Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Markov Chain Monte Carlo (MCMC) 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 more direct methods for random number ...

0
votes
0answers
11 views

Questions about approximate inference and calculating the posterior predictive

As I understand, computing the exact posterior of parameters $p(\theta|x)$ is nearly always impossible since we need to compute the evidence $\sum_\theta p(x|\theta)p(\theta)$ with every possible ...
1
vote
1answer
26 views

Monte Carlo maximum likelihood vs Bayesian inference

I recently heard about MCMLE (Monte Carlo maximum likelihood estimation) for finding $$ \hat\theta = \underset{\theta}{\text{argmax}} \frac{\exp\left(\theta^TT(y)\right)}{c(\theta)} $$ when the ...
1
vote
1answer
27 views

ABC: Population Monte Carlo (PMC) convergence statistics?

I'm using the abcpmc code: Approximate Bayesian Computing (ABC) Population Monte Carlo (PMC) implementation based on Sequential Monte Carlo (SMC) with Particle Filtering techniques. described in ...
1
vote
0answers
16 views

Definition of the integrated autocorrelation time

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\pi$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ $(X_n)_{n\in\mathbb N}$ be a real-valued stationary stochastic ...
2
votes
1answer
39 views

How worried should I be about low acceptance rate in cold chain (parallel tempering MCMC sampler)

I have a very noisy/multimodal likelihood function for a 6-parameter model. The popular emcee sampler fails miserably (no matter how many chains I use and for how ...
0
votes
0answers
12 views

Slice sampling of a model with continuous and discrete parameters

I have a model with 5 continuous and 1 discrete parameter. I am using PyMC2 to implement slice sampling. I have a custom likelihood function that returns the log likelihood value that gets passed to ...
0
votes
0answers
14 views

Is there a good text book on serial tempering?

I've read that serial tempering is an approach for "MCMC sampling from a sum of parametrized distributions". I've only found two papers (Marinari and Parisi and Geyer and Thompson) introducing this ...
10
votes
2answers
129 views

Proposal distribution for a generalised normal distribution

I am modelling plant dispersal using a generalised normal distribution (wikipedia entry), which has the probability density function: $$ \frac{b}{2a\Gamma(1/b)} e^{-(\frac{d}{a})^b} $$ where $d$ is ...
4
votes
1answer
29 views

Techniques for improving mixing when sampling from a multidimensional posterior

I'm currently trying to use the Metropolis-Hastings algorithm to sample from a posterior distribution of the form $$p(\theta | y ) \propto \prod_{ij} \phi (\theta_{ij}) \times \prod_{i=1}^n \pi_{y_i}...
2
votes
1answer
40 views

Bayesian MCMC: use the burn-in phase to find an appropriate scale factor for the likelihood?

In a previous question I asked if I could scale the likelihood as my MCMC process advanced, to keep the acceptance fraction within a reasonable range (~0.2-0.5). I was told that this is not a valid ...
2
votes
1answer
42 views

ABC: Population Monte Carlo (PMC) vs Sequential Monte Carlo (SMC)?

I'm reading about the Approximate Bayesian Computation (ABC) method, and I came across two rather popular approaches: Sequential Monte Carlo (SMC) methodology to sample sequentially from a ...
0
votes
0answers
38 views

Metropolis sampling for Bayesian networks

Gibbs sampling is a profound and popular technique for creating samples of Bayesian networks (BNs). Metropolis sampling is another popular technique, though - in my opinion - a less accessible method. ...
0
votes
0answers
23 views

HMM - Approximate log likelihood using Gibbs sampling

I am studying MCMC approaches to HMMs and Factorial HMMs. I am reading this paper 'introduction to hidden markov models and bayesian networks': http://mlg.eng.cam.ac.uk/zoubin/papers/ijprai.pdf In ...
0
votes
1answer
62 views

How to interpret Zero-Inflated Poisson in WINBUGS?

I have Winbugs code for a zero-inflated Poisson (ZIP) model. I obtained this code from my lab at university and the person who wrote it is not accessible for me to ask questions. Here is the code: <...
1
vote
1answer
17 views

How to interpret zero-inflation model for Bayesian regression?

