= Barker acceptance probability
{c}
{title2=$\beta(x,y)=\dfrac{\pi(y)q(y,x)}{\pi(x)q(x,y)+\pi(y)q(y,x)}$}
= Barker acceptance rule
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{synonym}
The <Barker acceptance probability> accepts a proposal using the displayed ratio. If $a=\pi(x)q(x,y)$ and $b=\pi(y)q(y,x)$, the accepted flux is $ab/(a+b)$, symmetric in the two states. This proves <detailed balance>; define acceptance as zero when $a=b=0$. Rejected proposals contribute to the diagonal <transition probability>. Replacing $\pi$ by an unnormalized target leaves the ratio unchanged. For a common proposal, $t/(1+t)\leq\min(1,t)$ shows that its acceptance <probability> never exceeds the <Metropolis–Hastings acceptance probability>. The name refers to <A. A. Barker>'s work on <Monte Carlo methods> for a plasma; the initials are retained without an unverified expansion.<Monte Carlo Calculations of the Radial Distribution Functions for a Proton-Electron Plasma by A. A. Barker>[Australian Journal of Physics 18, 119–134 (1965), bibliographic provenance]{ref}
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