# 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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### Tuning MALA (Metropolis-adjusted Langevin) proposal

I'd like to implement a version of Metropolis-adjusted Langevin sampling, but I'm unsure how to go about tuning the parameters of the proposal density. My understanding is that in MALA, a proposal ...
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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 $\mathrm{K}:E\times \... 1answer 95 views ### Erroneous expression for Metropolis-Hastings acceptance ratio in a paper Let$(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{\... 0answers 105 views ### How to implement a M-H step in a Gibbs sampling I am having trouble implementing a Metropolis Hastings step in a Gibbs sampling problem. The following code was taken from https://www.stat.colostate.edu/computationalstatistics/ Details: It is a ... 0answers 336 views ### Metropolis sampling for Bayesian networks Gibbs sampling is a profound and popular technique for creating samples of Bayesian networks (BNs). Metropolis sampling is another popular technique, though - in my opinion - a less accessible method. ... 1answer 467 views ### MCMC acceptance rate decreases when proposal variance gets smaller I am drawing a sample Y of size n from a p-dimensional Normal (\mu, \Sigma). Typically, p is 5. I have \bar{Y} and V = YY', the sum of squares. Now I want to draw samples from this \bar{Y}, ... 0answers 71 views ### Problem with kernel estimates in Metropolis-Hastings I am performing an experiment on Metropolis-Hastings. I have written a proof that returns for stocks have to be a function of the Cauchy distribution and that the distribution of \beta in the ... 0answers 808 views ### Criteria in determining "step size" of Metropolis-hasting algorithms I am training a complex Bayesian model using Gibbs sampling and Metropolis-Hasting algorithm. Most of the parameters are directly sampled by using conjugate priors except for 3 params which are ... 0answers 131 views ### Singular proposal in MCMC Suppose we want to obtain samples of the density f(\mathbf{x}) where \mathbf{x} is a d-dimensional vector, i.e. \mathbf{x} = (x_1, x_2, \dots, x_d). To that end, we choose the Metropolis-... 0answers 875 views ### Efficiency in Metropolis Vs Gibbs sampling I have read that Gibbs sampling is more efficient than Metropolis algorithm. Why? Is this due only to the fact the in Gibbs sampling the acceptance rate is 1, so that the chain needs fewer ... 0answers 171 views ### Computing a Metropolis-Hastings target distribution? In implementations of the Metropolis-Hastings algorithm, how is the target distribution \pi(\mathbf{x}) = P(\mathbf{x}|\mathbf{e}) computed or estimated while ... 0answers 21 views ### 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 ... 0answers 34 views ### Gibbs updating algorithm (Gibbs steps) for computationally expensive likelihood I am looking for a good way to update steps in a Gibbs sampler where the likelihood function is computationally expensive. Here is what I tried so far: By default JAGS uses a slice sampler. However, ... 0answers 28 views ### How the Markov Chain Monte Carlo chains can be trusted through diagnostic plots I am given several diagnostic plots of parameter values from a generated MCMC chains. . It is asking me that Do the diagnostic plots suggest that the MCMC chains can be trusted? My attempt: I tried ... 0answers 81 views ### Asymptotic variance of Metropolis-Hastings estimates on a disjoint subdivision of the state space I'm running the Metropolis-Hastings algorithm on a state space (E,\mathcal E) which can be disjointly subdivided into regions E_1,\ldots,E_k, k\in\mathbb N (k\approx1e5). On E, I have a ... 0answers 71 views ### MCMC converges to MAP and stays at same value - what may go wrong? I am working on a Gibbs sampler which is complex and I would like to avoid giving all the details here. I will focus on the most necessary details. The Gibbs sampler involves parameters and latent ... 0answers 70 views ### Estimating the asymptotic variance of a specific Metropolis-Hastings estimator Remark: I've added a detailed description of the actual setting of the application to the end of the question. I'm running the Metropolis-Hastings algorithm with target distribution \hat\mu (see ... 0answers 98 views ### Can we use of the Metropolis-Hastings importance sampling estimator here? EDIT: Note that proposing according to \hat Q is equivalent to the following scheme: Given the current state \tilde x=(i,x')\in\tilde E:=I\times E' and x:=\varphi_i(x'), sample j\sim\tilde T(\... 0answers 37 views ### Asymptotic convergence of the Metropolis-Hastings algorithm with a not necessarily positive target density Consider the Metropolis-Hastings algorithm on a general state space. Let p denote the density of the target distribution \mu, (X_n)_{n\in\mathbb N_0} denote the generated Markov chain and \... 1answer 58 views ### Unbiased Metropolis-Hastings estimator of the form \frac{\sum_{i=1}^nW_if(Y_i)}{\sum_{i=1}^nW_i}. How do we need to choose W_i? Let (X_n)_{n\in\mathbb N_0} be the Markov chain generated by the Metorpolis-Hastings algorithm with proposal kernel Q and target distribution \mu and (Y_n)_{n\in\mathbb N} denote the ... 0answers 34 views ### Is it correct to perform multiple swaps simultaneously in Parallel Tempering? In Parallel Tempering, after every N_{swap} iterations of running Metropolis method, we randomly select a temperature T_i and swap its configuration with T_{i+1} with a probability min(1, exp (∆... 0answers 162 views ### Is the Metropolis-Hastings kernel always aperiodic, irreducible and geometrically ergodic? Let (E,\mathcal E,\lambda) be a measure space, Q be a Markov kernel on (E,\mathcal E) with$$Q(x,B)=\int_B\lambda({\rm d}y)q(x,y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$$for some ... 