# Tag Info

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### When would one use Gibbs sampling instead of Metropolis-Hastings?

Firstly, let me note [somewhat pedantically] that There are several different kinds of MCMC algorithms: Metropolis-Hastings, Gibbs, importance/rejection sampling (related). importance and ...
• 106k
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### In Bayesian models, can you use Uniform(-inf, inf) as a prior?

On this forum, there are a lot of related questions and answers about flat priors, like the ones above. They are not uniform priors because they are not distributions but $\sigma$-finite measures (...
• 106k
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### Proposal distribution - Metropolis Hastings MCMC

A1: Indeed the Gaussian distribution is probably the most used proposal distribution primarily due to ease of use. However, one might want to use other proposal distributions for the following reason ...
• 15.6k
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### For Hamiltonian Monte Carlo, why does negating the momentum variables result in a symmetric proposal?

One of the reasons why the original construction of Hamiltonian Monte Carlo can be tricky to understand is that it is more restrictive than necessary, if only to simplify the theoretical proofs. In ...
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### Acceptance rate for Metropolis-Hastings > 0.5

The acceptance rate depends largely on the proposal distribution. If it has small variance, the ratio of the probabilities between the current point and the proposal will necessarily always be close ...
• 1,614
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### Understanding the Typical Set for Markov chain Monte Carlo sampling

$\mathrm{d}q$ is uniform across the entire space and that's the problem! Unfortunately as we consider higher-dimensional spaces out intuition of uniform starts failing us and we end up in conceptual ...

### Compute the likelihood in Metropolis–Hastings: How does it relate to a posterior in Bayesian Analysis?

There is a lot of confusion. You want to evaluate the posterior $$f(\theta|\mathbf{y}) =\frac{f(\mathbf{y}|\theta)f(\theta)}{f(\mathbf{y})}$$ where I use $f()$ to indicate a density, to be as ...
• 1,260
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### What is the relationship between Metropolis Hastings and Simulated Annealing?

Simulated annealing is a meta-heuristic algorithm used for optimization, that is finding the minimum/maximum of a function. Metropolis-Hastings is an algorithm used for exploring a function (finding ...
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### MCMC in a frequentist setting

As indicated in the many comments, Markov Chain Monte Carlo is a special case of the Monte Carlo method, which is designed to approximate quantities related with a distribution via pseudo-random ...
• 106k
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### Why periodically skip updating a parameter in MCMC?

This type of fine-tuned (Gibbs) MCMC is appropriate for cases when one conditional distribution is most "sticky" than other conditional distributions in the problem. For instance, updating only one [...
• 106k
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### What is the deeper intuition behind the symmetric proposal distribution in the Metropolis-Hastings Algorithm?

1) the Normal and Uniform are symmetric probability density functions themselves, is this notion of "symmetry" the same as the "symmetry" above? Both distributions are symmetric around their mean....
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### Acceptance rate for Metropolis-Hastings > 0.5

An easy example of acceptance probability equal to one is when simulating from the exact target: in that case $$\dfrac{\pi(x')q(x',x)}{\pi(x)q(x,x')}=1\qquad\forall x,x'$$ While this sounds like an ...
• 106k

• 106k
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### Conditional distribution of $\exp(-|x|-|y|-a \cdot |x-y|)$

Disclaimer: although there is nothing to complain about Ben's answer (!), except maybe that the normalising constant of the conditional is not of direct use, here is what I wrote while being off-...
• 106k
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### How does the Metropolis Algorithm "get off the ground"?

The confusion stems from a misunderstanding of the notation $$V \sim f_V$$ which means both (a) $V$ is a random variable with density $f_V$ and (b) $V$ is created by a PRNG algorithm that ...
• 106k
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### Using all Metropolis-Hastings proposals to estimate an integral

Recycling proposed values in a Metropolis-Hastings algorithm goes under the name of Rao-Blackwellisation. For instance, we made such a proposal in Casella and Robert (1996) Rao-Blackwellisation of ...
• 106k

### Monte Carlo Methods:

Preliminaries: The book Introducing Monte Carlo methods with R (no exclamation mark in the title, even though the Springer book series is called Use R!) was co-authored by my late friend George ...
• 106k
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### How to draw from a uniform distribution over a large state space via MCMC

Since you want to sample uniformly, $p(\xi) = c I(\xi \in S)$ so the ratio $\frac{p(\xi^*)}{p(\xi)} = I(\xi^* \in S)$ for all proposals $\xi^*$. In MH you accept with $\frac{p^*}{p}\frac{q}{q^*}$ but, ...
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### Multi parameter Metropolis-Hastings

You actually have a single joint prior, which is a function of the parameter vector $\theta = [a_1, ..., a_d]$. If the parameters are treated independently, the prior factorizes into a product of the '...
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### Use of Metropolis & Rejection & Inverse Transform sampling methods

It's not entirely correct to say that inverse methods are impossible to compute. There are perfectly good numerical approximations to the inverse Gaussian CDF. As far as I'm aware, plenty of methods ...
• 13.9k