The Gibbs sampler is a simple form of Markov Chain Monte Carlo simulation, widely used in Bayesian statistics, based on sampling from full conditional distributions for each variable.

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How can I sample multivariate binary variables such that sum of them follows a gamma distribution?

I want to sample multivariate binary variables $\mathbf x$ ($x_d \in \{0, 1\}$) such that sum of them follows a given gamma distribution (i.e. $\sum_d x_d \sim \mathrm{Gamma}(\alpha, \beta) $ where ...
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327 views

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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30 views

MCMC sampling with sum constraints

I'm interested in sampling a collection of variables with a sum constraint on them. For a simplified example: Prior: $X \sim \mathcal{N}(0, 1)$ $Y \sim \mathcal{N}(0, 1)$ Observation: $X + Y = 1$ ...
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Gibbs sampling version for estimating the Dynamic Topic Model (DTM)?

The paper of Blei et Lafferty published at ICML'06 implements a (quite complicated) variational inference (VI) technique for estimating the parameters of the Dynamic Topic Model, see: ...
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27 views

Gibbs Sampling for Boltzmann Machines

David Mac Kay, in his book on machine learning talks about Boltzmann machines, and on pg. 3 here http://www.inference.phy.cam.ac.uk/itprnn/ps/521.526.pdf He says "the second equation ...
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41 views

When should we use Gibbs Sampling in a deep belief network? Before or after fine-tuning?

Gibbs sampling allows for sampling a vector with a deep belief network. There are two steps to training a DBN for a supervised learning task: greedy unsupervised pre-training and supervised ...
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41 views

How to derive Gibbs sampling?

I'm actually hesitating to ask this, because I'm afraid I will be referred to other questions or Wikipedia on Gibbs sampling, but I don't have the feeling that they describe what's at hand. Given a ...
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189 views

Marginal Likelihood from the Gibbs Output

I'm reproducing from scratch the results in Section 4.2.1 of Marginal Likelihood from the Gibbs Output Siddhartha Chib Journal of the American Statistical Association, Vol. 90, No. 432. (Dec., ...
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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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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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69 views

Need help deriving a gibbs sampler for a normal mixture model with two components

Let $\theta_i$ be an indicator that the i-th eruption is a long eruption. (i.e. $\theta_i = 1$ if the i-th eruption is long and $\theta_i = 0$ otherwise.) Assume the following model and derive a ...
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21 views

Gibbs Sampling with given Posterior Distribution

I'm trying to implement an algorithm from a paper which assigns three types of labels $m_i, m_d$ and $m_s$. Here $m_i$ labels a collection of documents $G_i , m_d$ a subcollection of them and $m_s$ ...
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Sampling from a portion of the normal distribution?

I have a a conditional distribution $p(X_1 | \theta) \propto MVN(\mu, \Omega) \pi(X_1)$ where $X_1=[x_1, x_2, \dots, x_n]'$ and $\pi(X_1)=1$ when all $x_i \in [0,a)$ and $0$ otherwise. Is there any ...
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67 views

Gibbs Sampling and Probability Notation

Problem 1 I am trying to implement Gibbs Sampling for the following problem: There is a grid measuring 3 x 3 sites, each "site" can be designated in a state, $X$, of 1 or -1. The sites are numbered ...
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23 views

Gibbs within Metropolis

Consider a model with two parameters, $\alpha$ and $\beta$. We want to sample these two parameters conditioning on two data points, $d_1$ and $d_2$. Is it possible to use an algorithm like this: 1) ...
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48 views

Methods of fitting a dynamic linear model

I'm taking a time series course and am learning about exchangeable time series form of dynamic linear models (DLMs). This is given by: \begin{align*} \mathbf{y}_t' &= ...
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Gibbs Sampling form Non-standard distribution in R

Gibbs sampling examples in R has involved initializing and updating iteratively from conditional that are in standard form. Has anyone performed on a gibbs sampling when the conditionals are in ...
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132 views

Does the Gibbs Sampling algorithm guarantee detailed balance?

I have it on supreme authority1 that Gibbs Sampling is a special case of the Metropolis-Hastings algorithm for Markov Chain Monte Carlo sampling. The MH algorithm always gives a transition probability ...
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84 views

Is my OpenBUGS / WinBUGS model well specified?

I've just started trying to use OpenBUGS for Bayesian analysis of stochastic volatility models. In particular, I'm trying to calculate stochastic covariance, similar to the DC-MSV model specified by ...
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126 views

Bayesian modeling using multivariate normal with covariate

Suppose you have an explanatory variable ${\bf{X}} = \left(X(s_{1}),\ldots,X(s_{n})\right)$ where $s$ represents a given coordinate. You also have a response variable ${\bf{Y}} = ...
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1answer
42 views

Posterior parameter distribution

I am considering the following non-linear state space model: $X_t=\frac{X_{t-1}}{2}+25\frac{X_{t-1}}{1+X_{t-1}^2}+8\cos{1.2t}+\epsilon_t, \epsilon_t\sim N(0,\sigma_x^2 ) $ ...
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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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STEP BY STEP approach to learn LDA (Latent Dirchlet Allocation)

I have an urgent need to understand Latent Dirchlet Allocation (LDA) for Topic Modeling. I tried several sources, but it seems I do not have required knowledge to learn this. My statistics knowledge ...
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57 views

Is Slice Sampling a special case of Gibbs Sampling?

