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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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Bayesian MCMC and Only Updating Some Variables at a Time

I want to do Bayesian MCMC on a Gaussian Mixture Model. But, I want to update the means, weights, and covariance matrix for a single component separate from the others. Would there be the issue of ...
LifeisGood94's user avatar
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My MCMC Simulation

I am new to MCMC Simulation and Bayesian Analysis, so I wonder if my simulation has converged. My posterior is highly correlated by nature, so I'm facing some difficulty to ensure a sufficient number ...
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Recycling MCMC samples for another data set from the same distribution

Suppose I'm given $\theta_0$ and I want to sample data from a density $f(Y|\theta_0)$ and then sample from the posterior of $\theta|Y$ (given, obviously, some prior). I want to do this lots of times, ...
Thomas Lumley's user avatar
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two-step gibbs sampling vs block gibbs sampling

While reading Bayesian-related technical articles, I can see algorithms such as two-step Gibbs sampling and block gibbs sampling ...
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Is this a reasonable way to check the quality of simulated data in MCMC inference?

I have a hierarchical Bayesian model that looks like this: $\alpha_i \sim \mathcal{N}\left(\mu_\alpha, \sigma_\alpha\right) \tag{1}$ $\beta_i \sim \mathcal{N}\left(\mu_\beta, \sigma_\beta\right) \tag{...
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Transformer model conditional probability distribution of sub-sentences

I have a simple transformer model (decoder only) which is trained on some dataset containing sentences to do next-word prediction. The model captures a probability distribution $P_{\theta}(\mathbf{a})$...
JazzJammer's user avatar
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Priors DURING Metropolis-Hastings-Random Walk chain (MCMC)

Suppose we are running a Metropolis-Hastings Random Walk chain (MHRW) targeting the unknown posterior distribution of a $\theta$, using data $Y$ and likelihood $L$. Since we do not know the posterior ...
Alecos Papadopoulos's user avatar
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Proving whether or not a markov chain is irreducible/recurrent? (Metropolis-within-Gibbs)

We want to generate samples from a standard normal distribution using a variation on slice sampling. To do this, the following Gibbs scheme is proposed to sample uniformly from the set $S = \{(u,x)\...
Bajas's user avatar
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How to tune the unadjusted Langevin algorithm?

I want to start investigating the (unadjusted) simulation of the Langevin process $${\rm d}X_t=b(X_t){\rm d}t+\sigma{\rm d}W_t,$$ where $$b:=\frac{\sigma^2}2\nabla\ln p.$$ I don't want to simulate ...
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Is it possible to use Particle Marginal Metropolis Hastings to estimate the transition matrix and input?

A state space model is defined as: $$x_{t+1} = A_tx_t + B_tu_t$$ $$y_{t+1} = H_tx_{t+1}$$ So my question is: is it possible to use Particle Marginal Metropolis Hastings to estimate the transition ...
William Zhao's user avatar
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Metropolis-Hastings algorithm doesn't converge to the global minimum

I calculated the total root mean squared error of 24 parameters that are estimated with metropolis hastings, I ran the algorithm for 100.000 iterations, and as the chain forward it reached a global ...
William Zhao's user avatar
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Sequential updating vs Marginalized updating

Suppose I need to sample a posterior $\pi(\theta|D)$, whose analytic form is not tractable (not even up to a normalizing constant). However, I somehow manage to obtain an augmented posterior $\pi(\...
rryan's user avatar
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Sum of powers (geometric series) of state transition matrix

I am working discrete time Markov chain analysis for some large state transition graph. I want to find the rewards/cost to reach from the init state to the terminal/accepting states. I have the state ...
JackDaniels's user avatar
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Which test do I use for checking chain convergence on an mcmc glmm with a factorial response variable?

I am running an mcmc glmm (mcmc package in R) with the following structure: continuous response variable + continuous response variable + factorial response variable ~ all of my covariates+etc. This ...
Juliette's user avatar
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Are there efficient sampling techniques to leverage if my function has a seperable structure of functions? [duplicate]

I'm considering the posterior of a parametric model via the Bayesian approach. More specificity, I have a parametric model $u(p_1,p_2, p_3) = u_1(p_1) \times u_2(p_2) \times u_3(p_3)$ and I want to ...
CC Kuo's user avatar
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BIC with non-negligible priors

I want to do model selection based on the best-fit/MAP/marginal posterior I find from an MCMC and likelihood maximization. I have a likelihood $\mathcal{L}(X|\theta)$, some informative priors $\pi(\...
ojima's user avatar
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How to do prediction (evaluate marginal likelihood) in generative latent variable classifier?

