**254 metropolis-hastings questions.**

I'm trying to understand how to estimate the parameter vector $\mathbf{\theta} = (\theta_1,\theta_2, \theta_3)$ of a model using the MH algorithm.
I am given a joint posterior density:
$p(\mathbf{\...

Let's say I have a $GARCH(2, 3)$ model with $$\nu_i = \sigma_i\epsilon_i$$ where $\epsilon_i \sim N(0, 1)$ and $$\sigma_i^2 = a_0 + \sum\limits_{k = 1}^{2} a_k\sigma_{i - k}^2 + \sum\limits_{l = 1}^{3}...

I'm trying to implement the Adaptive Metropolis-Hasting (AM) algorithm. I know there are many AM algorithms out there. The one I want to use is proposed by Haario et al. (2001) and later restructured ...

I have time-series data generated via Metropolis algorithm - Monte Carlo simulations. I need to know correlation between data points generated given by $r_k = c_k/c_0$ where $c_0$ is the variance of ...

The Metropolis-Hastings ratio is defined as
$$
\alpha(x'|x) = \min\left(1, \frac{P(x')g(x|x')}{P(x)g(x'|x)}\right)
$$
and the state $x'$ is accepted if $u \leq \alpha(x'|x)$, where $u$ is ...

I was going through the Stan documentation which can be downloaded from here. I was particularly interested in their implementation of the Gelman-Rubin diagnostic. The original paper Gelman & ...

I wanted to implement multinomial probit in Bayesian with random-walk Metropolis Hasting. To achieve the best numerical efficiency when drawing $\beta$, I need to use the hessian matrix of $\beta$. ...

what are the differences between M-H algorithm and M-H-within-Gibbs algorithm. If possible, upload for me the two algorithms please.

Additionally: Use a simple symmetric random walk as the proposal distribution.
Source: "Introduction to Stochastic Processes with R" - Robert P. Dobrow, Chapter 5 Exercises: Question 5.6
I know this ...

Can the acceptance rate in MH algo be greater than 1? When that case occurs the proposal will off coruse be accepted with probability 1. But is it "ok" to allow a acceptance rate greater than 1?

Basic question about MCMC Metropolis–Hastings algorithm. I am trying to understand the Metropolis–Hastings algorithm and it's connection to Bayesian Analysis. Suppose I want to construct an MCMC MH ...

I have a proposal distribution for one parameter theta_guesstheta_guess = guessleft(theta_accept(1,r-1), 0.01,0)which is a ...

Why does the indicator function is equivalent to the integral over the Dirac mass?
In my lecture notes the proof for the Kernel of the Metropolis Hastings is given as follows:
$$P(X^t \in \mathcal{X}...

I came across the following simulation problem: given a set $\{\omega_1,\ldots,\omega_d\}$ of known real numbers, a distribution on $\{-1,1\}^d$ is defined by
$$\mathbb{P}(X=(x_1,\ldots,x_d))\propto (...

I want to write a Metropolis sampler to sample independent rvs $x$ from the mixture model $X \sim \frac{1}{2}\big[\mathscr{N}(\mu_1, \sigma_1) + \mathscr{N}(\mu_2, \sigma_2)\big]$.
My algorithm is ...

Intuitively, if I want to update two parameters in one step, I have to come up with a proposal that are good for both parameters. Assuming that the parameters are independent, is it correct to ...

For an ergodic Markov chain, it doesn't necessarily have to be $Detailed\ Balanced $ when it converges to stationary distribution, which means that:
$\pi(\theta)\ P(\theta^{\prime}|\theta) \neq \pi(\...

Is this ok to choose the same proposal distribution as the prior in Metropolis algorithm?
Perhaps it's a simple question and to me, it's totally fine but as I always see people choose different ...

I am using MCMC with the Metropolos-Hasting algorithm to generate solutions of a non linear regression problem.
Likelihood
My likelihood is a gaussian distribution centered in 0 of the residuals ...

I have just been doing some reading on Gibbs sampling and Metropolis Hastings algorithm and have a couple of questions.
As I understand it, in the case of Gibbs sampling, if we have a large ...

I tried to simulate from a bivariate density $p(x,y)$ using Metropolis algorithms in R and had no luck. The density can be expressed as $p(y|x)p(x)$, where $p(x)$ is Singh-Maddala distribution
$p(x)...

If one has to sample (with replacement) from a population $(x_1,x_2,\ldots)$ with weights $(\omega_1,\omega_2,\ldots)$, possibly infinite (although this is asking too much without further details), a ...

I want to speed up my R implementation of a Metropolis Hasting procedure by replacing the slow parts with functions written in Rcpp.
There are already some examples online using Rcpp to speed up ...

Typically in Gibbs sampling we want to sample from a joint distribution $p(X_1, X_2, ..., X_N)$, but because the joint is hard to sample from directly, we instead achieve this by iteratively sampling ...

Consider a univariate normal model with mean $µ$ and variance $τ$ . Suppose we use a Beta(2,2) prior for $µ$ (somehow we know µ is between zero and one) and a $log-normal(1,10)$ prior for $τ$ (recall ...

