From the general problem to its implementation in JAGS

The most general problem would be to evaluate the joint conditional probability of the uncertain (`unobserved') quantities, conditioned by the `observed' (`known'/ `assumed'/`postulated') ones, that is, in this case⁴⁶ (see Appendix A),

$$f(p,n_I,n_{NI},n_{P_I},n_{P_{NI}},\pi_1,\pi_2\,\vert\,n_P,n_s,r_1,s_1,r_2,s_2)\,,$$ (79)

although in practice we are indeed interested in $f(p\,\vert\,n_P,n_s,r_1,s_1,r_2,s_2)$, and perhaps in $f(n_I\,\vert\,n_P,n_s,r_1,s_1,r_2,s_2)$ and $f(n_{P_I}\,\vert\,n_P,n_s,r_1,s_1,r_2,s_2)$. This is done marginalizing Eq. () i.e. summing (or integrating, depending on their nature) over the variables on which we are not interested (see Appendix A). As commented in the same appendix, Eq. () is obtained, apart from a normalization factor, from

$$f(p,n_I,n_{NI},n_{P_I},n_{P_{NI}},\pi_1,\pi_2,n_P,n_s,r_1,s_1,r_2,s_2),$$ (80)

and the latter from a properly chosen chain rule. The steps used to build up Eq. () by the proper chain rule are exactly the instructions given to JAGS to set up the model, if we start from the bottom of the diagram of Fig.  and ascend through the parents (see Sec. ):

model {
  nP ~ sum(nP.I, nP.NI)
  nP.I ~ dbin(pi1, n.I)
  nP.NI ~ dbin(pi2, n.NI)
  pi1 ~ dbeta(r1, s1)
  pi2 ~ dbeta(r2, s2)
  n.I ~ dbin(p, ns)
  n.NI <- ns - n.I
  p ~ dbeta(r0,s0)       
}

The differences with respect to the JAGS model of Sec.  are

• the last instruction there, `fP $<$- nP/ns', is here irrelevant;
• we have to add a prior to p, because all unobserved nodes having no parents need a prior (for practical convenience, as we have seen in Sec. , we shall use a Beta distribution, as indicated in the code);
• the sequence of the statements has been changed, but this has been done only in order to stress the analogy with the chain rule constructed ascending the graphical model of Fig.  (let us remind that the order is irrelevant for JAGS, which organizes all statements at the stage of compilation).

Hereafter we proceed using, very conveniently, JAGS, showing in Sec.  the steps needed from writing down the chain rule till the exact evaluation of $f(p)$ after marginalization.