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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567 views

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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63 views

Keeping track of the variance of a Metropolis-Hastings estimator

Let $(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces, $p,q$ be probability densities on $(E,\mathcal E,\lambda)$, and $\varphi:E'\to E$ be bijective and $(\mathcal E',\...
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585 views

Noninformative prior for variance, understanding and coding

I have three questions regarding the understanding behind and implementation of a noninformative prior for variance. I'm attempting to build a Metropolis sampler and I'm trying to sample from a ...
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38 views

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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179 views

Pareto optimality in Metropolis sampling

In the Metropolis sampling algorithm, we have some function $f(x)$ proportional to a probability distribution $P(x)$. To generate a random walk with stationary distribution $P(x)$, we generate a ...
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48 views

How to estimate the weight matrix in distribution $X = VWV^T$?

Suppose the 1 x N vector $V\in \{0,1\}^N$ comes from the pdf $f(V) = VWV^T$, where $W$ is a N x N positive definite matrix. If the weight matrix is given, I can use gibbs sampling to generate a ...
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381 views

Is this how a Bayesian bootstrap works?

I am a bit new to the whole nonparametric and Bayesian idea, so tell me if this is correct: to estimate, say, the mean of a dataset's population we do the following: We define a function $f(x)$ that ...
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5k views

Metropolis-Hastings within Gibbs sampling

Suppose we have the following classical normal linear regression model: $$y_i = \beta_1 x_{1i} + \beta_2x_{2i} + \beta_3x_{3i} + e_i$$ where $e_{i} \sim iid.N(0, \sigma^2)$ for all $i = 1, 2, \cdots,...
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15 views

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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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{\...
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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 ...
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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. ...
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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 $\bar{Y}$, ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(\...
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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 $\...
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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 ...
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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 (∆...
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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 ...
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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_{...
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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 ...
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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(\...
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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)\...
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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}...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...
2
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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 ...
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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 ...
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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 ...
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35 views

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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32 views

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 ...