Questions tagged [markov-chain-montecarlo]
Markov Chain Monte Carlo (MCMC) refers to a class of simulation methods for generating samples from a complex 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 generation (e.g. inversion method) are infeasible. The very first MCMC method was the Metropolis (et al.) algorithm, later expanded by Hastings.
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MCMCglmm package [closed]
the summary() for my ordinal model isn't returning all of my cutpoints. There are four levels in my response variable but I'm only getting two cut points. Does anyone know why?
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SMC sampler weights when $K_n$ leaves $\pi_{n-1}$ invariant
I cannot seem to find this proof anywhere. Suppose I choose $K_n$ to leave $\pi_{n-1}$ invariant, and $L_{n-1}$ to be the reversal kernel. I want to show that the incremental weights
$$
w_n(x_{n-1}, ...
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Adjustment needed for multivariate Dvoretzky–Kiefer–Wolfowitz inequality on MCMC samples?
I was thinking about studying bounds on the multivariate empirical cumulative distribution function for samples from an MCMC chains. The multivariate Dvoretzky–Kiefer–Wolfowitz inequality would seem ...
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How to diagnose HMC results like r-hat for a Mixture Model?
I have the following distribution $$
\begin{align}
\boldsymbol \pi&\sim\text{Dirichlet}([1,\cdots 1]\in R^K)\\
\boldsymbol \theta&\sim P(\boldsymbol \theta)
\\
\mathbf y&\sim \sum _{i=1}^K\...
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Jacobian and proposal ratio of Birth/death step in RJMCMC of Gaussian mixture model
I am asking questions regarding RJMCMC several times in this site. Some of my questions are answered and some are unanswered. It didn't clarify all of my unclear points but I am glad that I have ...
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Using mcmc to estimate parameters of Dirichlet distribution
We have a probabilistic model with two parameters, $\theta$ and $\eta$, both of which are uniformly distributed between 0 and 1. The model has five possible outcomes, and the probability of each ...
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Calculate acceptance ratio of Jacobian of split-merge RJMCMC
I am keep studying the RJMCMC and want to ask question regarding the acceptance ratio of split/merge step of RJMCMC
The split/merge step, suggested by Richardson and Green (1997) is following for w_j, ...
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Assumptions and setting for bayesian mixture model (for RJMCMC)
I want to understand about Bayesian mixture model discussed in RJMCMC paper (Richardson and Green, 1997) (https://academic.oup.com/jrsssb/article/59/4/731/7083042)
I also posted similar question ...
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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 ...
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Terms and assumptions in trans-dimensional MCMC (RJ-MCMC) for Green 1995 paper
I want to use Trans-dimensional MCMC in my research and for fundamental understanding, I am trying to learn from Green (1995) paper, which is foundation of RJ-MCMC.
In part of 3.3 'switching between ...
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Adaptive Metropolis For Multidimensional Parameter
Hi recently I want to implement the adaptive Metropolis algorithm. However I dont know how to deal with multidimensional parameters. The normal step of the adaptive MCMC is to update the covariance ...
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Which method should be used to determine the class ID of multiple SVM models?
I'm using Support Vector Machine(SVM) with image classification.
Each SVM model results a linear model
$$y = wx + b$$
Where $w$ and $b$ is the SVM parameters.
If I have multiple SVM models, I will get ...
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Sampling from a Gaussian mixture (toy example ) using MALA
Say I want to sample from a Gaussian mixture $$\pi=\sum_{i=1}^3w_i\mathcal N(x_i,\sigma_i^2I_2)\tag1$$ where he support of the 3 distributions are "effectively separated"; e.g.
$w_1=.1$, $...
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Trying to understand Reversible Jump MCMC
As stated in the foundational Biometrika paper of Green (1995) 'Reversible Jump Monte Carlo Calculation and model discrimination'
I am researching the inversion of the data in Geophysics and MCMC is ...
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Bayesian CI - isn't it by definition a bootstrapped or robust enough confidence interval? or should it be optimized?
As the title said, I've been wondering about bayesian methods, particularly MCMC, which seems to rely on sampling of the sample via chained iterations. Do we need an optimized/robust CI when running ...
