# 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).

420 questions
Filter by
Sorted by
Tagged with
9 views

### Meaning and importance of 'Gibbs update' in MCMC

I am studying MCMC by "Handbook of Markov Chain Monte Carlo" by Brooks, Gelman This book is nice to explaining many fundamental concepts regarding MCMC. Especially in first chapter, they ...
40 views

### Block sampling hidden state using forward algorithm only

In a hidden Markov model, I can't get my mind around why I can't sample the full hidden state $\vec x$ using only a forward sampling algorithm. Let $\vec y$ be the observed data and $\theta$ the model ...
56 views

### Bayesian model (Gibbs sampling) with frequency data for non-standard distribution

I have the following model. \begin{align} X &= D\,(F+S\,(1+G)) \\ D &\sim \mathrm{Exp}\,(\lambda) \\ F &\sim \mathrm{Unif}\,(0,1) \\ S &\sim \mathrm{Poi}\,(\sigma) \\ G &\sim \...
10 views

### When to stop MCMC within collapsed Gibbs?

I am setting up a Hierarchical model whose target distribution is $p(\theta,w|y)$, $\theta$ being a reduced set of high-level parameters, $w$ being a data augmentation of very high dimension, and $y$ ...
44 views

### Random scan Gibbs sampling as special case of Metropolis-Hastings

I am reading Blitzstein's Introduction to Probability and come across with the following proof that I don't really understand: Theorem: The random scan Gibbs sampler is a special case of the ...
12 views

### Is it possible to increase the Hastings ratio by combining and mixing elementary kernels?

Let's say I am working with a state $X$ split into three parts $U$, $V$, and $W$. I can efficiently sample from $W|U,V$, $U|V$, and $V|U$. My initial intuition was to do a variable-at-a-time ...
10 views

1 vote
30 views

1 vote
55 views

63 views

34 views

### How much randomness is required for Gibbs sampling?

I am attempting to parallelize a program that executes hundreds of calls to Mallet's getSampledDistribution method, which is essentially an execution of Gibbs sampling over a topic distribution which ...
1 vote
87 views

### Is it possible to estimate the parameters of a superposition of Poisson processes through Bayesian inference from a binarized sequence?

My question is complementary to a previous problem : Bayesian inference on binarized Poisson distribution. I retake the previous notations. Problem description : I am counting the number of balls ...
124 views

I am drawing samples from my posterior, $P(x,y|z)$, using Gibbs sampling. When I sample $x$, I use a Metropolis-Hastings step. My question is whether I am allowed to use a proposal distribution for $x'... 2 votes 1 answer 248 views ### Distribution of conditional posterior for Gibbs sampling The following is a description of how the authors (Yongning Wang & Ruey S. Tsay) of this (2019) paper Clustering Multiple Time Series with Structural Breaks want to perform Gibbs sampling to ... 1 vote 0 answers 109 views ### Mean of skew normal distribution with normal prior obtained with Gibbs sampling I would like to obtain a new mean$\mu$of a skew normal distribution with a normal prior of the form$N(\delta,\tau)$on$\mu$, and a given standard deviation$\sigma$and shape parameter$\alpha. ... 2 votes 2 answers 273 views ### Bayesian updates for Dirichlet-multinomial with Gamma prior Let \begin{aligned} X_i &\sim \text{Dir-multinom}(X\mid\lambda)\\ \lambda_{j} &\sim \text{Gamma}(\lambda_j\mid\alpha,\beta)\\ \end{aligned} wherei$iterates over observations,$j$... 2 votes 0 answers 45 views ### Recovering samples from a density estimation with an additional prior on the samples. Used for Gibbs sampling Abstract Idea: Given a noisy measured density ($d_j$at position$p_j$) and a density model, sample from the model parameters under the following stochastic model: Stochastic Model: Prior for model ... 0 votes 0 answers 169 views ### Hyperprior in Gibbs Sampling Following up from this question, I have managed to derive the following posterior distributions$$\lambda_z | \boldsymbol{y}, \Theta^{(-\lambda_z)} \sim Gamma(a + \sum_{i=1}^{n_z} y_{ij}, \quad a + ... 0 votes 1 answer 520 views ### Poisson-Gamma Hierarchical Model I am fairly new to Gibbs Sampling and I am trying to build a Gibbs Sampler for a Poisson-Gamma hierarchical model. In this model, there are$m$restaurants in a city, with$n_z$number of observations ... 0 votes 1 answer 121 views ### Question about paper on Bayesian Shrinkage Estimation I am reading the paper Bayesian Shrinkage Estimation of the Relative Abundance of mRNA Transcripts Using SAGE, and I am trying to work out the calculations for the complete conditionals for the Gibbs ... 2 votes 0 answers 538 views ### Gibbs Sampling - why converge to stationary distribution Currently, I am going through Chapter 12.3 of Probabilistic Graphical Models - Principles and Techniques which talks about MCMC sampling methods. In Chapter 12.3.4.1, it states the following theorem: ... 1 vote 2 answers 380 views ### Question about a mixture dirichlet MCMC model I am self-learning Bayesian statistics using the book Computational Bayesian Statistics by Turkman et al. and I am currently stuck on Chapter 6 Problem 10. It can be found here on page 124. I am ... 0 votes 1 answer 542 views ### Beta-Binomial Gibbs Sampler I am self-studying Bayesian statistics from the book Computational Bayesian Statistics by Turkman et al, but I am stuck on Problem 6.3 from the book: Suppose we want to consider a Binomial (unknown$\... 319 views

### Metropolis-Hastings algorithm for logarithmic probability density

Similar question to posted here: Metropolis-Hastings using log of the density however my question is around sampling a random number from a uniform distribution. I am following the steps outlined in ...
386 views

### Why Gibbs Sampling for mixture models?

I am studying MCMC and in the book I'm reading there is this example on Gibbs algorithm for inferring the posterior of a gaussian mixture. I understand how the algorithm works and the fact that its ...
195 views

### Bayesian analysis example with convergence under Gibbs but not Metropolis-Hastings

Having a conceptual understanding of algorithms such as Metropolis-Hastings, Gibbs and Hamiltonian Monte Carlo can provide ideas of remediation to apply when models do not converge. This question ...
1 vote
Suppose $\textbf x \sim\mathcal N\left(\begin{pmatrix}1\\1\end{pmatrix}, \begin{pmatrix}1&-1/2\\-1/2&1\end{pmatrix}\right)$. Derive the full conditionals $p(x_1|x_2)$ and $p(x_2|x_1)$. ... 