has a domain of $(n, \infty)$. I tried using a Gaussian proposal density and ran into the ...
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Random Walk Metropolis: acceptance probability with truncated normal proposal

I want to draw from my target density $p(\theta)$ using Random Walk Metropolis. $\theta$ has domain $[2, +\infty)$, and I am using as proposal a truncated normal, namely: $$q(\theta_t') \sim N(\theta_{...
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Monte Carlo Methods: [closed]

Can someone explain to me the following statement from “Introducing Monte Carlo methods with R!” By Robert Christian. “If the exploration mechanism has enough energy to reach as far as the boundaries ...
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Computing the acceptance rate (empirically) of samples from the Metropolis algorithm, where samples are "thinned"

I have a number of queries about computing the acceptance rate of samples generated from the Metropolis (symmetric random walk) algorithm empirically, that is, in the presence of burning-in and ...
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Thinning and burn-in in Metropolis-Hastings algorithm

I have written a Metropolis-Hastings algorithm manually in Julia language for a customized distribution, and i want now to know how can i perform the thinning and the burn-in to increase the ...
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How do we define the kernel to calculate the acceptance ratio for Metropolis-Hastings Markov Chain Monte Carlo?

I am having a lot of difficulty understanding how to apply the algorithm to a real scenario. The part that confuses me is that we are looking for a target distribution (the real distribution of our ...
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Metropolis-Hastings with non-centered Proposal

I am trying to draw samples from the Laplace distribution $\pi^* = \text{exp}(-|\theta|)$, using Metropolis Hastings algorithm with a noncentered proposal, meaning that regular Metropolis wont work.. ...
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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, ...
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Metropolis-Hastings algorithm for a completely specified distribution

Consider a random variable $X\sim f(x)$, such that $$ f(x)=\frac{1}{c}\times K(x)\propto K(x), $$ where c: normalizing constant, K(x): the kernel of the distribution (ie the part which involves $x$). $...
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Application of Metropolis Hastings

I am trying to implement the Metropolis Hastings algorithm for Bayesian analysis. In this case, the parameter of interest is the scale parameter for a Weibull distribution. The context is for ...
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