Correlations among the values jointly inferred in the global sceptical analysis

Let us conclude showing further details concerning the MCMC sampling. Here is the output of the R command summary(chain) concerning the ten model parameters:²⁴

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

         Mean      SD  Naive SE Time-series SE
mu   493.6769 0.01285 1.285e-05      3.564e-05
r[1]   0.8101 0.48421 4.842e-04      5.354e-04
r[2]   1.0698 0.69051 6.905e-04      9.878e-04
r[3]   0.8171 0.48464 4.846e-04      5.161e-04
r[4]   0.8903 0.53044 5.304e-04      5.713e-04
r[5]   0.8175 0.49282 4.928e-04      5.288e-04
r[6]   4.5338 3.03359 3.034e-03      4.704e-03
r[7]   1.1908 0.70179 7.018e-04      7.371e-04
r[8]   0.8121 0.48643 4.864e-04      5.035e-04
r[9]   2.0938 1.76421 1.764e-03      3.675e-03

2. Quantiles for each variable:

         2.5%      25%      50%      75%   97.5%
mu   493.6497 493.6680 493.6778 493.6872 493.697
r[1]   0.3588   0.5322   0.6889   0.9332   1.986
r[2]   0.4098   0.6661   0.9014   1.2591   2.736
r[3]   0.3617   0.5381   0.6958   0.9420   1.999
r[4]   0.3928   0.5865   0.7584   1.0274   2.175
r[5]   0.3608   0.5370   0.6954   0.9426   2.007
r[6]   1.3815   2.7852   3.8487   5.4038  11.809
r[7]   0.5333   0.7871   1.0145   1.3696   2.905
r[8]   0.3644   0.5374   0.6922   0.9338   1.979
r[9]   0.4660   0.9894   1.6476   2.6319   6.396

As we can see, at this point the most suspicious measurement is the 6-th of the complete list, as we can better judge from the quantiles indicating that, for example, there is only about 2.5% probability that $r_6$ is below $1.38$. Let us also give a look at the correlation matrix:²⁵

        mu  r[1]  r[2]  r[3]  r[4]  r[5]  r[6]  r[7]  r[8]  r[9]
mu    1.00 -0.15  0.33  0.07  0.15 -0.01  0.40 -0.11 -0.10 -0.61
r[1] -0.15  1.00 -0.04  0.00 -0.02  0.01 -0.06  0.02  0.02  0.10
r[2]  0.33 -0.04  1.00  0.04  0.05  0.02  0.14 -0.03 -0.03 -0.20
r[3]  0.07  0.00  0.04  1.00  0.01  0.02  0.03 -0.01  0.00 -0.03
r[4]  0.15 -0.02  0.05  0.01  1.00  0.00  0.06 -0.02 -0.01 -0.09
r[5] -0.01  0.01  0.02  0.02  0.00  1.00  0.00  0.00  0.00  0.02
r[6]  0.40 -0.06  0.14  0.03  0.06  0.00  1.00 -0.04 -0.04 -0.25
r[7] -0.11  0.02 -0.03 -0.01 -0.02  0.00 -0.04  1.00  0.01  0.07
r[8] -0.10  0.02 -0.03  0.00 -0.01  0.00 -0.04  0.01  1.00  0.06
r[9] -0.61  0.10 -0.20 -0.03 -0.09  0.02 -0.25  0.07  0.06  1.00

The $r_i$ with some sizable correlation with the mass are, in the order, $r_9$, $r_6$ and $r_3$, i.e. those related to the individual results that mostly differ from the barycenter of the final pdf of the mass got by the MCMC sampling (see Fig. 15 in order to get an idea of the reason for the sign of each correlation coefficient).