Algorithm 2.

Consensus Monte Carlo and Markov Chain Monte Carlo

Input: positive group $P$, unlabeled group $U$, number of splits $S$ in consensus Monte Carlo, number of iterations $T$ and burn-in size $B$ in MCMC, prior distributions $p\left(\theta \right)$ for the set of all parameters $\theta$ in the model

Output: posterior distributions $p\left(\theta \mid \right.\text{Data})$ and classification probabilities $p\left(x_u\in P\mid \text{Data})\right.$

Split $U$ into $S$ subgroups $\left\{U^{\left(1\right)},\dots ,U^{\left(S\right)}\right\}$ with equal sizes

for $s=1$ to $S$ do:

Combine $P$ and $U^{\left(s\right)}$ as data $D^{\left(s\right)}$

Perform MCMC on data $D^{\left(s\right)}$ to obtain $T$ posterior samples for ${\theta }_t^{\left(s\right)},t=1,\dots ,T$ in the loop:

for $t=1$ to $T$

For the collection $\theta =\left\{{\psi }_1,\dots ,{\psi }_m\right\}$, where each ${\psi }_j$ represents a parameter in the model

Draw a sample from its full conditional distribution $p\left({\psi }_j\mid {\psi }_{-j},D^{\left(s\right)}\right)$, denote ${\psi }_{j,t}^{\left(s\right)}$

Let ${\theta }_t^{\left(s\right)}=\left\{{\psi }_{1,t}^{\left(s\right)},\dots ,{\psi }_{m,t}^{\left(s\right)}\right\}$

end for

Discard $B$ burn-in samples and combine the rest posterior samples from all subgroups

${\theta }_t={\sum }_1^S{w}^{\left(s\right)}{\theta }_t^{\left(s\right)}/{\sum }_1^S{w}^{\left(s\right)}$ according to certain weights $w^{\left(s\right)}$ for the subgroups

end for

Return $p\left(\theta \mid \right.\text{Data})$ approximated by posterior samples ${\theta }_t$ and $p\left(x_u\in P\mid \right.\text{Data})$ computed using posterior samples ${\theta }_t$