which it ergodically converges. A number of recipes for suitable transition operators can be constructed, such as Metropolis sampling [@metropolisequation] and Gibbs sampling[@gemanstochastic]; most of the results we present rely on variations on Gibbs sampling. More details on efficient implementations of stochastic transition circuits for Gibbs sampling and Metropolis-Hastings can be found elsewhere [@VMEJ-circuits-supplemental]. Note that if the input $X$ represents observed data and the state $S$ represents a hypothesis, then the transition circuit implements Bayesian inference.

We can scale up to challenging problems by exploiting the composition laws that stochastic transition circuits support. Consider a probability distribution defined over three variables $P(A,B,C) =
P(A)P(B|A)P(C|A)$. We can construct a transition circuit that samples from the overall state $(A,B,C)$ by composing transition circuits for updating $A|BC$, $B|A$ and $C|A$; this assembly is shown in Figure 2C. As long as the underlying probability model does not have any zero-probability states, ergodic convergence of each constituent transition circuit then implies ergodic convergence of the whole assembly[@andrieuintroduction]. The only requirement for scheduling transitions is that each circuit must be left fixed while circuits for variables that interact with it are transitioning. This scheduling requirement — that a transition circuit’s value be held fixed while 