Questions tagged [gibbs]

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 or group of variables. The name comes from the method being first used on Gibbs random fields modeling of images by Geman and Geman (1984).

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help interpreting plot of MCMC sample

I am estimating a model using MCMC (Gibbs Sampling). Because of the complexity of the model, I have been running two chains with many iterations. A plot of the draws for each parameter reveals a ...
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Gibbs sampling and Bayesian inference

I wanted to get a more in-depth understanding of sampling algorithms, and so I thought I could start with the very simple example of a binomial or bernoulli likelihood with a beta prior (since it is ...
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Burning and Sampling in Gibbs sampler [closed]

I am confused about the concept of sampling in the Gibbs sampler after the burn-in loops. This is the basic problem, I have a picture composed of 1's and 0's. This is a noisy version and I am trying ...
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300 views

When can the collapsed Gibbs sampler be implemented?

I understand Gibbs sampling is a means of statistics inference, and it seems that sometimes certain variables can be integrated out in the sampling process, known as collapsed Gibbs sampling. I really ...
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How to know whether a Gibbs sampler is irreducible? [duplicate]

How to know whether a Gibbs sampler is irreducible? I know that the Gibbs sampler in e.g. two variable case constructs a sequence of r.v.s $(X_1^{(i)}, X_2^{(i)})$ by sampling from the related ...
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How is the accuracy of the results of Gibbs sampler measured?

How is the accuracy of the results of Gibbs sampler measured? Most resources merely say to iterate it $k$ times. But how does one infer the accuracy of the result?
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Metropolis-Within-Gibbs sampling with only marginal distribution known for a subset of variables

Typically in Gibbs sampling we want to sample from a joint distribution $p(X_1, X_2, ..., X_N)$, but because the joint is hard to sample from directly, we instead achieve this by iteratively sampling ...
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Missing data in Gibbs sampling for dynamic linear models

Suppose I have the following DLM: $x_t = \Phi x_{t-1} + w_t$ $y_t = A x_t + v_t$ $x_0 \sim N(\mu_0,\Sigma_0)$ $w_t \sim N(0,Q)$ $v_t \sim N(0,R)$ Let $\Theta = \{\mu_0,\Sigma_0,\Phi,Q,A,R\}$. I ...
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1k views

Explanation regarding Gibbs Sampling

I am new to MCMC and reading a intro paper regarding Gibbs sampling. However, there are two parts in the paper I cannot understand and get stuck. The first part is equation 2.3 in page 168. It says ...
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19 views

Averaging Across Gibbs Sampling Runs with Reduced Dimensions

I need help thinking through my approach to Gibbs Sampling of many parameters and I'd like to know if there is literature on this topic: I have a dataset with 3 dimensions: ...
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339 views

Why does detailed balance not provide a stopping criterion in MCMC?

Like I undestand MCMC sampling, the fulfillment of the detailed balance equation guarantees that our MC has reached its stationary distribution (given we ensure ergodicity). Detailed Balance is: $\...
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544 views

deriving posterior conditionals for gibbs sampling

I'm new to Bayesian inference and Gibbs sampling in general, and I'm struggling trying to derive the conditional posteriors for a particular data generating process I'm trying to model. The model I ...
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270 views

Convergence of approximate Gibbs sampling

We have a bivariate random variable $(X,Y)$ for which sampling is challenging. If we were to know how to sample from the conditionals $(X|Y)$ and $(Y|X)$, we could get samples from the joint using ...
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1answer
66 views

Generating samples for $p(\theta_{i}|\pmb{x})$ if samples from $p(\phi|\pmb{x})$ are known

Suppose $X_{i}|\theta_{i} \sim D_{1}(\theta_{i})$ and $\theta_{i}|\phi \sim D_{2}(\phi)$. Moreover $\phi \sim D_{3}(c)$ where c is known. How would I generate samples for $p(\theta_{i}|\pmb{x})$ if I ...
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213 views

Gibbs sampler transition kernel proof [duplicate]

I'm trying to understand how the Gibbs sampling algorithm works. I've simplified it into a bivariate case to help my understanding but I'm unsure how to go from conditioning on $X^{t-1},Y^{t-1},X^t$ ...
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97 views

Sampling from conditional posterior - continuous and discrete terms

I'm working through Hierarchically Supervised LDA by Perotte et all (2011). The conditional posterior I'm supposed to sample values $z_i$ from, however, is zero almost everywhere. To see why, lets ...
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186 views

Sampling from a posterior with Gibbs sampling

In an image processing class, I dont really get behind the idea how to 'sample from a posterior' with Gibbs sampling. We have a posterior distribution: $f(z_1, .. ,z_n \mid x_1,.. ,x_n) := f(z \mid x)...
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Gibbs sampling for parameter estimation