0answers 64 views ### Minimization of the asymptotic variance in MCMC Suppose (X_n)_{n\in\mathbb N_0} is a Markov chain generated by the Metropolis-Hastings algorithm. Assume (X_n)_{n\in\mathbb N_0} is stationary and consider the ergodic averages$$A_n:=\frac1n\sum_{... 0answers 39 views ### Is there a reason why we should run the Metorpolis-Hastings algorithm with a target density approximating the density we're actually after? Let$(E,\mathcal E,\lambda)$be a measure space,$p:E\to[0,\infty)$be$\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ denote the ... 0answers 179 views ### Normal-Gamma: Metropolis-Hastings on log-scale, but no Jacobian? I am reading the paper by Griffin and Brown (2010) where at one step in their MCMC procedure they need to sample from the following conditional posterior: $$p(\lambda|\gamma, \Psi)\propto \pi(\... 0answers 21 views ### Using some objective priors (for unbounded space) in a Metropolis-Hastings MCMC I'm doing some simulations using a M-H MCMC, and I was thinking of using some objective priors for some parameters. These parameters must be in \mathbb{R}^+. I was thinking of using \pi(\theta)\... 0answers 37 views ### GARCH(2, 3) model with Metropolis-Hastings algorithm Let's say I have a GARCH(2, 3) model with$$\nu_i = \sigma_i\epsilon_i$$where \epsilon_i \sim N(0, 1) and$$\sigma_i^2 = a_0 + \sum\limits_{k = 1}^{2} a_k\sigma_{i - k}^2 + \sum\limits_{l = 1}^{3}... 0answers 250 views ### Designing better Metropolis-Hastings proposal distributions (for correlated parameters) Question: Is there a rule of thumb for setting a non-diagonal covariance matrix for your Metropolis-Hastings proposal distribution? References are appreciated. Background: Say I have some posterior ... 0answers 54 views ### Proposal distribution on a pair of ordered continous parameters I'd like to sample a pair of continuous parameters which has the constraint that one has to be smaller than the other one. I understand one approach is by rejection sampling by rejecting the samples ... 0answers 53 views ### What's a "corner" in a parameter space? I'm learning about Hamiltonian Monte Carlo and one of the stated benefits is that it can move around a parameter space more efficiently and that it can (from Bayesian Data Analysis, 3rd ed.) turn ... 0answers 134 views ### Metropolis Hastings: What motivates the use of Metropolis-Hastings? I am confused with metropolis hastings. This is a simple question. In the metropolis hastings, it is assumed that we know the un-normalised posterior,$\pi(x)$. We can obtain the density by ... 0answers 44 views ### In Metropolis Algorithm, if draws$\theta^{t-1}$and$\theta^t$have the same marginals, why is the target is the same as the stationary distribution? In the Metropolis algorithm, suppose I start my algorithm at time$t-1$with a draw$\theta^{t-1}$from my target distribution$p(\theta|y)$. It can be shown that$\theta^t$and$\theta^{t-1}$are ... 0answers 51 views ### Diagnosing Markov chain mixing time from time series? If we have a markov chain with the aim of generating a sample from some distribution$f(x)$, how can we diagnose whether the mixing of the chain is 'good' or 'bad'. As I understand it, mixing is how ... 0answers 104 views ### Conditional Density for Sigma (Bayesian Lasso) I found that in Bayesian Lasso commonly$\beta \sim N(0,\sigma^2*diag(\tau))$and$\sigma,\tau \sim \pi(\sigma,\tau)$is used. Whereas$\pi(\cdot)$is a product of Laplace distributions. Is it ... 0answers 93 views ### Can I use Adaptive MCMC in any setting? In time series econometrics and finance, most Bayesian authors approximate their models with a Gibbs Sampler, this is especial true for state space models, SV and so forth. The dimensionality of ... 0answers 150 views ### Understanding a measure of convergence of MCMC simulations I am trying to better understand better the Gelman/Rubin measure of convergence of MCMCs. The method starts off by defining two quantities:$B$and$W$.$B$is said to be the between chain variance (... 0answers 70 views ### Metropolis Algorithm for a 3 variated function I need to implement a program that generate a sample from a 3 variated$pdf$using the Metropolis algorithm (and its variations). I was thinking to use a 3-variated normal distribution as my proposal ... 0answers 673 views ### Random walk with bivariate normal distribution Let$X$be a random variable from$f(x; \theta)$, where$\theta =(\theta_1,\theta_2)$. I want to: take a sample from this distribution using Metropolis Hastings algorithm and update the parameters ... 0answers 93 views ### Metropolis-Hastings to sample from dependent random variables Imagine the goal is sampling from$p(X,Y)$and X and Y are dependent real-valued random variables, i.e.$p(X|Y)\neq p(X)$. Now the question is how can we apply Metropolis-Hastings algorithm on the ... 0answers 192 views ### Can I adapt a MCMC proposal using a parallel chain? I am running two MCMC chains (say chain A and chain B) in parallel, using the Metropolis-Hastings algorithm with acceptance probability:$P(accept\ x_t) = \min\{1, f(x_t)/f(x_{t-1})\}. I would like ... 0answers 180 views ### Estimation of a state-space model using Bayesian analysis with the Metropolis-Hastings algorithm I have the following state-space model: \begin{aligned} y_t&=c+Ax_t+q_t, &q_t \sim \mathcal N(0,Q), \\ x_t&=\mu+Bx_{t-1} + v_t, &v_t \sim \mathcal N(0,R), \end{aligned} where the ... 0answers 35 views ### MCMC algorithm for Hierarchical Bayes model with variable number of mixture components I am trying to develop an MCMC algorithm for clusteringn$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 ...
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 ...