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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Metropolis and Barker Algorithms: Irreducibility

A question from Brémaud, Markov Chains Gibbs Fields, Monte Carlo Simulation and Queues. Exercise 11.5.2. Metropolis and Barker Algorithms: Irreducibility how do you show explicitly that $$p_{i,j}= \...
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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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Updating prior in MCMC with new estimates for parameters

I'm new to doing Bayesian analysis and I wanted to learn by using baseball data. I took a group of players and found their hits and at bats for various years and I want to be able to get estimates for ...
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MCMC "for dummies" || how to estimate parameters and what limitations should I consider?

I'm interested in parameter estimation. In a nutshell, I have an expression, e.g., $$f(x,y,w,z; t) = \frac{x\cdot\,y}{w\cdot\,z} \, \frac{k_1}{k_2}\,\frac{j_1}{j_2}\,t$$ in which I know variables $x,y,...
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What is a good introductory guide to Bayesian MCMC Analysis in R?

I am trying to perform Bayesian Analysis in R, using of Monte Carlo Markov Chains to calculate the probability of, in a set of data, there being a gaussian peak at a certain location $x_0$ (my 'target'...
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How to map a sequence to a transition matrix

I have the following transition matrices, one for Maria and one for Anna: ...
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How to write the derivative of the inverse gamma function?

I have recently been writing an R program on the inverse of the gamma function and the derivative of the inverse function. Now there is some confusion I would like to ask for advice. I have written ...
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How does detailed balance relate to (conditional) expectation?

Let $(\mathsf{X}, \mathcal{X})$ be a measurable space and $\pi$ be a probability distribution on it. Let $\mathrm{K}:\mathsf{X}\times\mathcal{X}\to[0, 1]$ be a Markov kernel. We say that $\mathrm{K}$ ...
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How to analyze a time series of categorical data?

I have some data that look like this: ...
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Markov Chain Monte Carlo doesn't converge

I have a synthetic measurement model that looks like this, $$ x(t) = e^{j u t \frac{4}{\lambda}}, $$ $\lambda$ is a constant. $$ z(t) = x(t) + n(t) $$ The quantity $j = \sqrt{-1}$, the imaginary unit. ...
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Sufficient ESS in MCMC

Is there any rule of thumb to determine whether the ESS are sufficient to conclude that the sample are independent ? I know how to calculate ESS but not really sure how to interpret them or give a ...
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BMA and RJMCMC predictive performance

We have a family of statistical models, with parameter spaces of different dimensions, which we aggregated through standard Bayesian Model Averaging (BMA). Experiments using training and test sets ...
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MCMC Sample should be i.i.d

I'm a bit not sure how to show that MCMC samples are i.i.d. In my opinion the trace plot should behave like white noise model because white noise model has a strong stationary properties i.e. the ...
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Application of Time reversibility property of stochastic processes

Are there any theoretical or applied consequences of a stochastic process being time reversible? I know a Markov chain being time-reversible implies the existence of a stationary distribution. But in ...
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ERGM with missing tie data: Interpretation of the mcmc.diagnostics regarding autocorrelation

I am running an ergm model on network data that contains some missing tie variables. I am running this model to later use it to impute the missing data. I am currently investigating the mcmc....
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Do the following normalizing constants cancel out in Reversible Jump ratio?

We know that a Strauss Point Process has density $$p(x_{1}, x_{2},..., x_{K})\propto \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}$$ where ...
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Hamiltonian Monte Carlo vs. "Metropolis-Hastings with a Hamiltonian step"

In Hamiltonian Monte Carlo the proposal is accepted with probability: $$ \alpha\left(\mathbf{x}_n(0),\mathbf{x}_n(L\Delta t)\right) = \min\left(1, \frac{\exp\left[-H\left(\mathbf{x}_n(L\Delta t),\...
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What are good options for deriving probability distributions of transition matrices when data lacks the memoryless property of a Markov chain?

I have a dataset which I initially believed was suitable for running Markov chain simulations, in that there is a finite number of readily identifiable states that all population elements fall into ...
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Monte Carlo for Dirichlet Multinomial Model

Problem I am trying to implement Markov Chain Monte Carlo for the Dirichlet Multinomial mixture, described in this reference (where one used the expectation maximization algorithm). The model is as ...
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Confidence Interval for Markov Chain Probability

I have a simple transition model I am trying to use to predict the probability of two states. $$ \begin{bmatrix} p_{1,t+1}\\ p_{2,t+1} \\ \end{bmatrix}= \begin{bmatrix} p_{11} & p_{12} \\ ...
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Interpret and assess the output of MCMC in high dimension?

In one of my recent presentations I used MCMC to generate 50,000 samples from a 13-dimensional random variable. And the audience of this presentation is largely made up of layperson that has no ...
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Question about paper on Bayesian Shrinkage Estimation

I am reading the paper Bayesian Shrinkage Estimation of the Relative Abundance of mRNA Transcripts Using SAGE, and I am trying to work out the calculations for the complete conditionals for the Gibbs ...
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Gibbs Sampling - why converge to stationary distribution

Currently, I am going through Chapter 12.3 of Probabilistic Graphical Models - Principles and Techniques which talks about MCMC sampling methods. In Chapter 12.3.4.1, it states the following theorem: ...
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How probable is the best fit of a model to lie outside 68% limit constraints on model parameters?

