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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254 views

What does it means of Normalization term of Gibbs distribution?

I am studying about Gibbs distribution concept and I am confusing a one term in that concept that is normalization term. According to the Hammersley–Clifford theorem, an random $x$ can equivalently be ...
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300 views

Gibbs sampling for inferring the parameters of a GMM

I came across the following in Kevin Murphy's "a probabilistic perspective on machine learning". I am struggling to understand the derivation of the conditional probability for $z_i$. I tried ...
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Can I use Adaptive MCMC in any setting?

In time series econometrics and finance, most Bayesian authors approximate their models with a Gibbs Sampler, this is especial true for state space models, SV and so forth. The dimensionality of ...
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250 views

Gibbs sampling version for estimating Hierarchical Double Dirichlet Process Mixture of Gaussian Processes

I'm trying to implement Gibbs sampling to estimate the parameters of the following non-parametric model: $$\begin{align*} \beta|\gamma & \sim \text{GEM}(\gamma)\\ k_t|\beta & \sim \beta\\ \pi|\...
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212 views

In MCMC simulation, how to deal with very small likelihood values that couldn't be represented by computer? [duplicate]

I am working on a Bayesian project based on Stagnant data from a OpenBugs example, which is a changepoint problem. Basically we assume a model with two straight lines that meet at a certain ...
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207 views

Conjugate prior for multivariate with known mean and covariance known to a constant

I have a linear trend model (evolving mean and slope) embedded in a larger state space time series model that I would like to constrain to be a spline. With that assumption, the mean and trend ...
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142 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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179 views

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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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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423 views

Practical Implementation of Gibbs Sampling in Latent Dirichlet 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. http://people.cs.umass.edu/~wallach/courses/s11/...
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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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Gibbs sampler on the precision (with a gamma prior) in a hierarchical Bayesian model doesn't converge

I am deriving a Gibbs sampler with a model similar to the model in this paper (a graphical model is shown in page 4). To put it simple, my question only concerns $w_i$ (a $K$-dimensional vector drawn ...
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243 views

Mathematical reference for the convergence in distribution of the Gibbs sampler

This question is in some sense the intersection of this question and this question. I have read up on the Gibbs sampler, and am now asking for an introduction to the Gibbs sampler for mathematicians. ...
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185 views

Gibbs algorithm using negative binomial produces NAs

I have the following full conditionals distributions: $$ X_2|X_1=x_1\sim Bin(x_1,p)\\ X_1|X_2=x_2\sim NegBin(x_2,p) $$ So I'm using the following code to generate a sample from each one: ...
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248 views

Gibbs sampling for reducible chain

I am new to Gibbs sampling and I ran into a problem with irreducibility. For the Gibbs sampler to work the Markov chain has to be irreducible. But that assumption is not satisfied in my probability ...
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231 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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425 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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328 views

Expected value of an estimate done using Gibbs sampling

This is related to this question. I am concerned about the expected value of a function estimated using sampling. If the samples were obtained by a sampling method such as Gibbs sampling, and if I ...
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984 views

Implementing Latent Dirichlet Allocation - notation confusion

I am trying to implement LDA using the collapsed Gibbs sampler from http://www.uoguelph.ca/~wdarling/research/papers/TM.pdf the main algorithm is shown below I'm a bit confused about the notation ...
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101 views

Deriving full conditionals from joint distributions?

In this link (https://www.youtube.com/watch?v=a_08GKWHFWo), the author derives the conditional distributions from the joint; but I got lost in the mechanics of what happened, the process was overly ...
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181 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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Gibbs sampling in R

Let X a continous variable and Y a binary variable with joint distribution : $$p(x,y;\beta,\rho_1,\rho_2,\phi_1,\phi_2)=\frac{1}{Z(\beta,\rho_1,\rho_2,\phi_1,\phi_2)}\exp(-0.5 \beta x^2+1_{y=0}\rho_1 ...
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124 views

Truncated prior leads to non-intuitive posterior

I am setting up a linear regression model for continuous data that is Normally distributed. For this model, I want to assume that my $\beta$ predictor is truncated to be positive, that is $$\beta \sim ...
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1k views

How to incorporate parameter constraints in Metropolis Hastings

I am working on parameter estimation of GARCH model with Metropolis Hastings. But the results I have got doesn't look reasonable, actually it is quite different from what I have got from Gibbs sampler....
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Unsupervised Bayesian naive Bayes

