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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Computation of the marginal likelihood from MCMC samples

This is a recurring question (see this post, this post and this post), but I have a different spin. Suppose I have a bunch of samples from a generic MCMC sampler. For each sample $\theta$, I know the ...
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1answer
23 views

Predictions from BSTS model (in R) are failing completely

After reading this blog post about Bayesian structural time series models, I wanted to look at implementing this in the context of a problem I'd previously used ARIMA for. I have some data with some ...
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2answers
45 views

Markov chain Monte Carlo (MCMC) for Maximum Likelihood Estimation (MLE)

I am reading a 1991 conference paper by Geyer which is linked below. In it he seems to elude to a method that can use MCMC for MLE parameter estimation This excites me since, I have coded BFGS ...
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27 views

Hierarchical modelling - partial pooling with correlation

I am doing a Bayesian regression. I have groups of data $(y_1 ~X_1), (y_2~X_2),...$, where each $y$ and $X$ is a vector. The subscript is regarded as group number. The completely unpooled regression ...
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1answer
19 views

What is the difference and relationship between posterior distribution function and likelihood function in MCMC?

I am learning MCMC in class, and I encounter one question about the relationship between posterior probability and likelihood function. In our lecture, the professor asked us to take samples from ...
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31 views

Population Monte Carlo Algorithm

I am trying to wrap my head around the Population Monte Carlo Algorithm. I want to implement it for a mixture model, but I am uncertain on how to proceed. I am mostly looking for references or ...
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0answers
13 views

Warning message after a MCMCglmm call in R

I have a Warning message after a MCMCglmm call in R. Let me first write about the aims of the study : 1) describe whether a set of seven different behavioural traits vary (or are consistent) in time ...
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47 views

What are examples where only a single sample is needed?

Consider the following setup: Let $\Omega$ be a finite (but humongous) state space and $\pi:\Omega\to[0,1]$ be a probability mass function. It seems to me that when people want to "sample" in this ...
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7 views

How to determine and use the sampling lag in the collapsed Gibbs sampler?

I am implementing a collapsed Gibbs sampler for LDA model. According to this technical note's word, average a number of samples, and often it is desirable to leave an interval of L iteration ...
2
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1answer
47 views

Proposal distribution - Metropolis Hastings MCMC

In Metropolis-Hastings Markov chain Monte Carlo, the proposal distribution can be anything including the Gaussian (according to the Wikipedia). Q: What's the motivation for using anything other than ...
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15 views

How does GibbsLDA++ ensure that we are sampling from a good posterior?

This is an extending of this question, which asked that whether we should do some estimating to ensure that we are really using a likely topic assignment instead of the one happened with low ...
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70 views

How to check the convergence in the collapsed Gibbs sampling of LDA? [closed]

I am trying to implement the LDA model fit by collapsed Gibbs sampling by myself. I have go through this article. And there is a clear pseudo code (section 5.5), ...
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1answer
18 views

Symmetric PDFs in Metropolis-Hastings

My textbook says that a symmetric PDF satisfies $$f(x|y)=f(y|x).$$ Can anyone explain this? Is it equivalent to $f(x+a)=f(x-a)$?
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21 views

Dirichlet process mixture MCMC

I'm reading Markov Chain Sampling Methods for Dirichlet Process Mixture Models by Radford M. Neal. Equation (3.6) states that If $c=c_{j}$ for some $j\neq i $: $P\left(c_{i}=c\;|\;c_{-i}, y_{i}, ...
3
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1answer
30 views

MCMC using GIBBS sampling: can different burn-in be used for different parameters?

I have run a stochastic volatility model with 4 parameters. I have used the Heidelberg and Welch convergence diagnostic. The result shows 3 out of 4 parameters have passed the stationary and ...
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27 views

How can the (Ensemble) Kalman filter be viewed as a MCMC algorithm?

