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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Why model order selection is a big problem in statistics?

I’m learning statistical signal processing for my studies. I was doing a bit of literature review on model order selection and I didn’t want to miss out on techniques that I might not have seen. I ...
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The gradient vector in Hamiltonian Monte Carlo (leapfrog method)

Let $x_{t}, \omega_{t} \in \mathbb{R^{d}}$ The Hamiltonian Monte Carlo says this: Deterministic: it relies on the Hamiltonian dynamics so given an initial state, at any time $t$, specified by the ...
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What is a perfect distribution to consider for the step increase/ decrease for the reversible jump MCMC

I am trying to understand the hyper parameters in the paper [1] for the model order selection with reversible jump MCMC (RJ-MCMC). There is a hyper parameter $\Lambda$ (The parameter of the Poisson ...
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Binned resampling of correlated data with bootstrap method

The goal Compute the Binder cumulant defined as the estimator $$\text{B.C.}=\frac{\langle x^4\rangle}{\langle x^2\rangle^2} $$ and its statistical error on a sample of normally distributed data points ...
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Why Reversible jump mcmc has only one step increase/ decrease?

I was applying reversible jump MCMC for joint estimation of model order and parameter estimation. I've a conceptual question in my mind. First of all, the algorithm has 3 steps, namely the birth, ...
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Why do we sample from the uniform distribution in Metropolis-Hastings for acceptance?

For each iteration of the MH, sample $x'=q(x|x')$, then the acceptance probability is computed:$$A=\min(1,a)$$ where $$ \alpha=\frac{p(x')q(x|x')}{p(x)q(x'|x)} $$ Now, I've seen that the algorithm ...
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MCMC__ std of the posterior nearly 0

I am new to MCMC. I am trying to use Metropolis-Hastings MCMC to update a parameter set for a model based on measurements. But the posterior I got seems to be little bit wired as the std values for ...
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Updated book or review paper on MCMC methods (2022)

For a self-study course, I'm looking for bibliography that describes current MCMC algorithms. I'd prefer a book or a review paper. My background knowledge is at the level of Gamerman-Lopes or Gilks-...
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Is it appropriate to discretize conditional posteriors in an MCMC as an alternative to techniques like Metropolis-Hastings or slice-sampling?

Background Suppose I am interested in sampling the posterior distribution defined by $p(\theta_1,\theta_2|y)$, where $\theta_1,\theta_2$ are parameters of interest and $y$ is a vector of observations. ...
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Metropolis-Hastings or other MCMC method with an unknown asymmetric proposal distribution?

When working with the Metropolis-Hastings algorithm, we can work with an asymmetric proposal density $g(x^\prime | x)$ provided we know the distribution in order to calculate the ratio $\frac{g(x|x^\...
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MCMC: what is the problem with correlated parameters?

I'm new to Markov Chain Monte Carlo (MCMC). Various sources suggest investigating the joint distributions of all pairs of esimated parameters. Refer to the figure below for an example. alpha and beta[...
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How to evaluate the hypothesis that a set of samples belongs to the common general population if only the mean & stadard deviation are available?

I have n results of nested sampling analyses from prior distribution MCMC. Every one has its own estimation of marginal likelihood & its standard deviation. The analysis has a hyperparameter – ...
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MCMC model: how to study the influence of personality (repeated measures) on a trait

I am trying to investigate how frogs level of aggressiveness influence their plastic responses to a more or less big opponent. To do so, I presented an individual with a simulated opponent, ...
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The implications of using MCMC normalization algorithm output as input in a separate normal linear model (and similar combined approaches in general)

Background I am a statistics intern (not assigned a supervisor yet) in a medical research institute. I’m exploring methods for network analysis in the context of multi-omics data, focused on microbial ...
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ABC, make tolerance threshold $\epsilon$ adaptive

Briefly the Approximate Bayesian Computation instead of using the exact likelihood function $L(\theta;x)$ tries to approximate this function with the use of the observed summary statistics $s(x_{obs})$...
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About the paper "Deep Unsupervised Learning using Nonequilibrium Thermodynamics"

I have spent some time studying the paper Deep Unsupervised Learning using Nonequilibrium Thermodynamics. At page 5, the authors discuss the following integral: $$\int d\mathbf{x}^{(1\cdots T)}q(\...
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Is it circular reasoning to compute the ELBO using MCMC?

