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

Questions tagged [mcmc]

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
29 views

Clarification: Are Generative Adversarial Networks an alternative to MCMC sampling?

I have been reading the original Goodfellow, et. al. paper on Generative Adversarial Networks and the way that they can obtain estimates of the posterior distribution of a discriminative network or ...
0
votes
0answers
12 views

Non-deterministic MCMC model with updates

0 down vote favorite I have historical data for three variables : Y, X1, X2, for example 1000 points. The distribution of future values of Y depends from X1 and X2 and can't be expressed in ...
0
votes
0answers
22 views

Formulating likelihood for MCMC with observations (data) being randomly sampled

My problem is the following: I want to get a distribution of cumulative damage to a structure over its service life (say, 50 years) given observed extreme events. I have a function for cumulative ...
1
vote
2answers
34 views

How does one usually compute the gradient and the Hessian of a proposal in a MCMC algorithm?

In some proposals of a MCMC, the mean/location vector and the covariance/scale matrix are functions of the gradient/jacobian and hessian of the log-likelihood. I'm wondering how does one usually find ...
0
votes
0answers
14 views

Hamiltonian MCMC information gathering [duplicate]

I started gathering information about Hamiltonian MCMC and I would like to ask if someone knows some good papers or books.If it possible notes that give a detailed explanation of Hamiltonian MCMC. ...
0
votes
1answer
34 views

Proposal in MCMC lives in bigger space than parameter space. Which transformations should I choose?

I'm using a MCMC algorithm. The proposal is, due to lack of information on my part, a multivariate T-Student distribution, i.e. $\theta \sim \mathcal{MT}(\mu, \Sigma)$. However, some of the components ...
0
votes
1answer
27 views

Calculating Integral Using MCMC

Consider the integral $\int_{\Theta}f(\theta|\mathbf{x}) \Pi(\theta)d\theta$,where $\theta$ is a univariate parameter and $\Theta$ is the support of $\Pi(\theta)$. I need to evaluate the value of this ...
2
votes
0answers
23 views

Is the MC produced by HMC reversible?

I know that the deterministic dynamics in Hamiltonian Monte Carlo/Hybrid Monte Carlo are reversible and the numerical integrators one uses to approximate them are reversible too. But HMC consists of 2 ...
9
votes
2answers
221 views

Is Gibbs sampling an MCMC method?

As far as I understand it, it is (at least, that is how Wikipedia defines it). But I've found this statement by Efron* (emphasis added): Markov chain Monte Carlo (MCMC) is the great success story ...
2
votes
1answer
40 views

Is RJMCMC robust to overfitting?

I have noticed that RJMCMC is often described as robust to overfitting. I am struggeling a bit with the intuition for this. Why doesn't the Reversible jump Markov Chain Monte Carlo (RJMCMC) always ...
0
votes
1answer
146 views

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 ...
-1
votes
1answer
24 views

2 versions of Metropolis-Hastings : are they equivalent?

I have seen 2 different versions of Metropolis algorithm. First one : Second one : I don't understand the differences between the 2 versions, especially in the second one where I have to use the ...
0
votes
1answer
46 views

Metropolis Hastings - Acceptance ratio, proposal and lkelihood

From a previous post : First to explain the MH algorithm, it's used to approximate numerically a target distribution, in this case $p(\theta|D)$. At each stage of the algorithm: A value ...
0
votes
0answers
22 views

Multi parameters - Metropolis Hastings Algorithm - Concrete example [closed]

have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). Thanks to the Metropolis-Hastings algorithm, ...
3
votes
0answers
47 views

Posterior mean estimator with MCMC (Metropolis Hastings Algorithm) - Concrete example

I have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). I have the 6 parameters to estimate : $p=(\...
3
votes
2answers
114 views

Optimizing $\chi^2$ using MCMC

I have measurements of an object. Let's say I have its length $L$, mass $M$, and age $t$: $$\mathbf y = (10~\text{m},\ 0.01~\text{g},\ 5~\text{s}).$$ I also have the uncertainties on my measurements ...
1
vote
0answers
8 views

How to perform a phylogenetically informed regression on individuals rather than species?