I am struggling with quotes like The EnKF applies a Markov chain Monte Carlo (MCMC) method ... (1, p. 6) or In fact, the Kalman filter is a MCMC algorithm in the case of a linear and ...
2
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1answer
50 views

MCMC Metropolis-Hastings initial values [closed]

my posterior values that I obtained via Metroplis-Hasting are always around my initial values. For instance if I chose $\theta_0 =(1,2)$ my posterior values, after either taking mean or median, are ...
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42 views

Normalization factor of truncated multivariate normal distributions (Normalization factor from samples)

I would like to evaluate an expectation of the form $\int p(x) \;\mathbb{1}_\left(Bx \leq 0\right) dx$, where p(x) is a multivariate normal distribution. I.e. the probability of a linear constraint ...
2
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32 views

MCMC efficiency and nonlinear reparametrizations

The efficiency (e.g., effective sample size per density evaluation) of most MCMC methods depends on the parametrization. However, so far I have come across little work in the MCMC literature that ...
3
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2answers
116 views

The Harris recurrence of a stepping-out slice-sampling-within-Gibbs MCMC

I want to use a multistage version of the MCMC here. That is, I want to use a Gibbs sampler to draw from a general joint distribution $p(x_1, x_2, x_3, \ldots)$ with a Gibbs step for each full ...
1
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1answer
26 views

dmvnorm produce 0 likelihood

I am implementing an MCMC algorithm in R using the "mvtnorm" package. The data is about 150 dimensions so the likelihood produced by dmvnorm is usually zero (or -inf if "log=TRUE" is set), which make ...
2
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1answer
36 views

Multiple-Try Metropolis question

I read Multiple-Try Metropolis from Wikipedia and I do not understand some points. Suppose the current state is $\mathbf{x}$. The MTM algorithm is as follows: Draw ''k'' independent ...
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3answers
80 views

How to generate the transition matrix of Markov Chain needed for Markov Chain Monte Carlo simulation?

I'm conducting a sensitivity analysis of a model using MCMC approaches. By reading the code of the sensitivity test procedure, I find the steps in Markov Chain is quite similar to random walk. Also, ...
2
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1answer
79 views

Are there analytically derivable posteriors that save from doing MCMC other than conjugate priors? [duplicate]

Posteriors for conjugate priors can be analytically derived and save us from doing MCMC. Conjugate priors simply have a posterior in the same family as the prior distribution. Are there other ...
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2answers
75 views

Performance benchmarks for MCMC

Have there been large scale studies of MCMC methods that compare the performance of several different algorithms on a suite of test densities? I am thinking of something equivalent to Rios and ...
3
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1answer
29 views

Combining transition operators in MCMC

Many MCMC papers usually present a new single transition operator (or a family thereof) such as different proposals for Metropolis-Hastings, new forms of slice sampling, etc. I am interested in ...
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25 views

Joint modelling or two-step sampling

I am interested in sampling from the joint posterior $\pi(\theta,\lambda, G \mid X,Y)$ using MCMC. $X$ and $Y$ represent unrelated data and they are assumed to be generated along a graph $G$ by two ...
2
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1answer
34 views

Why using MCMC, not LHS (Latin Hypercube Sampling)?

Let's say I have to find the posterior distribution of a Bayesian estimate. Why should I use an MCMC chain of length 10,000 but not do a Latin Hypercube Sampling of 10,000 samples and calculate* the ...
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1answer
36 views

What makes parallel/distributed probabilistic inference difficult to implement?

My knowledge of probabilistic inference is severely limited, so coming from a Computer Science background I'm trying to understand what makes probabilistic inference difficult to implement in a ...
3
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0answers
24 views

Calculating error of MCMC algorithms?

If for example the Transitional MCMC algorithm is used (or does it matter which one?), what are the common approaches for calculating an error (some sort of distance from the actual PDF), or ...
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2answers
42 views

Why can't PyMC3 fit a uniform distribution with a Normal prior?

My question might be slightly ill-posed. My confusion comes from the oft repeated "Markov Chain Monte Carlo is a technique to solve the problem of sampling from a complicated distribution." -- ...
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0answers
29 views

MCMC chain stuck in adaptive phase?

I'm using R,OpenBUGS and R2OpenBUGS package on Linux Mint and i'm fitting a bayesian multinomial logistic model. After dribbling a lot of usual errors on my way to get my model running, i came upon an ...
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42 views

Bayesian A/B-test having trouble converging in PyMC

I'm calculating the estimated improvement of a group over another (in terms of clicks per user). Much like a A/B-test, but I'm using PyMC to be nice and Bayesian about it. This data and code works ...
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25 views

Extreme high autocorrelation in MCMCglmm but good prediction

I am fitting a logistic regression model without random effects with options set to: list(R = list(V=1, fix = 1)) nitt=100000 burnin=30000 The data set is very ...
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0answers
19 views

Can a Markov chain be approximated with an AR process?

