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A. A. Barker authored a 1965 calculation of radial distribution functions for a proton–electron plasma. The associated Barker acceptance probability supplies a useful rule for equilibrium Markov chain Monte Carlo: if the probability-and-proposal ratio for a proposed move is $r$, accept with probability $r/(1+r)$. For reverse moves the ratio is $1/r$, so the acceptance probabilities have ratio $r$, giving detailed balance. The rule connects a microscopic sampling decision with the desired equilibrium distribution.^[ref]

Author credited under these initials on the 1965 Monte Carlo method plasma paper Monte Carlo Calculations of the Radial Distribution Functions for a Proton-Electron Plasma by A. A. Barker. The initials are not expanded because an exact identity beyond the publication credit has not been established here. University of Adelaide thesis addendum identifying the 1965 publication.

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, $t/(1+t)\le \min (1,t)$ 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.^[ref]