Let's imagine a scenario in which I have 100 individuals from five different species of the same genus. Their phylogenetic relationship is known (a tree is available). I want to look at the overall ...
2
votes
1answer
34 views

Finding the Poisson rate parameter with PyMC3

I'm trying to compute the rate parameter of fake set of poisson data, where I set the parameter. When I run PyMC the posterior distribution always peaks around the true rate parameter, but never ...
5
votes
2answers
194 views

The meaning of Bayesian update

I'm new in Bayesian inference and I can't found the answer to this: In real life scenario people use MCMC to compute the posterior distribution given the likelihood and the prior. Analytical ...
3
votes
1answer
62 views

Bayesian MCMC methods that need to calculate the evidence / normalizing factor

I came across this answer which states that: NOT all the MCMC methods avoid the need for the normalising constant. I was under the impression that one of the strengths of the MCMC methods (usually ...
2
votes
1answer
23 views

Calculating DIC from MCMC (gibbs) output

I just want to make sure my understanding of how we estimate DIC using MCMC output is correct, as the wikipedia page is somewhat confusing. Defining DIC by $$DIC = p_D + \bar{D}$$ with $$D(\theta) ...
2
votes
0answers
20 views

Adaptive selection of Mass values in Hamiltonian Monte-Carlo?

I know there are good solutions for adaptive selection of path lengths and step-size for Hamiltonian Monte-Carlo (e.g. the NUTS sampler), but for the sampler to work efficiently we also require that ...
1
vote
0answers
29 views

Flatten the target density in the Metropolis-Hastings algorithm

Let $(E,\mathcal E,\mu)$ be a measure space $F$ be a $\mathbb R$-Banach space $f\in\mathcal L^1(\mu;F)$ $f^\ast:E\to[0,\infty)$ be $\mathcal E$-measurable with $$b:=\int f^\ast\:{\rm d}\mu\in(0,\...
1
vote
0answers
19 views

Bound for the bias of ergodic averages for non-stationary Markov chains

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\mathcal E)$ be a measurable space $X$ be a $(E,\...
4
votes
1answer
46 views

How are deltas chosen for the proposal distribution in multivariate metropolis hastings sampling?

Say I want to use Metropolis Hastings algorithm to get posterior draws of multivariate parameters. In the one variable case, you could manipulate delta until you found something that worked (gave 40% ...
2
votes
0answers
42 views

Quantify MCMC chain mixing

I'm running an MCMC process with several parallel chains (often hundreds). To improve the visibility of the final plot, I only want to plot two chains per parameter: ideally the "best" and "worst" ...
1
vote
1answer
40 views

Metropolis algorithm to Bernoulli likelihood and beta prior (Kruschke 7.3.1)

This question pertains to a specific line written in the book Doing Bayesian Data Analysis by John K. Kruschke. In section 7.3.1, he applies Metropolis algorithm to a case with: $prior = beta(\...
1
vote
1answer
38 views

Sampling from partial posterior

I'm reading a paper where the authors have something like the following as one step in their MCMC: $$ \begin{align} y&=\rho_1 x_1+\rho_2 x_2+\epsilon\\ z&=\beta\tilde{y}+u \end{align} $$ ...
0
votes
1answer
43 views

PyMC3: Mixture Model with Latent Variables

I have a rather basic knowledge of Bayesian inference and I'm somewhat new to MCMC and PyMC3. Can I model data that looks like this? ...
0
votes
1answer
25 views

how can this missing observation model be extended to include cases where sigma is a function of other variables?

Richard McElreath's blog entry Algebra and the Missing Oxen describes a simple missing observation model in RStan. At the end of the blog, he says it can be extended easily to cases in which the ...
0
votes
0answers
15 views

Permutation sampling with Gibbs sampling - When to apply permutation constraint?

We are sampling from a mixture posterior. The posterior is sampled from using Gibbs sampling. Since the likelihood is invariant under label switching, it is necessary to apply an identifiability ...
2
votes
1answer
49 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
78 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
41 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
36 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
46 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
14 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 ...
10
votes
2answers
145 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
39 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
54 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
74 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 ...
1
vote
0answers
69 views
+100

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
36 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
91 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
22 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
21 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
101 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
40 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
28 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
91 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 ...