I am trying to understand the zero-inflated poisson (ZIP) model used in Bayesian regression modelling. I came across code here for the ZIP model. My question is related to the 3rd line of code within ...
0
votes
0answers
16 views

MCMC dont converge in two level hierarchical model

I'm doing simulation in following framework. I have some responses $\theta_{ik}$ and since K is very large, I try to have a bayesian factor model to reduce the dimension. Following is a factor part, ...
1
vote
1answer
73 views

Scale log-likelihood as MCMC sampler advances, to improve acceptance rate

I am working with a rather noisy and multi-modal likelihood. I've found that in order to obtain reasonable results from my Bayesian MCMC sampler (emcee, an affine ...
2
votes
1answer
38 views

Is it always a requirement to declare a distribution model first before applying MCMC models/bayesian analysis?

I've read lot of articles that is using pymc python module to apply MCMC algorithms into solving real life problems. I found that all the examples are about to assume various kinds of distribution ...
1
vote
0answers
26 views

How to sample a vector from Multivariate normal with the last element constraint to positive?

I'm doing MCMC simulation and a posterior is hard to sample. Suppose I need to sample a vector $\beta \sim N(M_{\beta} , \Sigma_{\beta})1_{\beta_{K}>0}$, which mean $\beta$ is a vector with length ...
1
vote
1answer
78 views

Normalize likelihood for better MCMC performance?

I'm using the emcee package to sample the distribution of a single parameter, using a uniform prior and 8 chains. In this toy example, my likelihood is defined ...
0
votes
0answers
26 views

Distribution of posterior probabilities of samples from MCMC seems to be made up of several chi square components

I am running an MCMC sampler with a model that uses Cash's C statistic for the likelihood (along with gaussian priors), which is supposed to resemble a chi square distribution in the limit of large ...
1
vote
0answers
17 views

Plotting individual dimensions of posterior PYMC3 [closed]

I have used PYMC3 to perform inference on a Bayesian logistic regression model. I want to find the posterior over the weights $\beta \in \mathbb{R}^K$ given a Gaussian prior $\mathcal{N} \sim (0,100 \...
2
votes
1answer
38 views

12 independent samples are “enough” in MCMC sampling for what?

I'm reading Learning in Graphical Models (Jordan 1998) and in the chapter Introduction to MCMC method by D. J. C. Mackay (page 201) it says this: It's not clear to me to what this is referring to. "...
1
vote
1answer
31 views

Exact MCMC Logistic Regression Output

I am having trouble interpreting the output of an MCMC Logistic Regression run using R from the MCMCpack. Unfortunately I have had very little luck in finding sources on the web. I am assuming that ...
0
votes
1answer
54 views

What is the difference between monte carlo integration and gibbs sampling?

I am aware that both are methods of sampling from the posterior. MC integration replaces the integral by a sample MC sample. Is this sample independent? Gibbs sampling is a class of MCMC ...
0
votes
0answers
19 views

MCMC sampling from model with invalid regions (divergent / pathological behavior) in the parameter space

I currently am trying to figure out what would be the best option to perform MCMC sampling for a model which may show some kind of pathological behavior for some parameter combinations. The concrete ...
2
votes
1answer
27 views

Distorted hyperpriors when sampling from the prior only

I am currently testing some multilevel models in pymc3 and found that the hyperpriors get distorted when I run the level only to generate the prior. The hyperpriors I am using are generating ...
0
votes
1answer
38 views

Is the truncated normal distribution symmetric?

I am running a Metropolis-Hastings MCMC to find the distribution of a parameter that takes real, positive values. I was considering using the truncated normal distribution, and was wondering if I have ...
1
vote
0answers
40 views

How to reduce autocorrelation in MCMC

I'm using MCMC to simulation the distribution of some parameters in a Bayesian hierarchical model, which has the following form: $$\gamma_{ik} \sim Ber(\omega_{ik}).$$ Then I make a logit-...
1
vote
1answer
48 views

How to interpret the importance for a regression coeffcient in Bayesian regression from its posterior density?

I am trying to interpret the regression coefficients of a covariate in a Bayesian linear regression problem. More specifically, I am trying to determine if the regression coefficient have an important ...
1
vote
0answers
33 views

Sanity checking the effective sample size

When running MCMC sampling, a common measure of performance is the effective sample size (ESS). There are lots of different ways to estimate the ESS from samples e.g. https://arxiv.org/abs/1011.0175. ...
5
votes
2answers
349 views

Does multiplying the likelihood by a constant change the Bayesian inference using MCMC?

