Content deleted Content added
Add HMC; as a particular adaptation of the Metropolis-Hastings scheme. |
added PubMed |
||
Line 141:
If a Gaussian proposal density <math>g</math> is used, the variance parameter <math>\sigma^2</math> has to be tuned during the burn-in period.
This is usually done by calculating the ''acceptance rate'', which is the fraction of proposed samples that is accepted in a window of the last <math>N</math> samples.
The desired acceptance rate depends on the target distribution, however it has been shown theoretically that the ideal acceptance rate for a one-dimensional Gaussian distribution is about 50%, decreasing to about 23% for an <math>N</math>-dimensional Gaussian target distribution.<ref name=Roberts/> These guidelines can work well when sampling from sufficiently regular Bayesian posteriors as they often follow a multivariate normal distribution as can be established using the [[Bernstein-von Mises theorem]].<ref>{{Cite journal |last1=Schmon |first1=Sebastian M. |last2=Gagnon |first2=Philippe |date=2022-04-15 |title=Optimal scaling of random walk Metropolis algorithms using Bayesian large-sample asymptotics |journal=Statistics and Computing |language=en |volume=32 |issue=2 |pages=28 |doi=10.1007/s11222-022-10080-8 | pmid=35310543 |issn=0960-3174 |pmc=8924149}}</ref>
If <math>\sigma^2</math> is too small, the chain will ''mix slowly'' (i.e., the acceptance rate will be high, but successive samples will move around the space slowly, and the chain will converge only slowly to <math>P(x)</math>). On the other hand,
| |||