The dataset is $\{\boldsymbol x_t,y_t\}$ for $t=1,\dots,T$, where $y_t \in \{0,1\}$. Define a generative latent variable classifier whose plate diagram is shown above. For each data point, a local ...
Kaiwen's user avatar
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How to leverage the separable functions in MCMC sampling? [closed]

I'm considering the posterior of a parametric model via the Bayesian approach. More specificity, I have a parametric model $u(p_1,p_2, p_3) = u_1(p_1) \times u_2(p_2) \times u_3(p_3)$ and I want to ...
CC Kuo's user avatar
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Can MCMC sample any probability distributions?

I have three fundamental questions related to MCMC. I would appreciate the help on any one of those. The most fundamental question in MCMC field, which I can't find a reference, is: Can MCMC generate ...
George Lu's user avatar
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Particle Marginal Metropolis Hastings - How to multiply the proposal distribution by the distribution of x?

When we are using particle marginal metropolis hastings, we will approximate the distribution of x with particle filter, in this pdf written below says: In such situations it is natural to suggest ...
William Zhao's user avatar
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Expectation of Changes in Top K Elements Amongst Randomly Generated Numbers

I am conducting a Monte Carlo simulation where: I generate n random numbers uniformly. Select the top k of these numbers. Then regenerate c of the n numbers at random. I aim to compute the mean ...
Daniel's user avatar
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How to do Bayesian MCMC with Compositional Parameter Constraints

So I know that if you have parameter constraints and you were to do a random walk MH without them, you can use a truncated normal distribution as your proposal instead (and of course, this would be ...
LifeisGood94's user avatar
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Show that the total variation distance of the Metropolis kernel to its proposal kernel is equal to the rejection probability

Furthermore, let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $Q$ be a Markov kernel on $(E,\mathcal E)$ with density $q$ with respect to $\lambda$; $\mu$ be a probability measure on $...
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Posterior approximation following optimization methods

I'm trying to quantify the uncertainty in a high dimensional, and multimodal posterior space. We do not have a analytical solution for the forward model, and the forward model could be expensive to ...
Geooo's user avatar
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How can we compare the "performance" of different Markov chain Monte Carlo algorithms?

How can we judge the performance a Markov chain Monte Carlo (MCMC) algorithm? I guess we could consider one of the following: The variance of $X_t$ for a given $t\in I$; The asymptotic variance of $(...
0xbadf00d's user avatar
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generalized progressive hybrid censoring scheme type 1 and Muth distribution

I have data concerning failure of sodium sulfur battery cells with the following numeric values: 76,82,210,315,385,412,491,504,522,646+,678,775,884,1131,1446,1824,1827,2248,2385,3077. This dataset ...
imanattia's user avatar
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MCMC for correlated Posterior

I'm simulating the posterior of a (as it seems) highly correlated Posterior distribution using MCMC (DREAM Algorithm). My setting is that I have 7 parameters where x1/x3 and x2/x4 is highly correlated,...
Sobol's user avatar
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Sampling from a binomial and get stuck in boundaries

I am trying to use the Metropolis-Hasting in order to obtain a sample for X that is a vector of length N of values that go from 0 to K (in this case K=3). So X ~ Binom(K, p) and p ~ Beta(1,1). For ...
Bibi's user avatar
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Is MCMC on log-likelihood valid? [duplicate]

Hi it occurred to me that for numerical stability people often use the log-likelihood instead of a real pdf/likelihood. And while this might work fine for MAP & MLE it's not obvious that it works ...
profPlum's user avatar
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x value in MCMC - log(Obs/Exp)

My company recently implemented software that uses an MCMC method. In that program, a handful of randomly generated nuisance parameters are used to calculate expected values which are then compared to ...
Dan's user avatar
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Using PCA to check if parameters simulated from a hierarchical Bayesian model are close to real parameters

I have a hierarchical Bayesian model that learns a 5-parameter function for each of the N participants. The priors on each of the 5 parameters are parameterized by a scale parameter, so, it also ...
vishu's user avatar
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How can I determine if a system is equilibrated?

Cross-posted in SCSE and MMSE I am experimenting with a new MCMC protocol and new research. In the context of Monte Carlo simulation, a "state of equilibrium" refers to a condition where the ...
user366312's user avatar
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Is this a correct way of resampling the MCMC chain?

Please understand I am not familiar to the statistical languages. All I want is to resample a probability distribution from an existing sample drawn from another distribution using MCMC, without ...
Hojin Cho's user avatar
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1 answer
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MCMC seems very sensible to the evidence

currently starting to study bayesian ML, and specifically MCMC, in order to compute the posterior: $$ P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)} $$ Now, I see how the acceptance ratio makes sense ...
Alberto's user avatar
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How should you determine the probability returned by a flat uniform prior function

I am currently doing an analysis that involves fitting a model to a 1D graph. Following the example on the emcee documentation, I started with Maximum likelihood estimation and am now looking at using ...
shram's user avatar
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Is there a way to restrict the relationship of parameters in MCMC?