Assume I have a function $g(x)$ that I want to integrate
$$ \int_{-\infty}^\infty g(x) dx.$$
Of course assuming $g(x)$ goes to zero at the endpoints, no blowups, nice function. One way that I've been ...

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 ...

I think I am confused due to the lax notation typically used when dealing with probabilities and not having a formal probability background.
Bayes' Rule tells me that
$$Pr(X_t=a|X_{t+1}=b)Pr(X_{t+1}=...

I have been trying to get a sense of the different problems in frequentist settings where MCMC is used. I am familiar that MCMC (or Monte Carlo) is used in fitting GLMMs and in maybe Monte Carlo EM ...

I'm runing MCMC using Metropolis-Hasting algorithm to fit an equation with 6 parameters on a dataset composed of 30 instances. How will the fact that my dataset is so small impact the posterio ...

Like I undestand MCMC sampling, the fulfillment of the detailed balance equation guarantees that our MC has reached its stationary distribution (given we ensure ergodicity).
Detailed Balance is:
$\...

Consider a heteroskedastic model of the form, $y_i|x_i \sim \mathcal{N}\left(x_i, \text{exp}\{\boldsymbol\beta^\top\boldsymbol{x}\}\right)$ where $\boldsymbol{\beta}=\left[\beta_0,\beta_1\right]$ and $...

I started reading "A Conceptual Introduction to Hamiltonian Monte Carlo" today, and I've gotten stuck on understanding Betancourt's explanation of what a "typical set" is.
If $q_1, q_2, \ldots, q_n$ ...

Let $X \sim \text{Normal}(\mu, \sigma^2)$. Define $Y = \frac{e^X -1}{e^X+1}$. The inverse transformation is $X = \text{logit}\left(\frac{1+Y}{2}\right) = \log\left(\frac{1+Y}{1-Y} \right)$. By the ...

I'm working through a book called Bayesian Analysis in Python. The book focuses heavily on the package PyMC3 but is a little vague on the theory behind it.
Say I'm looking at a model like this
My ...

I have been going through Radford Neal's excellent HMC book chapter in detail. However, there is one detail that I'm really obsessing with now, and I'm not sure if I'm thinking about it right. When ...

I'm attempting to implement a Metropolis-Hastings Algorithm to evaluate integrals of the following form:
$$I =\frac{1}{\sqrt\pi}\int_{-\infty}^{\infty} {f(x)\exp(-x^2)} \text{d}x$$
Now we can ...

There are multiple versions of adaptive Metropolis Hastings algorithms. One is implemented in the function Metro_Hastings of R ...

$X$ is a multivariate Gaussian, whose confidence region I can derive.
$Y$ is a function of $X$, specifically $Y = (x_1, x_2 - b x_1^2 + 100b, x_3, \dots, x_n)$. I can use change-of-variable ...

I want to sample from a multidimensional gaussian distribution to perform a metropolis hastings algorithm updating by blocks the $n$ parameters. In order to do this, I know that I require a vector $\...

Suppose that we are given the matrix,
$$ A = \begin{pmatrix}6/5 & 3 & -3/10 & -4/10\\
7/5 & -7/10 & 7/10 & 14/5\\
-6/10 & -7/10 &-1/2 & 3/10\\
12/5 & 1 & ...

Suppose $X_{i}|\theta_{i} \sim D_{1}(\theta_{i})$ and $\theta_{i}|\phi \sim D_{2}(\phi)$. Moreover $\phi \sim D_{3}(c)$ where c is known. How would I generate samples for $p(\theta_{i}|\pmb{x})$ if I ...

I apparently do not understand correctly how nodes are updated in PyMC. I want to count the number of times that certain nodes are computed, in order to understand where CPU time will be spent when I ...

I am optimising a model with parameters with uniform bounded priors using Metropolis-Hastings sampling.
When any of the parameters in the proposed parameter set is outside of these bounds I reject ...

Let $p(m,n|\pmb{X}) \propto f(m,n)$. Now using a metropolis Hasting's algorithm, I need to sample values for $(m,n)$. I plan on using a Bivariate normal distribution as the proposal function. I have ...

I'm trying to use MCMC to fit different models (e.g. auto-regressive, mean-reverting etc) to some time series data. In the MCMC examples that I can find, the observations at different time points are ...

Suppose the 1 x N vector $V\in \{0,1\}^N$ comes from the pdf $f(V) = VWV^T$, where $W$ is a N x N positive definite matrix.
If the weight matrix is given, I can use gibbs sampling to generate a ...

I am trying to make a GIBBS/MH sampler for a bayesian model for football goal counts. (Without any specific packages(winbugs,jags etc)). Similar to this:
I have the priors for the hyperparameters and ...

I'm working on a project using Monte Carlo. The function below is giving me trouble with the metropolis algorithm. Does this function not satisfy the features required to run the metropolis algorithm ...

I need to implement a Metropolis Hastings where the acceptance probability $\alpha$ is not a probability but a logarithm of a score. The logarithm of a score is a negative float.
In original ...

- mcmc
- bayesian
- gibbs
- sampling
- monte-carlo
- markov-process
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- self-study
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- r
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- mathematical-statistics
- likelihood
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- pymc
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- distributions
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