I am running MCMC analysis on a model using an observation data point. After the MCMC run has converged, when I check the best fit parameters and compare it to the constraints of the model parameters, ...
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About "Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation"?

I am reading the Yoshua Bengio et al, Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation. After read this paper, I wonder how to establish a GFlowNet if I don't know ...
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Hamiltonian trajectory stays in the typical set?

I'm currently studying Hamiltonian MCMC by reading Betancourt's 2014 and Neal's 2011 pedagogical papers, but I still don't understand why following a Hamiltonian trajectory for our proposed update ...
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Question about a mixture dirichlet MCMC model

I am self-learning Bayesian statistics using the book Computational Bayesian Statistics by Turkman et al. and I am currently stuck on Chapter 6 Problem 10. It can be found here on page 124. I am ...
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Statistical significance and angular comparison (similarity test) between two vectors, MCMC and resampling approaches

I want to compare statistically if two vectors of PC1 loadings differ in terms of their direction which can be measured by the angle between them (e.g. this CV thread). I would like if someone can ...
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Is this Categorical Distribution a valid proposal for Metropolis Hastings?

Is it possible to use a categorical distribution as proposal distribution for Metropolis-Hastings? For example, suppose that at the current iteration we are at the state $x_{i}=1$, from which there ...
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Ergodic theorem for Markov chains

I am reading Robert and Casella (2004) on Markov Chain Monte Carlo methods and, in particular, Section 6.7. This contains the ergodic theorem, which is stated as follows, where $S_n(f)$ denotes a ...
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Modeling Urns and Balls System as a Markov Chain

Suppose I have $q$ urns each of which hold up to $n$ distinguishable balls, but only $1$ of each type of ball (there being $n$ types of balls too). I would like to make any particular configuration of ...
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Can I use RMSE to diagnose overfitting in a Bayesian Calibration?

I am fitting a simulation model using Bayesian Calibration (DREAMZS MCMC using the BayesianTools packages in R). I have several time series I am calibrating to (e.g. log stream flow and nitrate ...
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multivariate potential scale reduction factor less than one

I am attempting to implement the multivariate potential scale reduction factor (PSRF) mentioned in this answer and originally described by Brooks and Gelman (1998). When I use a basic Metropolis ...
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Simulations based noisy likelihood function

I have a problem where I have a measured data vector $D$ with Gaussian uncertainties (covariance matrix $\Sigma$). I am now trying to model this data with a generative model with parameters $\phi$. ...
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Is the Markov property important in the Metropolis algorithm?

I’m taking a class in Bayesian statistics, and we’re learning about the Metropolis algorithm. Suppose for simplicity that we just have one parameter we’re trying to estimate: $\theta$. According to ...
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Proposed transition matrix for MCMC in two-state Markov Chain

Suppose we would like to model the weather (either sunny $S$ or cloudy $C$) using a two-state Markov Chain, given a set of data collected from 10000 days: $$CCCSSSSSSCCCSSSSSSCCCC...$$ We can use the ...
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Can we generate HPD regions from MCMC draws using convex hulls?

I thought of a procedure to generate high probability density regions with probability $1-\alpha$ from $n$ MCMC draws: Find the $\lfloor(1-\alpha)\cdot n\rfloor$ draws with the largest probability ...
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Best method(s) to estimate the parameters of a stochastic process with a hybrid (i.e. switching) random input variable?

I'm looking for the best approach to and/or methods of solving the following inference problem. I have tried searching for similar questions but don't have enough knowledge on the various methods (...
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Why don't we see Copula Models as much as Regression Models?

Is there any reason that don't see Copula Models as much as we see Regression Models (e.g. https://en.wikipedia.org/wiki/Vine_copula, https://en.wikipedia.org/wiki/Copula_(probability_theory)) ? I ...
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Effective Sample Size for a cycle or mixture of kernels

The Effective Sample Size (ESS) of a univariate Markov Chain can be used to assess it performance. Is there a version of the ESS for when the Markov Kernel is a mixture $\alpha K_1 + (1-\alpha) K_2$ ...
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What is the relationship between model uncertianty and model parameter count?

I'm looking for references, information and/or existing theory behind the relationship between the uncertainty in a given model vs its complexity/parameter count. The situation I have in mind is using ...
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How are copulas used in the real world?

I have been reading about copula models. Essentially, copula models seem to be a creative method for creating a joint probability distribution from several variables, in which each individual variable ...
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What kind of bayesian approach should be used for expenditure reasearch?

I must find determinants of expenditure which using baysian approach. Dependent variable: LN education expenditure Independent variables: continuous: LN income, study year of mom, commuting time to ...
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Why is my proportion estimate 0? (BEST MCMC Bayesian inference)

I am trying to run a simple Bayesian inference on my y1 vector as shown below. ...
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is there a fmcmc equivalent R package for Gibbs sampling?

I was searching for a package like this fmcmc for Gibbs sampling, but with no luck. I've tried gibbs.met, LearnBayes among others, but I would like to try a few more.
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How to find 95% credible interval of a posterior predictive distribution?

I obtained a posterior predictive with 1,000 samples using MCMC, and I need to quantify the 95% credible intervals. I know that the difference between confidence intervals and credible intervals. One ...
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