Let's say we have a posterior distribution $q(\theta) = p(\theta \mid D, \mathcal{M})$ over parameters $\theta$ given data $D$ and a model $\mathcal{M}$. As is often the case, computing $q$ is hard, ...
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Should I specify priors to increase the effective sample size for multimembership MCMCglmm?

I am wanting to use a multimembership MCMCglmm with the following model ...
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Computing the Hastings ratio for multinomial distribution as a proposal distribution in Metropolis-Hastings accept-reject step

I have a question concerning calculating the Hastings ratio in a specific case (multinomial proposal distribution). I consider a discrete vector $M$ with integer values that sum up to some number $N$. ...
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I don't understand how doing Bayesian Inference using MCMC is not considered optimization?

I've read many documents/articles on Bayesian Inference using MCMC and they always mention that this is not optimization. Now, I understand that MCMC is only used to compute integrals and that this ...
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Bayesian Logistic Regression to Optimize Model Weights

I am new to Bayesian Inference and I want to understand how Bayesian Logistic Regression optimizes the weights of a regression model. To elaborate on a specific example I came across weights, coef1 ...
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Bayesian structural equation modelling - do all parameters have to show good diagnostics (e.g., autocorrelation, trace plot etc.)?

Would a Bayesian SEM model only be accepted by a scientific journal if all parameters show good diagnostics ? (e.g., autocorrelation, trace plot etc.). Is it realistic to build such a model? Or is it ...
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How to find the probability of a number being the mean of a normal distribution given a sample and SD?

The question is related to computing the likelihood function of a simple problem for MCMC. The full problem can be found here on page 2. The question in the title is written more succinctly in the ...
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Hiearchical Beta-Binomial model via rjags: How to draw posterior sample/do inference on posterior exactly?

I have the following code for bugs model which I want to use with rjags: ...
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How can I obtain a conditional distribution from a joint distribution using a MCMC sample?

Suppose that $\theta = (\theta_1,\theta_2) \in{\mathbb R}^2$ are the parameters of a model, and that I can obtain a MCMC sample from the posterior distribution of $\theta \mid {\bf x}$. Using the MCMC ...
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Parametrizing and Sampling Multivariate Garch Parameters Metropolis-Hastings MCMC

My question is how to sample multivariate GARCH parameters from a proposal distribution (multivariate normal) for a Metropolis-Hastings algorithm. Considering the different dimensions of the parameter ...
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Compute the Likelihood of binomial data

Say we have to following data: p = 0.95 -> rate of true positive result of pcr test. q = 0.1 -> rate of false positive result of pcr test. s = 0.2 -> rate of total patients in the population ...
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Fitting the mean and variance of 1D Gaussian using the Metropolis algorithm

This probably some beginners mistake, but I have been stuck on this for a few hours now and cant spot the mistake. The scenario: From a predefined 'true' distribution with a defined mean $\mu_{true}$ ...
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Interpreting Chart of Transition Densities from MSBVAR Gibbs Sampling

I've run the MSBVAR package on 3 time series, and then did a Gibbs Sampling and used the plotregimeid function based on the elements of the transition matrix Q. I'm trying to get an understanding of ...
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Simple nonlinear regression for gaussian peak

I want to do non-linear Bayesian Regression, my setup is almost exactly the same as the linear case: I have a signal encoded as $y_i \in \mathbb{R}$ given $x_i \in \mathbb{R}$ for $i = 1\dots N$ and I ...
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Esitmate of minimal of a function changed after transforming the variable

I want to perform MCMC or HMC for solving minimization problem of a function $f(x)$, then define the corresponding density $$g(x) = \exp\left(-f(x)\right)$$ Because the function of the future apply is ...
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How to evaluate likelihood in MCMC for arbitrarily shaped distributions?

