Inferring $p$ and $n_I$ with our `standard parameters'

Let us start using as $n_P$ the expected value of positives of $\approx 2010$, obtained from what has been our starting set of parameters through the paper, that is $p=0.1$ with $n_s=10000$, with the uncertain parameters $\pi_1$ and $\pi_2$ modeled by Beta distributions with $(r_1=409.1,\,s_1=9.1)$ and $(r_2=25.2,\,s_2=193.1)$, respectively. Also for the prior of $p$ we use a Beta, starting with $r_0=s_0=1$, that models a flat prior, although we obviously do not believe that $p=0$ or $p=1$ are possible. We shall discuss in Sec.  the role of such at a first glance an insane prior (see also Sec. ).

These are the R command to set the parameters of the game, call JAGS and show some results (for the complete script see Appendix B.10).

#---- data and parameters
nr = 1000000
ns = 10000
nP = 2010
r0 = s0 = 1
r1 = 409.1; s1 = 9.1
r2 = 25.2;  s2 = 193.1 

# define the model and load rjags (omitted)
# ......................................... 

#---- call JAGS ---------
data <- list(ns=ns, nP=nP, r0=s0, s0=s0, r1=r1, s1=s1, r2=r2, s2=s2)  
jm <- jags.model(model, data)
update(jm, 10000)
to.monitor <-  c('p', 'n.I')
chain <- coda.samples(jm, to.monitor, n.iter=nr)

#---- show results
print(summary(chain))
plot(chain, col='blue')

Here are the results shown by `summary(chain)'

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

         Mean        SD  Naive SE Time-series SE
n.I 991.12477 225.85901 2.259e-01      16.079460
p     0.09919   0.02278 2.278e-05       0.001601

2. Quantiles for each variable:

         2.5%       25%      50%       75%     97.5%
n.I 506.00000 838.00000 1012.000 1153.0000 1389.0000
p     0.05046   0.08372    0.101    0.1155    0.1396

So, for this run we get $p=0.0992 \pm 0.023$ and a number of infectees in the sample equal to $991\pm 226$, in agreement with our expectations. The results of the Monte Carlo sampling are shown in the `densities' of Fig. ,

Figure: Plots showing some JAGS results (see text).

together with the `traces', i.e. the values of the sampled variables during the $10^6$ iterations.⁴⁷As it is easy to guess and as it appears from the two traces of the figure, there is some degree of correlation between the two variables, because they are obtained in a joint inference. The correlation is made evident in the scatter plot of Fig. 

Figure: Scatter plot of $p$ vs $n_I$, showing the very high correlation between the two variables.

and quantified by $\rho(p,n_I) = 0.9914.\,$⁴⁸