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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Does MCMC Gibbs sampling algorithm first build a steady Markov Chain, then does the sampling to build the posterior distribution?

I am currently studying MCMC Gibbs sampling and while reading this part, a question has come into my head if MCMC Gibbs sampling first build a steady Markov Chain and does the sampling or does ...
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Gibbs sampling for simple posterior distribution?

I have a likelihood function, $$ p(x) = \theta^{\sum x} (1- \theta)^{n-\sum x} $$ and prior distribution, $$ p(\theta) \propto \theta^{\alpha - 1} (1- \theta)^{\beta - 1}$$ then the posterior ...
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Belief Propagation vs Gibbs Sampling

In general what are the cons and pros of using Gibbs sampling to estimate a complex posterior (assuming we can sample from the conditionals) over belief propagation (using a factor graph)?
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sampling question in Gibbs sampling for a Gaussian mixture model

I have some confusions regarding the Gibbs sampling step for the following mixture model: consider a mixture model of the following generative process: $\theta \sim Dir(\alpha) $ (global hidden ...
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the convergence speed for a Markov chain

For a metropolis hastings algorithm, suppose that the stationary distribution is defined as the Gibbs Boltzmann distribution $\pi_T(x)= \frac{1}{Z_T}e^{-\frac{V(x)}{T} }$ where $Z_T = \sum_{y\in V} e^{...
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Dirichlet Process mixture model with independent features

I'm trying to construct a Dirichlet process mixture model for clustering where the samples have independent features. In other words, to evaluate the likelihood of sample $x_i$, I would compute $\...
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21 views

Gibb's sampling where target prob distribution is itself a conditional joint distribution - p(x,y|t)

I'm new to Gibb's sampling and need basic guidance. Say p1,p2,q-> are Gaussian variables. p1->q<-p2 and q->x where x is a discrete variable (either 1 or 0). How do I go about Sampling (using ...
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sampling from joint distribution to recover marginal distribution

I'm going through Bayesian Core and have gotten stuck at this remark on page 233: " A first remark that motivates the use of the Gibbs sampler is that, within structures such as $$ \pi(x_1) = \int \...
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Any relationship between MAP full conditionals and MAP joint?

Context: Bayesian model that one can draw posterior samples from via Gibbs sampling of the relevant full conditionals. Question: Can anything be said (i.e. bounds, conditions for equality) about the ...
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Gibbs Sampling attempt at a simple Coxian distribution

I have the following Coxian model for inter-arival times ($x_i$) that has $C_x^2 < 1$: $$ p(x_i\mid \lambda,\theta) = \theta \lambda^r x_i^r e^{-\lambda x_i} + (1-\theta)\lambda e^{-\lambda x_i} $...
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Gibbs Sampler for GMM

In Rasmussen's paper it is introduced a Gibbs sampler to make inference about a standard Gaussian Mixture Model. To simplify, assume the 1-d case with basic hierarchical structure, that is: $x_i|...
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Shape of parameters marginal posterior in hierarchical Bayes model

Consider a generic hierarchical Bayes model with data $y_i\sim p(y|\theta_i)$, dependent of parameters $\theta_i\sim p(\theta|\phi)$ and hyperparameters $\phi\sim p(\phi)$. Furthermore, assume that $\...
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Inference of the collapsed Gibbs sampling for LDA

I am trying to understand the inference procedure of collapsed Gibbs sampling for LDA model. I refer to this document and LDA wiki page. I cannot figure out how does it simplify the sample equation ...
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Can Gibbs Sampling be used to generate synthetic data from the posterior distribution?

I am working with a few datasets. I would like to expand some of them somehow by creating synthetic data. Is it possible to use a Gibbs Sampler to, by sampling a given distribution of the original ...
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Metropolis-within-Gibbs for parametric inference of a regressive model

I have a regressive model of this form \begin{equation} y=f(\theta)+\varepsilon \end{equation} to describe observations $y$, with noise $\varepsilon$ and a parametric function $f$ with parameters $\...
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Conjugate prior for DPGMMs using Gibbs sampling

I am using Gibbs sampling to infer DPGMMs. The prior for multivariate Gaussians is Normal-inverse Wishart. But it turns out that the covariances are not estimated accurately. Here is codes and results....
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precision or variance of a Gamma distribution in a Gibbs Sampler?

