Reweighting of conditional inferences

The values of the influence variables and their uncertainties contribute to our background knowledge $I$ about the experimental measurements. Using $I_0$ to represent our very general background knowledge, the posterior pdf will then be $p(\mu \,\vert\,{\mbox{\boldmath$d$}},{\mbox{\boldmath$h$}},I_0)$, where the dependence on all possible values of ${\mbox{\boldmath$h$}}$ has been made explicit. The inference that takes into account the uncertain vector ${\mbox{\boldmath$h$}}$ is obtained using the rules of probability (see Tab. 1) by integrating the joint probability over the uninteresting influence variables:

$$p(\mu \,\vert\,{\mbox{\boldmath$d$}}, I_0) = \int\! p(\mu,{\mbox{\boldmath$h$}} \,\vert\,{\mbox{\boldmath$d$}}, I_0) \,\mbox{d}{\mbox{\boldmath$h$}}$$ (72)

$$= \int\! p(\mu \,\vert\,{\mbox{\boldmath$d$}},{\mbox{\boldmath$h$}},I_0) \,p({\mbox{\boldmath$h$}} \,\vert\,I_0)\,\mbox{d}{\mbox{\boldmath$h$}} \,.$$ (73)

As a simple, but important case, let us consider a single influence variable given by an additive instrumental offset $z$, which is expected to be zero because the instrument has been calibrated as well as feasible and the remaining uncertainty is $\sigma_z$. Modelling our uncertainty in $z$ as a Gaussian distribution with a standard deviation $\sigma_z$, the posterior for $\mu$ is

$$p(\mu \,\vert\,d,I_0) = \int_{-\infty}^{+\infty}\!p(\mu \,\vert\,d,z,\sigma,I_0) \,p(z\,\vert\sigma_z,I_0)\,\mbox{d}z$$ (74)

$$= \int_{-\infty}^{+\infty}\! \frac{1}{\sqrt{2\pi}\,\sigma}\exp\left[ -\frac{(\mu-(d-z))^2}{2\,\sigma^2}\right] \times \,$$

$$\ \ \ \ \ \ \ \frac{1}{\sqrt{2\pi}\,\sigma_z}\exp\left[ -\frac{z^2}{2\,\sigma_z^2}\right] \,\mbox{d}z$$ (75)

$$= \frac{1}{\sqrt{2\pi}\,\sqrt{\sigma^2+\sigma_z^2}}\exp\left[ -\frac{(\mu-d)^2}{2\,(\sigma^2+\sigma_z^2)}\right] \, .$$ (76)

The result is that the net variance is the sum of the variance in the measurement and the variance in the influence variable.