I am reading the paper by Willemsen et al (2015), "A multivariate Bayesian model for embryonic growth", Statistics in Medicine, 34:8, 1351–1365 where they define the posterior distribution as, \begin{...
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566 views

Gibbs sampling an Ising model with 0s and 1s

One of my problems in one of my courses ask to sample a 20 dimensional vector of 0s and 1s, $\{0,1\}^{20},$ when they are distributed as $$ \pi(x) = \exp\left\{-\beta \sum_{i=1}^{19} |x_{i+1}-x_i| \...
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Gibbs sampling with expectations instead of sampling

I see there is something called Iterated Conditional Models (ICM), which is a sort of Gibbs sampling where, instead of sampling, we use the value that maximizes the conditional. That is: ...
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53 views

Priors on Taylor Expansion series

I'm wondering what priors can i choose for a Taylor series as follows: $\theta_{1}+\theta_{2} (y-\alpha) + \theta_{3} (y-\alpha)^2$ What priors should I use for updating these parameters ($\theta_{1},...
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221 views

Collapsed Gibbs sampler on Hierarchical Dirichlet Process Mixture Model

I am trying to design a collapsed Gibbs sampler on a mixture model based on Hierarchical Dirichlet Process ($g\sim DP(\gamma, b)$, $\pi\sim DP(\alpha, g)$ ). Should I resample from the posterior of ...
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114 views

Rewrite conditional formula with three variables using Bayes formula

In equation (5) on page 3 on this paper a conditional probability is rewritten using Bayes' formula. I started using this answer Can I rewrite conditional probability of three variables like this? ...
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769 views

Beta distribution parameter estimation: method of moments

In a paper: Topics over time, method of moments was applied to estimate $\alpha$ and $\beta$ for a Beta distribution. My question is that how $\alpha$ or $\beta$ should be calculated if there are no ...
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72 views

Sampling from multivariate normal conditional on a negative minimum

Let $X\sim \mathcal{N}(\mu,\Sigma)$, where $\mu\in\mathbb{R}^n$ and $\Sigma\in\mathbb{R}^{n\times n}$. How can I efficiently sample from $X | {\min{X}\le 0}$? (I.e. from the distribution of $X$ ...
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Gibbs sampler from from $p(x) = C g(x)$ with $C$ unknown and discrete elements in $X$

By using the Hasting-Metropolis method, is there a way to draw samples from a distribution of this form: $$p(\textbf{x}) = C g(\textbf{x})$$ For $x$ being two dimensional and discrete. The reason that ...
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290 views

Why does the redundant mean parameterization speed up Gibbs MCMC?

In Gelman & Hill (2007)'s book (Data Analysis Using Regression and Multilevel/Hierarchical Models), the authors claim that including redundant mean parameters can help speed up MCMC. The given ...
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466 views

Influence of word counts from DTM on LDA with Gibbs Sampling

I'm trying to wrap my head around Topic Modeling based on LDA with Gibbs sampling (Griffiths, Steyvers 2004: Finding Scientific Topics). What struck me when reading some Python implementations like ...
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200 views

Gibbs sampling from conditional full posterior distribution

I am reading the paper by Willemsen et al (2015), "A multivariate Bayesian model for embryonic growth", Statistics in Medicine, 34:8, 1351–1365 where they define the posterior distribution as, \begin{...
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1answer
273 views

Transition probabilities for Gibbs Sampling in a Markov Random Field

I am currently reading this paper on Restricted Boltzmann Machines. On page 22, Given a Markov Random Field $\mathbf{X} = (X_1,\ldots,X_N)$ w.r.t a graph $G = (V,E)$ where $V = \{1 \ldots N\}$ and $...
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350 views

which R package is good for gibbs sampler when the likelihood function is complex

I see a lot of examples using MCMC to solve for posterior distribution when the likelihood is simply one of linear regression. What if the likelihood is an ugly, complex function. Which R package can ...
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153 views

use inverse Wishart for variance in MCMC

When you have a posterior that looks like as one step in Gibber Sampler $P(\xi | \Sigma_\xi, \theta) ∝ exp\{-1/2 \xi\Sigma_\xi^{-1}\xi\}P(data | \xi, \theta)$ Do you always assume inverse Wishart ...
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Estimation of Bayesian Models

I'm trying to get into Bayesian model estimation (I'm interested in posterior parameter distributions). I could get away with Metropolis-Hastings and Gibbs Sampling for models with few parameters (<...
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302 views

Gibbs sampling in the Hierarchical Dirichlet Process

For an inference problem using a Dirichlet Process prior, one can derive a "basic" Gibbs sampling scheme, where we have a conditional for any parameter $\theta_i$ given the samples $x_i$ and all the ...
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Get different results with different sampling order in Gibbs sampling: what could be wrong?

