# 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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### How to simulate samples using MH from multi-variate Gaussian proposal distribution?

I have the task of obtaining samples {$\theta ,. . .,\theta_N$} using the Metropolis-Hastings algorithm, where the proposal distribution $q(\theta)$ is a multi-variate distribution The proposal ...
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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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### 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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### 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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### How should I implement an adaptive Metropolis algorithm for a Gibbs sampler with two Metropolis steps?

Consider the Gibbs sampler Sample $\theta' \sim p(\theta|\tau, D)$ Sample $\tau' \sim p(\tau|\theta', D)$ Both conditional distributions are sampled with a Metropolis step. The joint distribution is ...
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### Metropolis-Hastings exercise with Cauchy and normal distributions [self-study]

I have the following exercise to solve and would appreciate some help. Consider a linear regression model $y = X\beta + \varepsilon$, where $y = (y_1,...,y_T)'$, $X = (x_1,...,x_T)$, $x_t$ is a single ...
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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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### Metropolis-Hastings: target distribution with two modes; deterministic transformation

I'm trying to construct a Metropolis-Hastings algorithm to sample a target distribution $p(x)$ with two different and isolated modes. The example I'm working with is \begin{equation} p(x) = \frac{\...
Given an input space $X$ and a function $f: X\rightarrow \mathbb R$, we want to find $x^*=argmin_{x\in X} f(x)$. One way is to cast this problem as a sampling, where we define a distribution \$p(x)\...