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Epistemic Uncertainty for branch-and-bound methods
I want to ask some leads on how to quantify or propagate an epistemic type of uncertainty (e.g. statistical uncertainty) when performing a branch-and-bound (sort of fault tree analysis). The ...
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How to specify dyad and subject ID random factors in an MCMC model in R?
I am working on a model to analyze a categorical outcome variable (7 categories) for a design with two fixed factors (Condition and Group) as well as the random factors of Subject ID nested in Dyad ID....
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MCMC predict volatility with GARCH
This is using MCMC (MH) algo to predict volatility from GARCH model:
https://www.oreilly.com/library/view/machine-learning-for/9781492085249/ch04.html
For simplication, consider (0,10)-GARCH (only ...
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Writing MCMC Sampling Code by Hand
I am trying to understand the steps involved in the Metropolis-Hastings Algorithm and trying to learn how to implement it myself.
As an example, suppose I am interested in estimating the "...
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Simulate SDE without error
Let
$d,k\in\mathbb N$;
$\sigma\in C^1(\mathbb R^d,\mathbb R^{d\times k})$ be Lipschitz and $\Sigma:=\sigma\sigma^\ast$;
$(W_t)_{t\ge0}$ be a $k$-dimensional Brownian motion;
$\lambda$ denote the ...
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Why do we need to scale the variables in a Bayesian model?
In a Bayesian MMM model using pymc3 the variables are scaled. It is said that scaling helps in improving the efficiency of the MCMC algorithm. Also, it is stated that setting priors for the non-scaled ...
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What is the trace in a Bayesian Model?
I am studying a python library that uses Bayesian inference to identify the coefficient of a linear regression.
I have two questions, one very broad and one more vertical on MCMC and numpyro.
What is ...
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Is it okay to merge seperate MCMC chains, using different seed value? [duplicate]
I'm new to Bayesian analysis. I'm trying to estimate species's abundance.
As I know, when using MCMC sampling, it is recommended to make more than three chains. However, the function I use can make ...
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Combination of a discrete and a continuous Markov Chain in a MCMC
Recently, I've been questioning myself on the possibility of combining a discrete update and a continuous update on a single MCMC. Stephens (2000) in Algorithm 3.2 runs the process for a fixed amount ...
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Non-reversible MCMC based on diffusion dynamics
Given that the current sample location is $x\in[0,1)^d$, I would like to take the next sample $y$ as $$y=x+b(x)\Delta t+\sigma(x)\sqrt{\Delta t}\xi\;;\;\;\;\xi\sim\mathcal N_{0,\:I_d}\tag1$$ for some ...
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Nesting in bayesian inference
I am currently trying to develop a model that takes into account different data sets that are not directly comparable.
More specifically, my model needs to describe the amount of a drug in the blood, ...
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Arithmetic using summary means of the MCMC chains, differ from when I do the arithmetic directly using each row of the MCMC chains
I am trying to calculate the 'absolute risk difference' and 'needed to treat' (NNT). NNT is 1/(absolute risk difference) where the absolute risk difference is just the rate of an event in one ...
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Can I use the output of an MCMC algorithm as the input of an independent Metropolis-Hastings algorithm?
Can I and if so, how can I, use the output of a MCMC method as the input for an independent Metropolis-Hastings algorithm? Maybe this question reduces to: How can I get (independent? or at least "...
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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 $y$ ...
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What kind of sampling is using a sample from a Uniform distribution to accept and reject [closed]
Given a sampler from a uniform distribution $U(0,1)$ and the target Bernoulli distribution $b(p)$ with two targets $+1,0$ , what kind of sampling technique is:
Sample $u_n\sim U(0,1)$
If $u_n<p$: ...
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Single walker MCMC using emcee
Does anyone know if it is possible to use the MCMC implementation emcee in python with only a single walker given 15 parameters? Or do I have to use a different implementation? I want to do this as I ...
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Bayesian variant of Ljung-Box test?
I am using Bayesian MCMC estimation methods for GARCH models and I want to check, if model residuals are uncorrelated. In classical frequentist approach, one would use standard Ljung-Box test to check ...
