Markov Chain Monte Carlo refers to a class of methods for generating samples from a target distribution by generating random numbers from a Markov Chain whose stationary distribution is the target distribution. MCMC methods are typically used when less computationally expensive methods for random ...

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Why is specifying the distribution of the data different from specifying the data itself?

I am performing MCMC with pyMC on a nonlinear model, specified in Probabilistic modelling MCMC question with pyMC Imagine I have 2000 points of experimental data, normally distributed: ...
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Proving that Markov Chain Monte Carlo converges

I actually asked the same question in http://math.stackexchange.com/ as well at http://math.stackexchange.com/questions/753105/proving-that-markov-chain-monte-carlo-converges but since the question is ...
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Fitting multiple logistic functions at once with pymc

I have read a similar post here Fitting logistic function with pymc but it seems there is a rule that I shouldn't ask question in someone else's post. My approach is to fit 10 logistic functions to ...
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Why is Sampling Importance Resampling (SIR) better than Importance Sampling (IS)?

From what I understand, SIR is a mechanism for sampling from a distribution $p$ that works as follows: Approximate a target distribution $p$ using an importance sample $S$ from a proposal ...
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Can effective sample size in MCMC simulation be greater than the actual sample size?

I used coda package's effectiveSize() to find the effective sample size of my MCMC simulation. My effective sample size is greater than the actual sample size, e.g. ...
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Switchpoint detection with probabilistic programming (pymc)

I'm currently reading the Probabilistic Programming and Bayesian Methods for Hackers "book". I've read a few chapters and I was thinking on the first Chapter where the first example with pymc consist ...
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Can MCMC iterations after burn in be used for density estimation?

After burn-in, can we directly use the MCMC iterations for density estimation, such as by plotting a histogram, or kernel density estimation? My concern is that the MCMC iterations are not ...
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Advice on sensitivity analysis for priors in Bayesian statistics

I'm not clear on how to perform sensitivity analysis on the priors. Many sites have different answers. One site indicates to perform three non-informative, weakly informative and known priors. Another ...
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43 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
36 views

Probabilistic modelling MCMC question with pyMC

This is my first post and I am a newby in pymc. I am trying to model a non-linear system (see below for a further explanation). I create my synthetic data with: ...
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174 views

MCMC methods - burning samples?

In MCMC methods, I keep reading about burn-in time or the number of samples to "burn". What is this exactly, and why is it ...
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42 views

Analytical formulation of a Hierarhical Bayes problem

In the free online book Bayesian Methods for Hackers, the last figure shows the estimation of the expected value of $\lambda$ for any given day: It looks like the author is calculating the expected ...
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Metropolis Algorithm

I am currently doing my statistics thesis on modelling football data which requires quite a great knowledge of Bayesian Theory especially MCMC methods. However I have some minor problems regarding ...
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Problems with acceptance ratio of MCMC when using a Kalman filter and conjugate-prior for sampling

1. Model I am trying to build a MCMC estimation of the following model (simplified): $\log(P^{-1}(obs_t, \sigma)) = \log(Y_t) + \epsilon_t$ where $\epsilon_t \sim \mathcal{N}(0,\sigma_{e})$. ...
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1answer
148 views

Can we change the acceptance rate in random walk Metropolis algorithm by changing the parameter of the proposal distribution?

Can we change the acceptance rate in the random walk Metropolis algorithm by changing the parameter of the proposal distribution? Let the target distribution be $\pi$. Let $p(x_2 | x_1)$ be the ...
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What does “sampling period” mean in MCMC?

In MCMC, Use a single chain with a burn-in period of 100 iterations and a sampling period of 1000 iterations. If I am correct, the "burn-in period" is the length of the first part of a sample ...
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Correlation of fixed effects with multiple response variables in MCMCglmm

I'm working with a mixed model for which I have several response measurements for every individual. One goal is to determine the sampling variance/covariance of the fixed effect estimates for a ...
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1answer
35 views

Sampling of beta in bayesian regression (variable selection)

I am sampling a beta using a Gibbs sampling. It is variable selection model. So in different iteration of gibbs different covariates are included to the model (denoted by a variable selection ...
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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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Can I use one conjugate prior for samples from two normal distributions with same variance but different mean?

I have got a set $\{Y_t\}$ of observations consisting of two subsets $\{Y_{t,1}\}$ and $\{Y_{t,2}\} \subset \{Y_t\}$ with $\{Y_{t,1}\} \sim \mathcal{N}(\mu_1,\sigma^2)$ and $\{Y_{t,2}\} \sim ...
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70 views

Only allow positive MCMC-samples: Implications for credible interval

We calculate a Bayesian model and only expect positive values for our parameters. Our prior is however a uniform prior—we get negative samples from MCMC. For Bayes-factor calculations we use the ...
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71 views

Parameter fitting with STAN?

I have a model that produces data given a set of parameters. Now, given data, I'ld like to find out which parameters of the model are likely. I have an implementation in Matlab that uses Delayed ...
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Annealing MCMC with constant temperature?

Let's say I'm trying to sample from a posterior distribution, $p(\theta|x)$, which I suspect is very "bumpy" (10s to 1000s of small modes). Moreover, the evaluation of the posterior kernel ...
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1answer
60 views

Particle filters and loopy belief propagation

I want to implement a loopy belief propagation algorithm for factor graphs with continuous variables and messages represented using particles, that is vectors of samples for an empirical ...
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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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Specifying a multilevel model in MCMCglmm (R), that is heteroskedastic at level one

I am considering MCMCglmm as an alternative to MLwiN. The former package works perfectly fine, but I cannot figure out how to model heteroskedasticity at level one. For instance, if I have the ...
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Regarding three R packages for Bayesian analysis [duplicate]

There are several R packages for Bayesian analysis, i.e., RBugs, JAGS,MCMCPack. Are there ...
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1answer
61 views

MCMC estimation in mixed models: is there something like “significance”?