In some MCMC literature/source code, a Markov chain is often approximated with an AR(1) process. There is some theory to suggest that such an approximation is somewhat valid for a finite state space, ...
4
votes
2answers
52 views

Number of Markov chain Monte Carlo Samples

There is a lot of literature out there about Markov chain Monte Carlo (MCMC) convergence diagnostics, including the most popular Gelman-Rubin diagnostic. However, all of these assess the convergence ...
0
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1answer
26 views

Sampling from Posterior without MCMC

I want to sample from the posterior density $f(x \mid y)$ which is related to the prior density $f(x)$ and likelihood density $f(y \mid x)$ via Bayes' rule: $$ f(x \mid y) = \frac{f(y \mid x) f(x)}{c} ...
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0answers
50 views

Prior Parameters in Bayesian Hierarchical Linear model

I'm trying to fit a linear model to describe student performance in 2 different schools. My response variable is $$Yij= X{ij}*\beta+Z_{ij}*\gamma_j + \epsilon_{ij}$$ . $$i = 1,...,n $$ $$j = 1,2 ...
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0answers
14 views

How to compute initial weights of SMC, when you initialize the algorithm with MCMC draws

Let $$y_t | \alpha_t \sim N(0, \exp(\alpha_t)), \\ \alpha_t = \phi_0 + \phi_1\alpha_{t-1} + \sigma \epsilon_t, \\ \text{ where } \epsilon_t \sim N(0,1) \text{ i.i.d.},\\ t=1,2,\cdots,T$$ The unknown ...
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1answer
52 views

MCMC convergence, analytic derivations, Monte Carlo error

I'm trying to figure out some convergence statements on an MCMC example. The setup is: I'm generating data samples as observations from a (known) deterministic parameter, say $s$ (using a forward ...
0
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0answers
15 views

Derive Marginal Posterior to set up Gibbs-Sampler

I am currently trying to replicate a Hierarchical Model for multivariate returns proposed in the paper Portfolio selection using hierarchical Bayesian analysis and MCMC methods. However, in order to ...
0
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0answers
22 views

Inversely weighting posterior probabilities in the Metropolis-Hastings algorithm

During the Metropolis algorithm sampling procedure, given a stationary distribution, $\pi(\theta^{*}|\theta^{(T)})$, arising from a corresponding Markov chain, the following acceptance ratio is ...
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0answers
49 views

Singular proposal in MCMC

Suppose we want to obtain samples of the density $f(\mathbf{x})$ where $\mathbf{x}$ is a $d$-dimensional vector, i.e. $\mathbf{x} = (x_1, x_2, \dots, x_d)$. To that end, we choose the ...
3
votes
1answer
90 views

Fitting power function to data

I am trying to implement an MH algorithm to fit a power function to my data. The power function has the following form: $\hat{y} = a * x^b$ The data are assumed to be normally distributed ...
2
votes
2answers
51 views

Can the parameterization of the prior help with MCMC inference?

Say we are trying to estimate the posterior over a set of random variables using MCMC for a Bayesian model. We have prior knowledge about the variables and we can express this knowledge as a prior ...
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44 views

Priors on variable ordering and/or percentile ranking

Consider a set of variables $\mathbf{X}$ = $X_1 \ldots X_n$ where each variable is $\in [0,1]$. I am modeling an inference problem on these variables. Among other things, I have the following prior ...
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34 views

can Bayesian estimation be used to compare two posteriors estimated from MCMC?

If I have two posterior prediction distributions, each sampled 1000's of times using MCMC in rJAGS, can I use those MCMC samples (N = several thousand) to compare the two groups using Bayesian ...
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votes
1answer
20 views

Preventing Pareto smoothed importance sampling (PSIS-LOO) from failing

I recently started using Pareto smoothed importance sampling leave-one-out cross-validation (PSIS-LOO), described in these papers: Vehtari, A., & Gelman, A. (2015). Pareto smoothed importance ...
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0answers
18 views

How to implement MCMC sampling of a non-standard distribution

Given that I know the distribution to a constant. What sampling method? MH(SGMCMC) or slice sampling? More importantly, what tools/package can I use to impelement the sampling directly(input the ...
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26 views

Why don't we use Kernel Density Estimation instead of MCMC?

Markov Chain Monte Carlo has a goal of approximating the posterior in Bayesian inference. Kernel density estimation can be thought of regressing the probability density function. Why don't we use ...