We use Metropolis Hastings to sample from the posterior over logical forms, using one of the parser models as the proposal distribution.
if we choose the feedback parser as the proposal distribution, and initialize the Markov Chain with a logical form sampled from the task parser, it can be shown that the acceptance probability r for the first sample conveniently simplifies to the following expression: whereŷ t andŷ f are logical forms sampled from the task and feedback parsers respectively.