Refers to a Markov Chain Monte Carlo algorithm used to sample from probability distributions.

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Metropolis-Hastings Algorithm within Gibbs Sampling

I have this $f$ function below. $$ f(x_1,x_2)\propto \left(\dfrac{x_1}{x_2}\right)\left(\dfrac{\alpha}{x_2}\right)^{x_1-1}exp\left\{-\left(\dfrac{\alpha}{x_2}\right)^{x_1} \right\}I_{R^+}(x) $$ where ...
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51 views

Transformation of variables (Metropolis Hastings)

Say I have a bunch of data from a Poisson distribution and I want to find out my posterior i.e. I'm data fitting: $p(\lambda | X) \sim p(X|\lambda)p(\lambda)$ where $p(X|\lambda) = ...
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14 views

What sort of data would be appropriate to analyze under an MCMC method?

MCMC methods describe stochastic sampling but I'm not entirely sure the contexts in real datasets one would wish to apply MCMC methods. What kind of data could I gain insight into with MCMC methods?
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18 views

Comparison of MCMC methods? [closed]

Where can I find a good comparison of Gibbs, Metropolis, and Hybrid MCMC in R or Python? I have thus far found this ...
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52 views

Why is acceptance prob $A(x \rightarrow x')=min\left[1,\frac{\pi(x')Q(x'\rightarrow x)}{\pi(x)Q(x\rightarrow x')}\right]$ for Metropolis-Hastings?

In the Metropolis-Hasting algorithm we choose the acceptance probability $A(x \rightarrow x')$ as following: $$A(x \rightarrow x') = min \left[1, \frac{\pi(x')Q(x' \rightarrow x)}{\pi(x)Q(x ...
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23 views

Why accept Metropolis-Hastings sample if more probable than previous sample?

One step in the Metropolis-Hastings sampling algorithm is to determine whether a sample $x_i$ is to be accepted based on the previous sample $x_{i-1}$. My understanding is that $x_i$ is accepted with ...
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69 views

Sampling with Metropolis-Hastings

In Metropolis-Hastings sampling, if every draw of my proposal distribution (Q) is independent from the previous draw, is the convergence to the stationary distribution still guaranteed? To be more ...
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37 views

Systematic change of proposal density of metropolis algorithm - still a markov chain?

The problem is that I have a Gibbs-Sampler where one of the parameters has to be sampled via a Metropolis-Step from the respective FCD. I have problems finding a suitable scale for the Gaussian ...
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27 views

Monte carlo integrations with metropolis hastings step

Consider the following problem: Suppose we want to compute the following integral $$ f(y_1|y_2) = \int_{\theta} \int_{x_1} \int_{x_2} f(y_1| x_1,x_2,\theta, y_2) f(x_1,x_2|\theta,y_2) f(\theta | ...
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35 views

Metropolis-Hastings sample reusal

I have a posterior distribution from which I calculate some statistics using sampling, for example I calculate expectation. So I draw 1000 samples using Metropolis-Hastings and then I calculate their ...
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39 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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62 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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1answer
127 views

Gibbs Sampling Detecting Change point in time series

I was reading through this one page paper on using Gibbs sampling for detecting a change point in a time series like data. While I understand the part where the $\lambda$ and $\phi$ are chosen from a ...
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1answer
83 views

MCMC: examples of when direct sampling is difficult (but Metropolis Hastings is easy)

The Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. Would ...
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433 views

Island Hopping with Metropolis Algorithm

From John Kruschke's book, Chapter 7, Pg. 120 (summarised for succinctness): A politician is constantly travelling from island to island on a chain of islands... His goal is to visit all the ...
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30 views

Viterbi-like algorithm with a single observation

Consider a Metropolis-Hastings walk over an exponentially large, known state space S. The proposal/transition matrix P is known, as is the target stationary distribution. In fact, assume there is an ...
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26 views

Simultaneous multiple perturbations in Markov chain Monte Carlo

I'm coding a McMC algorithm for geophysical applications. Using the Metropolis–Hastings scheme to accept/reject the proposed models is something that I thought I understood completely, but I don't. ...
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1answer
100 views

burn in for Metropolis Hastings MCMC

I was wondering if there is a principled way to figure out how many samples to discard during the MH-MCMC burn-in stage. So, as I understand it, the initial samples can introduce bias in the ...
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1answer
67 views

Gibbs sampling versus general MH-MCMC

I have just been doing some reading on Gibbs sampling and Metropolis Hastings algorithm and have a couple of questions. As I understand it, in the case of Gibbs sampling, if we have a large ...
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67 views

How to implement a metropolis hastings algorithm to find the posterior pdf of a time-dependent parameter?

Assume that I have a time series observations denoted by Yi where i is from 1-5000. yi=β1 x1i + β2 x2i + β3 x3i + Ci; Here x is the input. I have to find the values of Betas. But, I'm taking all ...
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1answer
46 views

Simplex Random Walk Mean

Hi I have two questions related to a previous question I asked here: Simplex Random Walk In this link it describes how to perform a random walk on the simplex. ...
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20 views

What methodology should I choose? If hiearchical, what model design?

