Approximated solution for non-linear fits with normal errors

Linearity implies that the arguments of the exponential of the integrand in Eq. (28) contains only first and second powers of $\mu_{x_i}$, and then the integrals has a closed solution. Though this is not true in general, the linear case teaches us how to get an approximated solution of the problem. We can take first order expansions of $\mu_{y}(\mu_{x},{\mbox{\boldmath$\theta$}})$ around each $x_i$

$$\mu_y(\mu_{x_i};{\mbox{\boldmath$\theta$}}) \approx \mu_y(x_i;{\mbox{\boldmath$\theta$}}) \, + \, \mu_y^{\,\prime}(x_i;{\mbox{\boldmath$\theta$}})\cdot (\mu_{x_i}-x_i)\,.$$ (31)

The difference $y_i-m\,\mu_{x_i}-c$ in Eq. (28), that was indeed equal to $y_i-\mu_y(\mu_{x_i};{\mbox{\boldmath$\theta$}})$ in the general case, using the linear approximation becomes

$$y_i - \mu_y(x_i;{\mbox{\boldmath $\theta$}}) - \mu_y^{\,\pri... ..._y^{\,\prime}(x_i;{\mbox{\boldmath $\theta$}})\cdot x_i\,] \,,$$

i.e. we have the following replacements in Eqs. (28)-(30):

$$m \rightarrow \mu_y^{\,\prime}(x_i;{\mbox{\boldmath$\theta$}})$$ (32)

$$c \rightarrow \mu_y(x_i;{\mbox{\boldmath$\theta$}}) - \mu_y^{\,\prime}(x_i;{\mbox{\boldmath$\theta$}})\cdot x_i\,.$$ (33)

The approximated equivalent of Eq. (30) is then

$$f({\mbox{\boldmath$\theta$}}\,\vert\,{\mbox{\boldmath$x$}},{\mbox{\boldmath$y$}},I) \propto\approx \prod_i \frac{1}{\sqrt{\sigma_{y_i}^2+{\mu_y^{\,\prime}}^2 (x_i;{... ...math$\theta$}})\cdot\sigma_{x_i}^2] } \right]}\, f(\theta\,\vert\,I)\,,\ \ \ \$$ (34)

where the unusual symbol ` $\propto\approx$' stands for `approximately proportional to'.