I'm reading a paper Gibbs sampling for the uninitiated. In this paper, the authors try to use Gibbs sampling for a bayesian naive bayes model. They formalize the model as a graphical model in page 8. ...
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2k views

Need help on Gibbs sampling with truncated normal and gamma

I am trying to use Gibbs Sampling to simulate a random sample from a joint distribution $f(\beta ,{{Z}_{1}},...,{{Z}_{75}},{{\lambda }_{1}},...,{{\lambda }_{75}})$, where the fully conditioned ...
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564 views

Posterior expression for Gibbs sampling

I am trying to estimate parameters of a two dimensional Normal distribution using Gibbs sampling. While it was very easy transform the posterior equation for mean vector to a single dimension normal ...
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209 views

Gibbs sampler for Dirichlet Process concentration parameter

I am trying to implement a Gibbs sampler for Hierarchical Dirichlet process, but I cannot seem to correctly estimate the concentration parameters. I therefore started testing just this part of a ...
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55 views

Distribution sampling when no analytic expression

The goal of gibbs sampling is to sample the joint distribution when this latter has not an analytic expression, by deriving the conditional distribution of each variable. So it is supposed that the ...
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517 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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116 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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664 views

How to do MC integration from gibbs sampling of posterior

I'm a beginner of MCMC. Two questions confused me. If I know the posterior distribution, and from the Gibbs sampling, I got the sampled parameter, so How to draw the histogram with y axis as ...
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377 views

Conditional posterior probability density (steps)

I'm trying to understand how to condition a probabilistic posterior distribution. Consider the following probability density: $$ p(\alpha, \beta | y) = \prod_{i=1}^n (\alpha+\beta t_i)^{y_i}e^{-(\...
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436 views

Collapsed Gibbs Sampling in Mixture Models

I tried to learn how Gibbs sampling works on Mixture models by studying David Blei's notes: http://www.cs.columbia.edu/~blei/fogm/2015F/notes/mixtures-and-gibbs.pdf In the equation 28: $p(z_i = k| ...
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947 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 $<x_ix_j>...
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777 views

Gibbs Sampling for Gaussian Mixtures

Does the Gibbs sampler converge to a global maximum in the presence of multiple modes? For example in case of a Gaussian mixture distribution?
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Sampling covariance matrix using Gibbs sampling

I am sampling covariance matrix from a Inverse Wishart distribution. In one dimensional case, after doing sufficient iterations I am taking the mode value for variance (after removing the burn-in ...
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46 views

Multi-steps direct forecasting in AR(2) model through bayesian estimation of the model

I'm estimating an AR(2) model using Bayesian methods through Gibbs sampling and I want to perform 4 step ahead multi-steps direct forecasts. Inside the MCMC loop in each iteration I'm drawing the ...
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52 views

Gibbs sampling example of a bivariate normal with unknown correlation

I'm looking for an example of using Gibbs sampling with a bivariate normal, where the correlation parameter is not fixed or known. In other words, what is the conditional distribution of the ...
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67 views

Stationarity of coefficients when sampling VAR by Gibbs sampler

I am using Gibbs Sampler for VAR and have noticed that some researchers check the stationarity of $\beta$ coefficients while drawing. I am not sure why do they do that? Bayesian VARs do not require ...
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150 views

What is the role of simulated annealing in Gibbs sampling?

While I was reading about Gibbs sampling, I happened to see "simulated annealing" but what is it doing in Gibbs sampling? Although I don't understand the full context of simulated annealing, I am ...
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44 views

Does Gibbs sampling require to know the partition function?

As the title suggest, does gibbs sampling require to know the partition function? For example, if I want to sample variable $a$ and I have worked out $p(a|rest) \propto f(a|rest)$ where $rest$ ...
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346 views

Conditional distribution in this Gaussian Mixture Model

Say I observe $N$ observations $\{x_1, \dots, x_N\}$ from a $k$ component Gaussian Mixture model, with $k > 0$ known and is such that each $x_i|\boldsymbol{\pi}, \boldsymbol{\mu} \sim \sum_{j=1}^{k}...
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814 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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110 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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1answer
238 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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628 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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733 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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478 views

Why Gibbs sampling?

I am new to Gibbs sampling and sampling in general, so here is a basic question. I am reading this tutorial. Equation (40) is our complicated joint probability and equation (49) the less complicated ...
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550 views

Generating Sample Path of HMM via Gibbs Sampling

I have a question regarding section 7.1.1 here. I have two questions to ask you. What does the following sentence mean? we shall be drawing values for $C_T,C_{T-1},\cdots,C_1$ in order. How the ...

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