I want to confirm my thinking on a quick question I have regarding the Normal-Gamma Gibbs sampler that we see so often, but I am unable to find a satisfactory answer. If we are interested in ...
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Gibbs sampling for drawing samples and estimating parameters

I'm learning Bayesian inference by myself and having a difficulty for understanding Gibbs sampling. From what I understood, Gibbs sampling is to draw samples from a given probability distribution $p(...
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Gibbs sampler: how can thinning equal to the number of iterations work?

I fit an LDA topic model, using the R package topicmodels. No hiccups and everything runs smoothly, my question here is conceptual. When controlling the Gibbs sampler, the default value (in the ...
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Gibbs sampling where I can only find the mode of conditionals?

I'm trying to solve a problem with Gibbs Sampling, so I'm trying to do: $$ x_1^1 \sim p(x_1 | x_2^0, x_3^0)\\ x_2^1 \sim p(x_2 | x_1^1, x_3^0)\\ x_3^1 \sim p(x_3 | x_1^1, x_2^1)\\ x_1^2 \sim p(x_1 | ...
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Determine frequent states

I have a distribution from which I can sample (namely, a Boltzmann Machine). Which methods exists to determine frequent states (states with high probability) / the most frequent state (state with the ...
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forward sampling for Bayesian network with continuous variables and equation-based causal relationships

I have a physical system which can be represented by the following Bayesian network. It has the following characteristics 1) The encoded variables are continuous variables 2) The causal ...
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Gibbs sampler for ARIMA AR(1) parameters: division by zero

Suppose the following AR(1) model: $$ y_t = \mu + \phi (y_{t-1} - \mu) + \epsilon_t $$ with $\epsilon_t \sim \mathcal{N}(0,\sigma^2)$. Following issue arises when sampling from $P(\mu_i \;|\; \phi_{...
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bayesian decision making - comparing expected loss

The problem is like this: Suppose that I am considering which country should I invest on, country A and country B, based on their GDP growth rate $\alpha$. There are two possible choices for each ...
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How to create a distribution and sample?

Suppose we are given some small set of data on bundles of electrical wires and increasing voltages run through them, and we note how many of the individual wires fail. So for example, a large data ...
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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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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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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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How does GibbsLDA++ ensure that we are sampling from a good posterior?

This is an extending of this question, which asked that whether we should do some estimating to ensure that we are really using a likely topic assignment instead of the one happened with low ...
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Uniform sampling of constrained binary vectors by Gibbs sampling

General statement of the problem: Let $x,y$ be two binary vectors, connected by the following constrains: $$y=f(x),\qquad x=g(y)$$ That is, $x$ determines $y$, and $y$ determines $x$. There are many ...
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Is it possible for Metropolis sampling to converge to the wrong value?

I have simulated data under three parameters of interest, say a, b, c. The prior I put on c was a Gamma, so it only takes positive values. The full conditionals of a and b are known distributions, but ...
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Gibbs sampling: ancillary and sufficient parametrization

After asking a question about Gibbs sampling earlier, I have another one for you. I have not been able to find laymen's background on this, the only referenced use I've found for this is in ...
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How does Shuffled-Complex-Evolution-Metropolis algorithm compare to other adaptive samplers (e.g. NUTS)?

I recently heard of the Shuffled-Complex-Evolution-Metropolis algorithm and am curious how it compares to other adaptive MCMC sampling algorithms. Unfortunately I am still learning about optimizing ...
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108 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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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 $Pr\left(\beta_i^{(k)}|\...
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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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How can I sample from the conditional distribution?

I am learning Gibbs Sampling, in which there is a step named sampling from conditional distributions. I don't understand: 1. where is the conditional distribution from? From a general case, how can I ...
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104 views

How to sample via blocked Gibbs the conjugate normal model?

i am trying to sample the two dimensional parameter $\theta=(\mu,\sigma^2)$, for the Normal model. I have the full conditionals being for the mean: $\pi(\mu_{j}|\ldots) \sim N\left(\frac{\xi_{j}\...
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Test for convergence within Gibbs sampler

I am running a Gibbs sampler for Multivariate Normal times Inverse Wishart posterior distribution with missing data imputation step. I am trying to check if my step of simulating covariance matrices ...
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Inferring GMM parameters with Gibbs Sampling

On my book, "Machine Learning A Probabilistic Approach". It's stated that is straightforward to derive a Gibbs sampling algorithm to fit a mixture model, especially if we use conjugate priors. So ...