I'm very confused with the use of MCMC to estimate distributions that have a complex shape, like multiple peaks, or that aren't generated from a known distribution. In particular, when calculating the ...
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Doubt in rewriting of equation to approximate expectation w.r.t posterior

I'm reading Speagle's A conceptual introduction to Markov Chain Monte Carlo Methds to try to learn the contexts where MCMC sampling is needed (in this case the paper focuses on Bayesian inference). We ...
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Relation between typical set and stationary condition in MCMC

In Betancourt's paper "A Conceptual Introduction to Hamiltonian Monte Carlo", I don't quite see the relation between imposing the stationary condition on the Markov Chain and finding and ...
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M-H algorithm for scale parameter ($\sigma>0$)

Suppose I have a continuous distribution with a location parameter $\mu\,[\mu\in \mathbb{R}]$ and scale parameter $\sigma(\sigma>0)$. I obtain full conditional posterior of $\mu$ and $\sigma$ and ...
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Metropolis - Hastings algorithm on a set of countable sequences

I want to simulate $\sigma$ from a measure $\pi(\sigma)$ through the Metropolis-Hastings algorithm, where $\sigma$ is a sequence of 0's and 1's on $S = \{0, 1\}^n$, the set of all sequences of 0's ...
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Checking the results of the Metropolis - Hastings generated sample

I've generated a sample for the distribuition $f(x) \propto cos(x), x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ through the Metropolis - Hastings algorithm. For that I've used as candidate distribuition $q ...
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Can the proposal distribution for Metropolis-Hastings within Gibbs be conditioned on other variabless?

I am drawing samples from my posterior, $P(x,y|z)$, using Gibbs sampling. When I sample $x$, I use a Metropolis-Hastings step. My question is whether I am allowed to use a proposal distribution for $x'...
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Estimating priors and marginal likelihood of Gaussian distribution

I am new to Bayesian Inference and MCMC, and I am currently reading the "A Conceptual Introduction to Markov Chain Monte Carlo Methods" by Joshua S. Speagle. The paper can be found here: ...
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Using non-positive (semi)definite matrix as "covariance matrix" for MALA

If the target distribution is a Gaussian distribution, MALA (metropolis-adjusted langevin algorithm) becomes: $$X_{t+1} = A X_t + \sqrt{2\tau}\xi$$ where $ A= I - \tau\Sigma^{-1}$. where $\tau$ is a ...
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Simulating a Strauss Point Process with Birth-Death Algorithm

Suppose that I have the following two unconditional Strauss Process $$f_{1}(x_{1}, x_{2},..., x_{n};a,\delta) \propto \prod_{i}^{n}\phi(x_{i}) \prod_{1\leq i \leq j \leq n}a^{\left | x_{i}-x_{j} \...
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Distribution of conditional posterior for Gibbs sampling

The following is a description of how the authors (Yongning Wang & Ruey S. Tsay) of this (2019) paper Clustering Multiple Time Series with Structural Breaks want to perform Gibbs sampling to ...
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What is the probability of acceptance for this algorithm?

What is the distribution of $Y$ from using the Rejection Sampling algorithm? Repeat Sample $X$ with distribution function $F_X = (1-(1+x^\alpha)^{-1})1_{x\ge 0}(x)$ Until $X>x_0$, where $x_0$ is a ...
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ERGM: How come `ttriple` is degenerate but `transitiveties` is not?

This title is an effort to elicit a simplified explanation as to how come the transitiveties term causes less numerical instability than ...
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Sampling a Parameter Space with a Neural Network versus other Methods [closed]

I have heard a lot about neural networks as an approach to sampling a large parameter space in aim of finding the space of model parameters that match a dataset. I've done some research online, but ...
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MCMC simulations in R for a simple poisson model [closed]

I would like to put together my own R function for MCMC simulations rather than using a R package or Winbugs. This is purely to ensure I understand what is going on. I have used the R Nimble package ...
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Does Geyer's mode selection heuristic perform no-worse than burn-in?

In Charles Geyer's Burn-In is Unnecessary he writes Another possible rule is to start at a point, like the mode, known to have reasonably high probability. If no such point is known, this rule is ...
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How to use the parameters estimated by MCMC?

Considering this example, taken from the coursera course "Bayesian Statistics: Techniques and Models", Dataset: ...
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Simple alternative to reversible jump for nested models?

Suppose that we have a model such that $p(y\mid k, \theta_1,\dots,\theta_{k_\text{max}})$ depends only on $k,\theta_1,\dots,\theta_k$. Hence, as $k$ assumes the values $1,\dots,k_\text{max}$, we have ...
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Markov Chain Monte Carlo with known normalisation

I would like to compute the expectation value $\langle O \rangle = \sum_x P(x) O(x)$ of some random variable over an extremely large sample space that I cannot simply exhaustively go through. Usually ...
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