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Results tagged with markov-chain-montecarlo
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user 7224
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.
13
votes
Why should we care about rapid mixing in MCMC chains?
In completion of both earlier answers, mixing is only one aspect of MCMC convergence. It is indeed directly connected with the speed of forgetting the initial value or distribution of the Markov chain …
- 108k
1
vote
Accepted
MCMC - reject value or keep the same
Rejecting until an acceptance without repeating the current value induces a bias in the algorithm since it does not simulate from the proposal but from another distribution. Many questions on this for …
- 108k
1
vote
Examples of errors in MCMC algorithms
A very clear case (connected with the marginal likelihood approximation mentioned in the first answer) where true convergence is the example of the problem of label switching in mixture models coupled …
- 108k
6
votes
When is MCMC useful?
When you are given a prior $p(\theta)$ and a likelihood $f(x|\theta)$ that are either not computable in closed form or such that the posterior distribution $$p(\theta|x)\propto p(\theta)f(x|\theta)$$i …
- 108k
3
votes
Accepted
Why do we require aperiodicity in MCMC?
Aperiodicity or the absence thereof is a minor nuisance: when a chain is periodic, it does not converge to the stationary distribution, strictly speaking. This does not prevent Birkhoff's ergodic theo …
- 108k
3
votes
Accepted
In MCMC, why can throwing only initial data away improve the estimate?
Without proposing here a complete recap of Markov chain Monte Carlo methods, let me point out that the fundamental concept behind these methods is that they converge to the distribution $\pi^\star(\cd …
- 108k
2
votes
Sufficient ESS in MCMC
The theoretical ESS of a Markov kernel is
measures the modification in the asymptotic variance due to autocorrelation
is unknown in most cases, thus need be estimated
varies with the parameterisation …
- 108k
2
votes
Using autocorrelation to measure a Markov chain's mixing time
This is a consequence of a CLT established by Kipnis and Varadhan (1986):
If the Markov chain $(X_n)$ is aperiodic, irreducible, and reversible
with invariant distribution $\pi$, the CLT applies …
- 108k
7
votes
Understanding MCMC: what would the alternative be?
Calculating the denominator does not help in understanding the nature of the posterior distribution (or of any distribution). As discussed in a recent question, to know that the density of a d-dimensi …
- 108k
5
votes
Accepted
Can the Acceptance rate for Metropolis-Hastings be greater than 1?
It all depends what you mean by "acceptance rate". If this means the ratio $$\dfrac{\pi(\theta^\text{prop})}{\pi(\theta^\text{current})}\times \dfrac{q(\theta^\text{current}|\theta^\text{prop})}{q(\th …
- 108k
2
votes
How to generate a random draw from the joint posterior density after MCMC has converged?
Assuming $\beta^{(t)}$, the value of the Markov chain after $t$ iterations, is demonstrably generated from the stationary distribution with density $f$, generating$$\beta_1^{(t+1)}|\beta_2^{(t)}\sim f …
- 108k
4
votes
Accepted
Scale log-likelihood as MCMC sampler advances, to improve acceptance rate
In short, this method is not statistically valid without further steps.
The function
sample = mcmc_sampler(log_lkl, log_prior, lkl_scale)
should be detailed because, as presented, the method does …
- 108k
2
votes
Hastings ratio for ensemble samplers
To simplify the algorithm, consider the case when a fixed point $\xi\in\mathbb R^n$ is used as a pivot and the proposal is
$$Y_t=\xi+Z_t(X_t-\xi)\qquad Z_t\sim p(z)\propto 1/\sqrt{z}\quad z\in(a^{-1}, …
- 108k
2
votes
Is the result of Bayesian Inference with MCMC reliable since there maybe a big variance?
The question seems to be mixing two issues:
Is using an MCMC approximation reliable?
Is returning the average of the MCMC sample a good Bayesian estimate?
Concerning point 1., MCMC may fail to con …
- 108k
1
vote
Metropolis-Hastings simulation of independent geometric random variables
If you consider each Metropolis-Hastings scheme separately with a separate acceptance probability, i.e., distinguish adding from removing as two separate Metropolis-Hastings schemes, the chain associa …
- 108k