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 ...

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Metropolis, simulated annealing and proposal distributions

I'm trying to understand the physical meaning of the proposal step generating function in Metropolis algorithm. In the original paper, and most derivations I found, it seems that it's not much ...
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State-of-the-art sampling methods for different information about target density

I was wondering what are the current state-of-the-art methods (aka your favourite methods, if you are an expert) for Monte Carlo sampling from a target density function $f(x)$ with $x \in ...
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markov blanket for gibbs sampling in graphical model

below is a bayes net where the nodes are discrete. I want to Gibbs sample each of the $S_t$ nodes. for example, to Gibbs sample $S_1$ conditioned on rest of variables, it should be sufficient to ...
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38 views

Bespoke MCMC priors & likelihoods, & feeding a posterior joint pdf back in as the prior next time

We're looking at PyStan, PyMC3 and emcee. (switching to R could also be an option, if need be). We have a lot of bespoke priors and bespoke likelihood functions: they are bespoke in the sense that ...
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12 views

how to obtain burn-in value in Markov Chain Monte Carlo Methods [duplicate]

I am working on MCMC methods in my thesis. Please is there a definite way or method of obtaining the burn-in value of MCMC methods? But firstly, I would love to see a particular problem solved using ...
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28 views

Ones trick in BUGS gives node inconsistent with parents error [closed]

Edit: This issue doesn't come up if I use OpenBUGS. But I can't use it for my bigger problem as it seems "very slow" compared to JAGS at least on my machine. I am using JAGS as my BUGS flavor to run ...
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15 views

Full conditionals for the parameters of a Bayesian regression with dependent components

Let $\mathbf{y}_i=\{y_{ij}\}_{j=1}^p$, $i=1,\dots ,n $ be a $p-variate$ vector and $$ y_{i,j} = \alpha_{j}+\beta_{j}x_{ij}+\epsilon_{ij}, $$ where $x_{ij}$s are known constants and ...
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36 views

For Metropolis-Hastings algorithm, should target density and proposal distribution have the same distribution?

I watched some youtube videos about the Metropolis-Hastings algorithm. They used a Gaussian as a proposal function to estimate an unknown Gaussian, or used a Gamma function as the proposal function to ...
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10 views

MCMCGLMM for continuous data

while MCMCGLMM specifies as a generalised mixed effect model (and is therefore unsuitable for analyzing continuous data with a Gaussian distribution), the function specifically offers ...
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25 views

Notation in Markov Chain and MCMC literature

In the MC and MCMC literature one commonly finds statements of the following form (see e.g. Roberts & Rosenthal, 2004): $$ \int_{x \in \mathcal X} \pi(dx)P(x, dy) = \pi(dy). $$ What is the ...
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38 views

model fitting of data to multiple distributions

I have a set of numbers $ X = \{x_1, x_2,\ldots,x_n\}$ and I am interested in finding the most fitting combination of these numbers to multiple exponential distributions. Using predefined rules, I ...
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44 views

slice sampling within a Gibbs sampler

Questions My questions are: Is the following slice-sampling-within-Gibbs approach valid? Is there a good reference out there that uses, or better yet, justifies it? Context I'm trying to sample ...
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2answers
54 views

Monte Carlo Simulation of Complex Dynamical System

Assume that $\vec{z}(t)$, the state at time $t$ of a particle in a two-dimensional space, can be fully described by its position and velocity: $\vec{z}(t) = [r_x(t)\ r_y(t)\ v_x(t)\ v_y(t)]$. ...
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21 views

Acceptable range for standard error in MCMC simulation

Is there a convention when it is acceptable to accept an estimate like an empiric mean generated by a MCMC run based on the value of the standard error, similar to Cohen's effect size categories?
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17 views

Reporting MCMC autocorrelations of variables in a hierarchical model

I implemented a a Gibbs Sampler for a hierarchical model with priors and hyperpriors that has around 16 variables. When it comes to autocorrelation plots, I have seen in some papers that they do not ...
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68 views

Gibbs sampler gets stuck in local mode

I am very new to statistics and trying to implement a Gibbs sampler. However, according to wikipedia https://en.wikipedia.org/wiki/Gibbs_sampling and this discussion thread ...
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15 views

MCMC regression model - fixed or stochastic variance

I'm trying to fit a regression model to my data. I have apriori stochastics $a,b,d,\omega$ and a formula: $$\mu_t = a + b*t + d*t*cos(\frac{2 \pi}{\omega$})$$ I want to fit this to my data using $Y_t ...
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20 views

Switchpoint test with many customers

Im just reading the wunderful book Probabilistic Programming and Bayesian Methods for Hackers. In the very first chapter they inspect whether the habits of one single customer change over time, for ...
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How to sample random variables (x,y) from a bivariate Cauchy distribution using a Gibbs sampler?

