Editing Markov chain Monte Carlo (section)

====Proposal tuning and adaptation====
Another approach to reducing correlation is to improve the MCMC proposal mechanism. In [[Metropolis–Hastings algorithm]], step size tuning is critical: if the proposed steps are too small, the sampler moves slowly and produces highly correlated samples; if the steps are too large, many proposals are rejected, resulting in repeated values. Adjusting the proposal step size during an initial testing phase helps find a balance where the sampler explores the space efficiently without too many rejections. 

Adaptive MCMC methods modify proposal distributions based on the chain's past samples. For instance, adaptive metropolis algorithm updates the Gaussian proposal distribution using the full information accumulated from the chain so far, allowing the proposal to adapt over time.<ref>{{cite journal
 | last1  = Haario | first1 = Heikki
 | last2  = Saksman | first2 = Eero
 | last3  = Tamminen | first3 = Johanna
 | title  = An adaptive Metropolis algorithm
 | journal= Bernoulli
 | volume = 7 | issue = 2 | pages = 223–242
 | year   = 2001
 | doi    = 10.2307/3318737
 | jstor = 3318737
 | url    = https://www.researchgate.net/publication/38322292
}}</ref>