I read on this thread the following: If you can use both the gibbs sampler and slice sampling to sample from a posterior I would use the Gibbs sampler as the slice sampler seems unnecessary to ...
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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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References on deterministc augmented variable for Gibbs sampling

According to Wikipedia, It is also possible to extend Gibbs sampling in various ways. ... It is ... possible to incorporate variables that are not random variables, but whose value is ...
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Gibbs sample from AR(1) of exogenous input

I am trying to fit a model where there is a sequence of exogenous "shocks", $X_1, X_2, ..., X_T$, and a AR(1) of these shocks explain $Y_1, Y_2, ..., Y_T$. Specifically, Data (known): $X_1, X_2, ...
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63 views

Mixture of probits: understanding truncated-based likelihoods

I am trying to implement a mixture model of probits to infer the best decision boundary for every latent subpopulation. When doing Gibbs sampling, we eventually have to compute $P(y^* | w_c)$ where ...
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53 views

Gibbs Sampling Inserting Some Known Predictors

Imagine you would like to use a simple Gibbs sampling to resample from a joint probability distribution which is difficult to model (but you know all the conditionals ...
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46 views

MCMC for Bayesian Inference (Gibbs Sampling) Varying Observed Data

At every step $k$, a Markov chain Monte Carlo algorithm for Bayesian inference with Gibbs sampling draws a parameter of the model to fit, $\beta_i^{(k)}$, from the conditional ...
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50 views

Gibbs sampling for correlated random variables

Short summary Suppose two latent variables of a hierarchical model are correlated. Let $1-\epsilon$ be the degree of correlation. As $\epsilon\rightarrow 0$ the variables become perfectly correlated ...
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69 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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71 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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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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44 views

Practical Implementation of Gibbs Sampling in Latent Diriclet Allocation

In the collapsed Gibbs sampling version of LDA, the posterior distribution of topic assignments for each word is sampled. From what I have read (e.g. ...
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80 views

Kalman filter conceptual question

I'm using the function dlmGibbsDIG (Gibbs sampler) in the dlmpackage from R to estimate the unknown variances. The output are ...
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1answer
125 views

Can I subsample a large dataset at every MCMC iteration?

I have a large dataset from which I want to perform a bayesian probit regression using Gibbs sampling 1. Since the dataset has one milion rows, and variables from a truncated normal must be sampled ...
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58 views

What will happen if we sample the most probable value in the Gibbs sampling?

I am now working with the Gibbs sampling. One problem that puzzled me is that when we use the Gibbs sampling, we always sample randomly from the conditional probability. What will happen if we sample ...
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59 views

Gibbs sampling with Log-Normal observations

I am writing a Gibbs sampler for data that is Log-Normal (LN) distributed, with unknown mean and variance. There is a wealth of information on inference for LN models when either the mean or variance ...
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265 views

Gibbs sampling for Ising model

Homework question: Consider the 1-d Ising model. Let $x = (x_1,...x_d)$. $x_i$ is either -1 or +1 $\pi(x) \propto e^{\sum_{i=1}^{39}x_ix_{i+1}}$ Design a gibbs sampling algorithm to generate ...
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350 views

Gibbs sampling how to sample from the conditional probability? Bayesian model

I want to learn Gibbs sampling for a Bayesian model. How can I sample the variable from the conditional distribution? In this example, arrow means dependent; for example, ...
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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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1answer
129 views

Gibbs sampling with Dirichlet Likelihood

I have a sequence of observations that I am representing as proportions: X1 X2 X3 X4 X5 0.10 0.20 0.50 0.12 0.08 0.07 0.24 0.55 0.04 0.10 ... ...
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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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44 views

EM on product of multinomials

I have the following conditional density: $$ P(x | \theta, \pi) = \prod_{i=1}^I \prod_{j=1}^J t_{ij}! \prod_{k=1}^K \frac{1}{x_{ijk}!}(\sum_{l=1}^L \theta_{il} \pi_{jkl})^{x_{ijk}} $$ Here, $x$ is ...
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93 views

Gibbs measure and normal distribution

On Wikipedia, the Gibbs measure defines the probability as: $$ P(X=x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$ Now, the familiar form of the normal distribution is: $$ P(x) = ...
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65 views

Gibbs sampler for conditionals that are exponential: Example from Casella & George paper

I am trying to work out Example 2 from Casella and George's paper "Explaining the Gibbs Sampler" in R. The example is: ...
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440 views

Gibbs sampling for LDA — does a small Dirichlet concentration parameter make a difference?

I'm using a Gibbs sampler for Latent Dirichlet allocation as described by Griffiths and Steyvers (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC387300/). The sampling of a new topic $j$ for word $i$ is ...
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198 views

How can I estimate the precision of a normal using a Gibbs sampler?

I am trying to estimate the precision $\tau$ of a normal distribution with either WinBUGS or OpenBUGS: $c \sim \text{normal}(\mu,\tau)$ $\mu \rightarrow \lambda \cdot t^{-\beta}$ $\tau \sim ...
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1answer
159 views

What are the main differences between classical and Gibbs sampling Latent Dirichlet Allocations?

In these weeks I have been studying the classical Latent Dirichlet Allocation (LDA) algorithm by David Blei and colleagues (2003), and the LDA variant based on Gibbs sampling introduced by Tom ...