A bivariate Cauchy distribution is equivalent to a bivariate t-distribution with 1 degree of freedom.
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32 views

How the Markov Chain Monte Carlo ensure the stationary distribution converge to the target distribution?

I am reading about the MCMC but now I got a lot of questions. Firstly, It says we could construct a markov chain which satisfy the detailed balance:...
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52 views

Metropolis-Hastings simulation of independent geometric random variables

Consider the following Metropolis-Hastings scheme to sample independent geometric random variables $X = (X_1, \dots, X_N)$, where each $X_j$ has pmf $\mathbb{P}(X_j = x) = p(1-p)^x$ for $x \geq 0$. At ...
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58 views

Mathematical foundation of using MCMC in global optimization

MCMC is commonly used to compute the integral in the form of $$\text{Problem A.}~~\int F(x)\pi(x) $$ where $\pi$ is hidden. In the literature, it is explained why MCMC can handle problem A by ...
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52 views

MCMC advice: Ignoring some parameters in a MCMC scheme?

I am after some general advice regarding my MCMC scheme, which is causing me some grief. Essentially, I have a large (2N + 9 parameters) MCMC scheme which works great. However, the problem is that ...
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40 views

question about bayesian structural time series model

I am investigating the stability of results from a bayesian structural time series model using bsts package in R. The following code estimates a local linear trend model (using an example from the R ...
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314 views

How to sample from discrete distribution on the non-negative integers?

I have the following discrete distribution, where $\alpha,\beta$ are known constants: $$ p(x;\alpha,\beta) = \frac{\text{Beta}(\alpha+1, \beta+x)}{\text{Beta}(\alpha,\beta)} \;\;\;\;\text{for } x = ...
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35 views

how to do MCMC simulation

I've been diving through the internet, trying to get the best possible understand on how to do this type of simulation, most of the information is theory or MCMC in simple english, but its been hard ...
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22 views

Samples from the posterior - Markov Chain Monte Carlo

I am self-studying MCMC in the context of graphical models. I understand that if we run MCMC on a graphical model (e.g. a belief net) that the model will hopefully converge to a stationary ...
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47 views

Setting up a MCMC scheme for Multivariate Stochastic Volatility

I want to understand the survey of Lopes and Polson (2010) regarding MV stochastic volatility. Assume the $p$-dimensional vector $y_t$ follows $$y_t\sim N(\Theta,\Sigma_t).$$ In order to model the ...
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26 views

Comparing two Bayesian models under disjoint prior supports using MCMC

I have a Bayesian model involving three parameters $\theta_1$,$\theta_2$ and $\theta_3$. Experts think that $\theta_1 > \theta_2 > \theta_3$. So I would like to test the submodel $M_0$ ...
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63 views

Lift in Bayesian A/B-test with pyMC

I'm implementing an A/B-test in pyMC to determine which of two groups to bet on in terms of pageviews per uniqe user. Working code, but I would love some feedback ...
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25 views

Can someone explain Metropolis Hastings algorithm with simplest possible example? [duplicate]

I am just beginning to learn about Metropolis Hastings algorithm and MCMC techniques. I have a basic understanding of Markov chains and stationary distributions and need for the Metropolis Hasting but ...
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How to model discrete co-occurence data

I have a dataset of transactions where in each transaction a class is tagged with a number of labels. I would like to build a probabilistic model using this data to determine given a class how likely ...
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96 views

Understanding Monte Carlo sampling

In rejection sampling or Markov chain Monte Carlo methods, we usually have a target distribution $p(x)$ whose form makes it difficult or impossible to draw samples directly, but we can evaluate $p(x)$ ...
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74 views

Using a black box MCMC algorithm as a proposal distribution

Let's say I have some MCMC algorithm implemented as a function, black_box(x), which generates samples from $P(x)$ when run in a chain. I would now like to sample ...
3
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23 views

Compute autocorrelation in a trace of covariances matrices sampled by MCMC

Imagine that we sample a covariance matrix from a Wishart distribution by MCMC. At every iteration, we get a new sample matrix $S_i$ from the Wishart distribution. Q: Given the trace that contains ...
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What is the difference between Metropolis Hastings, Gibbs, Importance, and Rejection sampling?