I am working on a problem that has can beyond my level of understanding. I am quite familiar with R, so that would be my preferred choice but I also have access to SAS. Data I have created a fake ...
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55 views

Simplex Random Walk

This link describes how to perform a random walk on the simplex using the Metropolis-Hastings algorithm: http://en.wikipedia.org/wiki/User:Skinnerd/Simplex_Point_Picking The description says: "The ...
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33 views

Analysis of mcmc output

I have a sequence of Bayesian Networks for del with more epochs. For example if i work with trafic on street I can have a BN for morning, one for afternoon e one for evening. If i collect data for ...
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157 views

Metropolis Random Walk Algorithm in Python

I am trying to implement random walk metropolis in Python for generate from a pdf $f$, which is defined in the domain: $$(0, \infty) \times [3, 4] \times (0, \infty)$$ I am using Metropolis Random ...
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72 views

Metropolis independence sampler

I need to implement an algorithm of Metropolis Independence Sampler, where the proposal distribution is a Multivariate Normal with 3 parameters. Since my function $f$ is really difficult to ...
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68 views

Proper proposal distribution in Metropolis-Hastings for a 2-state discrete process

This question is related to Understanding Metropolis-Hastings with asymmetric proposal distribution. I'm trying to draw sample paths from a 2-state markov chain using M-H. Unfortunately, I've only ...
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49 views

MC Standard Error for Metropolis Random-Walk Algorithm

I need to calculate some integrals using MCMC with Metropolis Random-Walk Algorithm. To decide the value I will accept as my integral, I will calculate 5 simulations of MCMC with size 2000, but I ...
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52 views

Metropolis Random Walk Algorithm - Covariance Matrix

I need to calculate an integral of a 3 variable density $f(x_1, x_2, x_3)$, defined in a domain $D$, in the region $$T=\{x \in R³|f(x) \geq M \}$$ using MCMC , where $M$ is a given positive constant. ...
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44 views

Metropolis Hastings when acceptance rate is not a probability

I need to implement a Metropolis Hastings where the acceptance probability $\alpha$ is not a probability but a logarithm of a score. The logarithm of a score is a negative float. In original ...
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111 views

Metropolis random walk using multivariate normal

I need to implement a program that generates a sample from a really complicated distribution $f$ of 3 variables. I need to implement it using Metropolis-Hastings Algorithm's and its variation and I ...
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33 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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170 views

MCMC with Metropolis-Hastings algorithm: Choosing proposal

I need to do a simulation to evaluate an integral of a 3 parameter function, we say $f$, which has a very complicated formula. It is asked to use MCMC method to compute it and implement the ...
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1answer
58 views

Simple question on acceptance rate in Metropolis Hasting

When we say acceptance rate in a MH algorithm (for examples, suppose it's recommended to have 40% acceptance rate), do we mean overall acceptance rate or we mean acceptance rate after burn in? It's a ...
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1answer
99 views

pymc random seed doesn't guarantee the same posterior samples?

If pymc.numpy.random.seed(0) guarantee the same random number sequence to initialize a stochastic variable (say a Uniform distribution), why does its posterior samples (from trace plot) don't have the ...
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113 views

Choosing an appropriate proposal distribution for metropolis hastings

So far the only constraint I've found for sampling from some target distribution $\pi(x)$ is that the proposal distribution must include the support of $\pi(x)$. That's very vague. What makes a ...
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65 views

Proper likelihood function in acceptance probability of Gibbs Sampler

I have a question about the acceptance ratio used when implementing a random walk M-H in a gibbs sampler to generate sample paths of an unobservable process. When computing the likelihood of a set of ...
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56 views

Metropolis Ergodicity

I have encountered one last problem with regarding to the Metropolis-Hastings algorithm. I know that ergodicity is needed in the algorithm to imply convergence to a unique stationary distribution. But ...
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63 views

Problems with acceptance ratio of MCMC when using a Kalman filter and conjugate-prior for sampling

1. Model I am trying to build a MCMC estimation of the following model (simplified): $\log(P^{-1}(obs_t, \sigma)) = \log(Y_t) + \epsilon_t$ where $\epsilon_t \sim \mathcal{N}(0,\sigma_{e})$. ...
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48 views

how to predict Yn value in this formula with Metropolis Hastings or Gibbs?

I have a model with this formula: $$ Y_n=aX_n^b + e_n $$ $$ X_n \in [0,2] \quad\quad a = 1.5 \quad\quad b = 0.5 \quad \quad e_n = N(μ = 0, σ^2 = 1) $$ I want to predict "$Y_n$" value with using ...
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73 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 ...
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136 views

Metropolis algorithm, what is the target distrbution and how to compose it?

When do Metropolis sampling or MCMC, we need a target distribution $P_{target}(\theta)$, and a proposal distribution $P_{proposal}(\theta)$, then a value $\theta_i$ is generated via ...
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602 views

Covariance matrix proposal distribution

In a MCMC implementation of hierarchical models, with normal random effects and a Wishart prior for their covariance matrix, Gibbs sampling is typically used. However, if we change the distribution ...
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60 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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1answer
175 views

Bayesian model averaging for variable selection in R

I am trying to use Bayesian model averaging for variable selection with a large number of variables. In R, the BMS package allows to apply the method, with the option of using MCMC sampler (Metropolis ...
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65 views

Random walk on simplex as part of Metropolis-Hastings

I would like to perform a random walk on a J-dimensional simplex. However, since this is part of a metropolis-hastings algorithm application, my understanding is that the steps need to be drawn from a ...
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745 views

How to sample using MCMC from a posterior distribution in general?

Assume one has the posterior distribution of a parameter, $p(\theta|y)$ and what I mean by having it is that for each point of $\theta$, one can use Monte Carlo method+MCMC to calculate the ...
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924 views

Implementing a Metropolis Hastings Algorithm in R

Consider a univariate normal model with mean $µ$ and variance $τ$ . Suppose we use a Beta(2,2) prior for $µ$ (somehow we know µ is between zero and one) and a $log-normal(1,10)$ prior for $τ$ (recall ...
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113 views

Perceived circularity in the Metropolis-Hastings algorithm - Where is my error in reasoning?

If I understood it correctly, the Metropolis-Hastings algorithm allows one to sample from a distribution without an analytical representation, which comes in handy, for instance in the Bayesian ...
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756 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, ...