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How can I pool Bayesian parameter estimates after multiple imputation?
After multiple imputation (imputed dataset = 20), I would like to conduct Bayesian Model Estimation with Adaptive Metropolis Hastings Sampling (amh) -- using the MCMC method.
How can I pool the ...
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How does Rao-Blackwellization of the Metropolis-Hastings algorithm work?
I've read the paper A vanilla Rao-Blackwellization of Metropolis-Hastings algorithms, but I don't get what their actually suggested estimator is.
To give some detail, we are considering the following ...
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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 ...
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Monte Carlo simulation to get stationary distribution of a complex system
I was wondering if I could get some help with my problem.
I have a complex Markov chain where I cannot track its transition analytically. Instead, I decided to simulate $N$ numbers of particles for $T$...
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Does Hamiltonian Monte Carlo explore the target distribution uniformly?
I am trying to incorporate HMC into an algorithm that requires me to generate samples that are uniformly distributed in the distribution being sampled. Ie, the samples are not necessarily uniformly ...
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Metropolis Hastings Algorithm and Breaking Reversibility in MCMC
If the goal is to sample from a distribution $\pi$ it is common to build a Markov Chain with stationary distribution $\pi$. Solving this problem using Markov Chain Monte Carlo is essentially ...
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non-reversible (irreversible) MCMC Langevin sampling methods. Intuition
Mathematically it is well understood that irreversible samplers are superior to reversible samplers in a number of ways. For example, if we want to sample from a distribution $\pi \propto e^{-U}$ by ...
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Could one use mixtures of Gaussians to turn MCMC posterior samples into a new prior?
Theoretically in Bayesian inference one could use one experiment's posterior as another experiment's prior, such that knowledge of the parameters accumulates from $p(\theta) \rightarrow p(\theta|\...
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Estimation with Random Walk 2 Priors
In (https://becarioprecario.bitbucket.io/inla-gitbook/ch-smoothing.html#sec:smoothterms), they show an example of a Random Walk 2 (RW2) prior being used on the LIDAR dataset. For the model set-up, we ...
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How can I sample a multivariate normal vector that satisfies a linear equality constraint?
Let $X \sim N_n(\mu, \Sigma)$, such that $AX=b$ where $A$ is a ($p \times n$) matrix, with $p \ll n$. How can I efficiently sample from this distribution?
I've seen techniques using elliptical slice ...
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In Bayesian linear regression Advantages of predictive posterior compared to posterior of model coefficients
In Bayesian linear regression, if we want to get confidence intervals for predictions of a new observation. I was thinking of the following two options.
Use the quantiles from samples sampled from ...
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Subtilities of MCMC method and more generally about covariance matrix and Samplers
i have difficulties to better understand about what we commonly called a sampler, especially how to produce a covariance matrix between parameters during a MCMC code run.
In MCM, I know that we start ...
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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 ...
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What is the intuition for the limited variation in potential energy for HMC?
In A General Metric for Riemannian Manifold Hamiltonian Monte Carlo (Betancourt, 2013), the author writes:
The first [5] and still most common choice of the conditional density, $\pi(p|q)$, is a ...
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Estimating integrated-out variables
An illustration
David Blei in his lecture notes considers a collapsed Gibbs sampler for a Gaussian mixture model. In this case $B = (\mu_1, \dotsc, \mu_K)$ is a latent vector of means and $A = (Z_1, \...
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Gibbs sampler, how to generate samples based on odds?
I am reading a paper that contains a Gibbs sampler for a regression model with parameters $\beta$ and design matrix $X$. One of the steps in the Gibbs sampler requires simulating from a binary random ...
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blue noise error distribution in (MC)MC estimation
In computer graphics, you have an (MC)MC estimate $Q_i$ of the color value of the $i$th pixel and a true value $I_i$. Now you take $\epsilon_i:=Q_i-I_i$, which can be thought of as an "error ...
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Gibbs Sampler Initialization
I am trying to implement a Gibbs sampler with three blocks, that's not actually a full conditional Gibbs sampler.
Say I partitition my parameter $X$ into three: {$X_1$, $X_2$, $X_3$}.
I inted to ...
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