Linear fit with normal errors on both axes

To apply the general formulas of the previous section we only need to make explicit $\mu_{y_i}(\mu_{x_i},{\mbox{\boldmath$\theta$}})$ and the error functions, and finally integrate over $\mu_{x_i}$. In the case of linear fit with normal errors the individual contributions to the likelihoods become

$${\cal L}_i(m,c\,;\,x_i,y_i) = k_{x_i} \int \frac{1}{\sqrt{2\pi}\, \sigma_{x_i}}\, \exp{ \left[ ... ...\frac{(y_i-m\,\mu_{x_i}-c)^2} {2\,\sigma_{y_i}^2} \right] } \,\, d{\mu_{x_i}}\,$$ (28)

$$= k_{x_i} \, \frac{1}{\sqrt{2\pi}\, \sqrt{\sigma_{y_i}^2+m^2\,\sigm... ...\frac{(y_i-m\,x_i-c)^2} {2\, (\sigma_{y_i}^2+m^2\,\sigma_{x_i}^2) } \right]}\,,$$ (29)

that, inserted into Eq. (25), finally give

$$f(m,c\,\vert\,{\mbox{\boldmath$x$}},{\mbox{\boldmath$y$}},I) \propto \prod_i \frac{1}{\sqrt{\sigma_{y_i}^2+m^2\,\sigma_{x_i}^2}}\, \ex... ...)^2} {2\, (\sigma_{y_i}^2+m^2\,\sigma_{x_i}^2) } \right]}\, f(m,c\,\vert\,I)\,.$$ (30)

The effect of the error of the $x$-values is to have an effective standard error on the $y$-values that is the quadratic combination of $\sigma_y$ and $\sigma_x$, the latter `propagated' to the other coordinate via the slope $m$ (this result can be justified heuristically by dimensional analysis).