I have been trying to learn MCMC methods and have come across Metropolis Hastings, Gibbs, Importance, and Rejection sampling. While some of these differences are obvious, i.e., how Gibbs is a special ...
3
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1answer
47 views

Measuring dependency of subsequent points from Markov chain

The question is about stimulating different type of species (coded 1-10) based on given species frequencies, and other parameters (eg. mean of normally distributed mass and ratio) using gibbs ...
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40 views

In a Bayesian Framework, can we ever replace nuisance parameters by their MLE?

I am wondering if it is permissible in a Bayesian analysis to replace nuisance parameters with their maximum likelihood estimates. I am dealing with a model which commonly has several thousand of ...
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Why does sampling from the posterior predictive distribution $p(x_{new} \mid x_1, \ldots x_n)$ work without having to average out the integral?

In a Bayesian model, the posterior predictive distribution is usually written as: $$ p(x_{new} \mid x_1, \ldots x_n) = \int_{-\infty}^{\infty} p(x_{new}\mid \mu) \ p(\mu \mid x_1, \ldots x_n)d\mu $$ ...
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Estimation of arithmetic Brownian motion volatility with transformed data

I want to estimate the volatility $\sigma$ of a process $(X_t)$ following an arithmetic Brownian motion, that is, for a constant time step $\Delta$, $X_{t+\Delta} = X_t + \sigma B_{\Delta}$ , where ...
2
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2answers
123 views

“True” answer for MCMC model

Theoretically, given a model with $N$ parameters and $\forall x \in \mathbb{R}.\; p(x)>0$ in the prior of all parameters. If i'm interested only in the end result and not in time-to-convergence, ...
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29 views

Given a marginal posterior distribution, when should I report an upper limit on the parameter?

I've been utilizing MCMC in the analysis of $\gamma$-ray spectra. For many of my model-parameters, the marginal posterior distributions, $p( \theta_i |X)$ where ($\theta_i\in(0,1]$), are normal ...
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35 views

How does one determine which variables can be collapsed in Gibbs Sampling?

I am going through the derivation for Gibbs Sampling update equations for LDA. The claim is that $\theta_{d}$ (document specific topic distribution) and $\phi_k$ the topic-word distribution can be ...
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44 views

Hamiltonian Monte Carlo with large parameter values fail to converge

I'm trying to learn about Hamiltonian Monte Carlo. Therefore I tried to infer the Parameters of a Multivariate Normal given some samples. My procedure is the following: Define $\mu$ and $\Sigma$ ...
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26 views

MCMC with dependent variables

I want to run Metropolis-Hastings on a problem which involves two parameters that are not independent. I.e. I want to estimate both of these parameters. At the moment I'm trying to understand if this ...
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15 views

Interpretation of Gibbs sampling in Dirichlet Process posterior calculation

I wonder if anyone who is familiar with Gibbs sampling in the context of Dirichlet Process semi-parametric models could please help clear this question up. In Radford M. Neal's 2000 paper "Markov ...
2
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1answer
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Using MCMC to sample from a posterior, are our posterior beliefs on parameters independent?

I've been given a classification problem in which MCMC (slice-sampling) is used to sample from a hierarchical posterior. After getting $n$ samples, the Monte Carlo method can be used to give an ...
2
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1answer
43 views

Efficiently sampling from mixture distribution posterior

I have the following model: $$ \begin{align} \pi_1\sim & \text{Unif}(0,1)\\ \lambda_1,\lambda_2\sim & \text{Ga}(1,1)\\ z_i\sim & \pi_1^{1(z_i=1)}\pi_2^{1(z_i=2)}\\ ...
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64 views

how to get the stationary distribution of MCMC (Markov Chain Monte Carlo) model

Based on my reading some course notes as well as the answers and comments to this SO post I started thinking about the general steps for creating a stationary distribution. Assuming my problem were ...
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19 views

Convergence in MCMC [duplicate]

I am using the gelman.diagfunction in R to check the convergence of MCMC. This is the result: ...