In sampling a complex spatio-temporal model by Gibbs sampling, I found if I change the order of sampling (for example, to sample $P(\theta_1,\theta_2|D)$, in one try, I sample $\theta_1\sim P(\theta_1|...
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Gibbs Sampling / Monte Carlo with Weights

Consider pairs of data and their population weights $(y_i, \omega_i), i = 1, 2, \dots$ alongside some hierarchical structure, $$y_i\leftarrow\theta_{i(j)} \leftarrow \gamma$$ where $i(j)$ is perhaps ...
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135 views

Eq. 45-46 in Gibbs Sampling for the Uninitiated

I am trying to figure out how Eq. 45 simplifies to Eq. 46 in the paper - "Gibbs Sampling for the Uninitiated" by Resnik and Hardisty.www.cs.umd.edu/~hardisty/papers/gsfu.pdf (page 15) Eq. 45 $$ \...
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141 views

Is Coordinate Ascent algorithm related to Gibbs Sampling in some way?

I wonder if only me feel there are certain connections between them, I googled it for a long time, but found no where mentioned these two method. But to me, they indeed looked so related, Could anyone ...
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331 views

Gibbs Sampler, Bivariate Normal, Subchain

In Example 7.1 of "Introducing Monte Carlo Methods with R", the authors write $(X,Y)\sim N\Bigg((0,0),\begin{pmatrix}1 &\rho \\ \rho & 1\end{pmatrix}\Bigg)$ Then, Given $x_t$, $Y_{t+1}\mid ...
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82 views

Gibbs Sampler Consistency

Simple question that I haven't found easily online: why are the estimates obtained from a Gibbs sampler consistent (converge to the true probabilities)?
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11k views

Gibbs sampler examples in R [closed]

How can I implement Gibbs sampler for the posterior distribution, and estimating the marginal posterior distribution by making histogram?
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51 views

Gibbs sampler -transformation of conditional posterior

If my conditional posterior $\pi(\sigma^{-2}|\mathbf{y },\mu)\sim Gamma(a,b)$, how can I get the conditional posterior $\pi(\sigma^{2}|\mathbf{y },\mu)$ with a transformation? The reason I ask is ...
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26 views

Sampling from a truncated random effects distribution

How would one sample observations $T_{ij} = U_i + \varepsilon_{ij}$, where the distribution of $U_i, \varepsilon_{ij}$ are known and mutually independent, condition on the fact that $L_{ij} \le T_{ij} ...
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1answer
27 views

Gibbs Sampler while fixing one parameter

I came across a problem for Gibbs sampler. Suppose I want to draw samples from $f(x,y,z)$, Can I use the following scheme to draw samples? Step 1, draw $f(x^{2t-1},y^t|z^{t-1})$. Step 2, draw $ f(...
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1answer
185 views

What is a hierarchical model that can estimated via the Metropolis-Hastings Algorithm but not the Gibbs Sampler?

My understanding of the differences between MH and Gibbs Samplers is that a Gibbs Sampler is usually used when the full conditionals are present to us. In other words, it is a known distribution, so ...
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342 views

Sampling from the joint distribution p(x,y) when y = f(x)

Suppose I want to sample from the joint distribution $p(X, Y)$, where $X$ is a random variable and $Y = f(X)$ where $f$ is a known function of $X$. However, sampling from $p(X,Y)$ directly is hard. ...
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367 views

Gibbs sampling on the use of gamma distribution

I am trying to reproduce a Gibbs sampling but some points are unclear to me. Let's assume a linear model of the following form $$y=x\beta + \epsilon$$ The prior distribution for the beta ...
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692 views

Bayesian estimation of Dynamic Regression with AR(1) parameters

I would like to draw (Bayesian) inference in a dynamic linear regression with regression parameters following independent AR(1) processes $\beta_{t,i} = \mu_i+\beta_{t-1,i}+w_{t,i}$. However, I ...
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2answers
156 views

throwing away all Gibbs samples after approximation

This is more of a theory question, consider: $$P(w_1|D)=\int P(w_1|S)P(S|D)d(S)$$ which we approximate via Gibbs sampling $S$ (assume the initial state of the Gibbs sampler is denoted by $M_0$), ...
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245 views

stationarity of vector autoregression and Gibbs sampling

I'm estimating a vector autoregression (VAR) using Gibbs sampling. At each iteration, I'd like to check the coefficients to ensure the VAR is stationary. An older, related question has been posted ...