Correlation in results caused by systematic errors

We can easily extend Eqs. (73), (77), and (79) to a joint inference of several variables, which, as we have seen, are nothing but parameters ${\mbox{\boldmath$\theta$}}$ of suitable models. Using the alternative ways described in Sects. 6.1 and 6.2, we have

$$p({\mbox{\boldmath$\theta$}} \,\vert\,{\mbox{\boldmath$d$}},{\mbox{\boldmath$h$}},I_0) \propto \ p({\mbox{\boldmath$d$}} \,\vert\,{\mbox{\boldmath$\theta$}},{\mbox{\boldmath$h$}},I_0) \, p_0({\mbox{\boldmath$\theta$}} \,\vert\,I_0)$$ (80)

$$p({\mbox{\boldmath$\theta$}} \,\vert\,{\mbox{\boldmath$d$}},I_0) = \int\! p({\mbox{\boldmath$\theta$}} \,\vert\,{\mbox{\boldmath$d$}... ...$}},I_0) \,p({\mbox{\boldmath$h$}} \,\vert\,I_0)\,\mbox{d}{\mbox{\boldmath$h$}}$$ (81)

and

$$p({\mbox{\boldmath$\theta$}},{\mbox{\boldmath$h$}} \,\vert\,{\mbox{\boldmath$d$}},I_0) \propto \ p({\mbox{\boldmath$d$}} \,\vert\, {\mbox{\boldmath$\theta$}},{\... ...h$}},I_0) \, p_0({\mbox{\boldmath$\theta$}},{\mbox{\boldmath$h$}} \,\vert\,I_0)$$ (82)

$$p({\mbox{\boldmath$\theta$}}\vert\,{\mbox{\boldmath$d$}},I_0) = \int\!p({\mbox{\boldmath$\theta$}},{\mbox{\boldmath$h$}} \,\vert\,{\mbox{\boldmath$d$}},I_0)\,\mbox{d}{\mbox{\boldmath$h$}}\,,$$ (83)

respectively. The two ways lead to an identical result, as it can be seen comparing Eqs. (81) and (83).

Take a simple case of a common offset error of an instrument used to measure various quantities $\mu_i$, resulting in the measurements $d_i$. We model each measurement as $\mu_i$ plus an error that is Gaussian distributed with a mean of zero and a standard deviation $\sigma_i$. The calculation of the posterior distribution can be performed analytically, with the following results (see D'Agostini 1999c for details):

• The uncertainty in each $\mu_i$ is described by a Gaussian centered at $d_i$, with standard deviation $\sigma(\mu_i)=\sqrt{\sigma_i^2+\sigma_z^2}$, consistent with Eq. (76).
• The joint posterior distribution $p(\mu_1,\mu_2,\ldots)$ does not factorize into the product of $p(\mu_1)$, $p(\mu_2)$, etc., because correlations are automatically introduced by the formalism, consistent with the intuitive thinking of what a common systematic should do. Therefore, the joint distribution will be a multi-variate Gaussian that takes into account correlation terms.
• The correlation coefficient between any pair $\{\mu_i,\mu_j\}$ is given by
  

  $$\rho(\mu_i,\mu_j) = \frac{\sigma_z^2}{\sigma(\mu_i)\,\sigma(\mu_j)} = \frac{\sigma_z^2} {\sqrt{(\sigma_{i}^2+\sigma_z^2)\,(\sigma_j^2+\sigma_z^2)}}\,.$$ (84)

  
  
  We see that $\rho(\mu_i,\mu_j)$ has the behavior expected from a common offset error; it is non-negative; it varies from practically zero, indicating negligible correlation, when ( $\sigma_z\ll \sigma_i$), to unity ( $\sigma_z\gg \sigma_i$), when the offset error dominates.