I am a bit confused in determining whether to keep or remove some of the random coefficients in a mixed model. There are quite few level two units, therefore I have to use MCMC estimation. As result, ...
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44 views

Conditionals for Gibbs sampling of relational clusters

I'm trying to implement a Gibbs sampler, but I'm having trouble to find some of the conditionals of this model. Model We have $A$ actors, $K$ classes or clusters, and a matrix $\phi$ that determines ...
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Detecting convergence in Random walk

I am trying to detect convergence of a random walk on a graph. After doing some preliminary research, the Geweke convergence diagnostic seems to be most commonly used for this. This diagnostic calls ...
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81 views

Detecting latent relational clusters (a.k.a blockmodeling) (PyMC)

By looking at the set of relationships within a community, we might discover that we can divide them in groups where people in the same group (a.k.a block or role) tend to relate to the same other ...
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60 views

How is the number of particles decided in particle filtering?

Say the observations are $x_i$ and the states are $y_i$ in a sequential model. I understand that particle filtering works by generating "particles" from $p(y_i | x_1,\ldots,x_i)$ for approximating ...
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Is this JAGS model ok, and can it be made faster?

I'm very new to Bayesian analysis, and I've come up with the following model. My goal is to get for each individual "test unit" a distribution that describes the lift in success rate under one of ...
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Speed of convergence of Markov chain Monte Carlo

How can I say better for a MCMC in terms of speed? I'm confused with several measures of speed of convergence. I know some people compute the second largest eigenvalue of transition probability ...
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Metropolis algorithm, what is the target distrbution and how to compose it?

When do Metropolis sampling or MCMC, we need a target distribution $P_{target}(\theta)$, and a proposal distribution $P_{proposal}(\theta)$, then a value $\theta_i$ is generated via ...
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1answer
62 views

Hamiltonian Monte Carlo: why is reparameterizing needed?

In the Stan user's manual (Version 2.0.1, page 157), it says A hierarchical model such as the above will suffer from the same kind of inefficiencies... [for a Hamiltonian Monte Carlo method] ...
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Formal statistical test for comparing likelihood distributions obtained via MCMC

I am trying to formally compare the distribution of the likelihood values generated using two different models with marginal posterior values of the parameters obtained using MCMC in order to assess ...
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Is the MCMC simply a probabilistic gradient descent?

I'm learning about Markov Chain Monte Carlo methods, and to my undifferentiated mind, they basically resemble gradient descent with a stochastic component replacing the gradient computation. Is this a ...
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60 views

How to sample from this dirichlet distribution with an L1 prior?

I'd like to draw a sample from a distribution with p.d.f $$f(p,q,r,s) \sim \mathrm{e}^{-w(|p+q-r-s|+|p-q-r+s|+|p-q+r-s|)}p^aq^br^cs^d \mathbb{1}_{p+q+r+s=1}$$ $w > 0$ is a free parameter (which ...
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Looking for an adaptMCMC (R package for adaptive MCMC sampling) tutorial

I need to fit a very complex Bayesian hierarchical model and I want to use the adaptMCMC package of Andreas Scheidegger to do the posterior sampling. I can only find the adaptMCMC package reference ...
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MCMC and degree of freedoms (small number of observations)

I want to estimate a simple Regression. I have 20 observations and 10 Regression-parameters. The degrees of freedom are too small to get reliable point estimates and p-values. I found the following ...
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Multilevel model and sample size

I want to estimate a complex hierarchical multilevel logit model with spatial structure. The model is a two-level-model. I have 20 regions (upper Level) which contain several districts (lower Level). ...
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1answer
62 views

Proposal distributions for covariance matrices in MCMC implementation of hierarchical models [duplicate]

In a MCMC implementation of hierarchical models, with normal random effects and a Wishart prior for their covariance matrix, Gibbs sampling is typically used. However, if we change the distribution ...
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How to find the Likelihood function of superposed processes for MCMC?

i'm trying to understand how MCMC works. In my special case I have to stochastic processes which are superposed. I observe data $Y=\{Y_1,...Y_T\}$ and assume, that there are some latent variables ...
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139 views

Covariance matrix proposal distribution

In a MCMC implementation of hierarchical models, with normal random effects and a Wishart prior for their covariance matrix, Gibbs sampling is typically used. However, if we change the distribution ...
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54 views

Metropolis-Hastings to sample from dependent random variables

Imagine the goal is sampling from $p(X,Y)$ and X and Y are dependent real-valued random variables, i.e. $p(X|Y)\neq p(X)$. Now the question is how can we apply Metropolis-Hastings algorithm on the ...
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MCMC sampling with increasing data size?

So I am dealing with a case that I need to sample from posterior distribution in an online manner, i.e. my data size increases in each step and therefore the conditional posterior will have one more ...
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When is MCMC required? [closed]

I am trying to understand when MCMC algorithms are utilized. I can do density estimation with MLE for non-linear cases just like linear cases, and EM for hidden variables and missing values. What are ...
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Thinning samples obtained using population MCMC methods

I am trying to use a Population Markov Chain Monte Carlo algorithm for parameter estimation in an ordinary differential equation model of gene regulation. The paper here gives a good summary of ...