For numerical Bayesian inference we have Posterior~Prior*Likelihood. In MCMC we do not need to calculate the denominator in Bayes rule. My question is that can I multiply the Likelihood by a large ...
3
votes
1answer
57 views

Interpretation of “scale function” in Foster-Lyapunov drift condition

I'm reading about Markov chains and I'm starting to bump into these drift conditions, and their relationship with a chain's ergodic properties. The drift condition is that there exists a "scale ...
0
votes
1answer
49 views

Metropolis-Hastings in a Bayesian Hierarchical model

I am trying to estimate a Bayesian Hierarchical model using the random-walk Metropolis-Hastings algorithm. While in a non-Hierarchical model, the algorithm is staight-forward, I am not sure I am ...
0
votes
0answers
45 views

How to get Historical prediction value from BSTS model in R

I have a BSTS model and need the forecast for the entire period. For example, My training set is between 2008 to 2016 and my testing is 2017 Jan to 2018 Jan. Now I need the predicted values for 2008 ...
1
vote
1answer
36 views

MCMC samples for constructing a histogram

I am interested in generating samples from a density $\pi(\theta)$ to construct a histogram for $\pi(\theta)$ and to use these samples to generate samples of $f(\theta)$ for some function $f$. I may ...
1
vote
0answers
20 views
2
votes
2answers
53 views

Joint credible regions from MCMC draws

Lets say I have $n$ posterior samples of $\theta_1$ and $\theta_2$. I suppose that any region $R$ which contains exactly $(1-\alpha)n$ of the points will be an approximate $(1-\alpha)\times100$ ...
3
votes
1answer
93 views

In exactly what sense do MCMC draws approximate the target?

Background We want to sample from some intractable density $\pi(\theta)$. Using an MCMC algorithm, we generate a sample of draws $\{\theta_i\}_{i=1}^N$ from a Markov chain that has $\pi(\theta)$ as ...
1
vote
2answers
59 views

What to do once states are rejected in MCMC?

I need to generate samples from a pdf given by $\frac{f_Z(z)\cdot 1_{Z \in B}}{P(Z \in B)}$ where $Z \in \mathbb{R}^d$ is a normal random vector with independent components. $Z \in B$ is a set that is ...
1
vote
1answer
42 views

How does WINBUGS determine the posterior density of a parameter with multiple chains?

I am a new user to WINBUGS. I am running a model with 2 chains. When my model has finished running I have the following posterior density plot of my parameter: The plot only shows one distribution (i....
1
vote
1answer
113 views

MCMC vs Bayesian Optimization Efficiency for MAP estimate

I believe MCMC could be utilized to estimate the MAP. At least there is an option in packages like PyMC. I just started reading about Bayesian Optimization, but the first thing that hit me was that ...
0
votes
0answers
11 views

How to interpret low posterior probability of covariate's positive or negative association?

I am using the following model in WINBUGS to run a hierarchical Bayesian regression where the beta are my covariates: If I modify this model by adding the ...
2
votes
1answer
24 views

Slice Sampling asks to draw from $f^{-1}]y,+\infty[$

Slice Sampling asks to draw uniformly from $f^{-1}]y,+\infty[$. Wikipedia page However, how can we be sure that a uniform defined over the set $f^{-1}]y,+\infty[$ is in fact proper? If I had to ...
1
vote
1answer
28 views

How can the support of proposal distribution impact convergence of RH-MH algorithm?

In the book Introducing Monte Carlo Methods by Casella and Robert, there's a sentence with which I'm having some trouble to understand. «If the domain explored in $q$ [proposal] is too small, ...
2
votes
1answer
55 views

Discrete and continuos parameters in MCMC sampler

I'm working with a 6-dimensional Bayesian model, and the affine-invariant sampler implemented in emcee. Four of those parameters are discrete, while the other two ...
0
votes
0answers
20 views

How to validate Bayesian hierarchical (mixed) model ?

I am new to Bayesian analysis and using the following WINBUGS example to understand Bayesian hierarchical modeling: This is a 'mixed' model with both fixed effects (covariates given by 'beta' terms) ...
1
vote
1answer
32 views

How do interpret a vague prior for hierarchical modeling?

I am new to Bayesian analysis and using the following WINBUGS example to understand Bayesian hierarchical modeling: I have 2 questions: 1) For the fixed effects terms, i.e., the beta0 and beta1 ...
3
votes
1answer
67 views

Would a simple Gibbs, or a Metropolis-Hastings algorithm work for a State-Space model?

I'm wondering if a MCMC algorithm, in a Gibbs or a Metropolis-Hastings style, work for a State-Space model. Would I also be able to learn about the state variable and not just the parameters? I've ...
1
vote
0answers
27 views

HOW TO - Applying MCMC to conditionally select random variables?

I am quite new to the using Graphical Models, so pardon me for the naivety. My intention is to have some fun for the weekend and impress my friends on Monday. I am trying to understand MCMC and ...