I have seen some discussion on the restriction on the parameter space such as MCMC on a bounded parameter space? this post, but I am wondering if there is a method to restrict the function or ...
math's user avatar
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Parallel Tempering and Bayesian Non-Parametrics

In parallel tempering, we run multiple MCMC chains in an ascending temperature ladder, where the posterior density of the $i$th chain is exponentiated to the reciprocal of the temperature of the $i$th ...
Faydey's user avatar
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Enumerating feasible solutions to the subset sum problem using Gibbs sampling

Given a set of $m$ strictly positive real numbers $W = \{ w_{1}, \dots, w_{m} \}$, I want to find subsets of $W$ whose sum is less than or equal to a maximum value $N$ using Gibbs sampling. To do this,...
scj's user avatar
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A recurrent Markov Chain implies its k-step version is also recurrent?

I am curious about whether a Markov Chain $X_n$ is recurrent implies that for any $k > 0$, $X_{kn}$ is also recurrent. Here are my observations. If $X_n$ is transient, $X_{kn}$ must be transient by ...
qqhgsjah8221's user avatar
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23 views

Metropolis-Hastings on domain $(2, \infty)$

I'm trying to understand the Metropolis Hastings algorithm in depth by solving some exercise problems. On one of them, I'm asked to use MH to generate samples from $$f(x) = c \frac{1}{\theta}e^{-\frac{...
Christina Kataki's user avatar
3 votes
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42 views

Does the mode of MCMC samples equal the MAP of the posterior?

If I had millions of MCMC samples from a posterior, should the most frequent value among those samples (i.e., the peak of a histogram of those samples) at least in principle always equal the maximum-a-...
Durden's user avatar
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2 votes
1 answer
68 views

Reversible-jump MCMC and Poisson processes

Suppose we have a time interval $t \in [0, T]$ in which events occur as a Poisson process with some arbitrary time-dependent rate $\lambda(t)$. These events occur at times $Y=(Y_1, Y_2, \dotso, Y_M)$ ...
Jordan's user avatar
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About paper "Deep Unsupervised Learning using Nonequilibrium Thermodynamics"

Here is the paper link related to the question from my title. In Appendix B, it computes the entropy of $p(X^T)$ and says "By design, the cross entropy to $\pi(x^t)$ is constant under our ...
user405729's user avatar
1 vote
0 answers
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Help finding Bayesian model with multi-modal posterior [closed]

Background: There is a paper (link) that concerns combining MCMC methods with a normalizing flow (a type of generative model). The basic idea is that the normalizing flow helps propose samples, which ...
caitlin's user avatar
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1 vote
0 answers
28 views

ESS with many short, parallel chains

I have a curious problem I would like to share and ask for references and advice about. I am working on a MCMC problem and I realized that I have big economies of scale when I am drawing many chains ...
SCS's user avatar
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160 views

Setting up a particle filter for a deterministic system with stochastic, time-discrete observations

I have a deterministic process $x(t)>0$ for $0 < t < T$, governed by an ODE for which I want to do parameter inference in a Bayesian sense. The process is hidden but I have $n$ stochastic ...
lmyt's user avatar
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The lower bound of acceptance rate for independent Metropolis–Hastings algorithm

In comparison with rejection sampling, for independent M-H algorithm, if there is a constant C such that$$f(x)=\frac{p(x)}{\int p(x)dx} \leqslant Cg(x)$$ for all x, then the acceptance rate is at ...
向洋杉's user avatar
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119 views

How to compute Expected Squared Jump Distance (ESJD) of a Metropolis-Hastings algorithm

The Expected Squared Jump Distance (ESJD) seems to be defined slightly differently in various papers, which makes this very confusing. For instance, Definition 2.2 of Optimal Scaling of Random-Walk ...
Euler_Salter's user avatar
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1 vote
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Bayesian estimation of nonlinear state space model using Gibbs Sampler

I want to estimate a state space model that looks like this: $y_t = exp(\beta_t)x_t + \varepsilon_t \hspace{1cm} \varepsilon_t \sim \mathcal{N}(0, \sigma^2_{\varepsilon})$ $\beta_t = \rho\beta_{t-1} +...
Giorgetto's user avatar
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Showing that the estimator of the log posterior used in stochastic gradient MCMC is unbiased

In the SGLD paper as well as in this paper it is claimed (paraphrasing) that the following estimator: $$\widetilde{U}(\theta) = -\dfrac{|\mathcal{S}|}{|\widetilde{\mathcal{S}}|} \sum_{{x}\in \...
Tan's user avatar
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