# Probability distribution modeling of discrete differential estimates using exponential family?

For one reason or another I started to estimate histograms of gradients of natural images.

Let us assume that I want to try to fit a function of the exponential family to this data.

For example $$f(t) = C \exp\left(-\sum_{\forall i}\left(\frac{{|t|}}{b_i}\right)^{a_i}\right)$$

To data where we have measured pairs $$(t_k,\hat f(t_k)), k = 1,2,\cdots,n$$ and we want to estimate $$C,a_i,b_i$$

So my question is, how can I find some method to try and optimize

$$\min_{C,a_i,b_i}\left\{\sum_{\forall k} (f(t_k) - \hat f(t_k))^2\right\}$$

Here is an estimation I made manually with just one term. In the present case $$f(t) = C \exp\left(-\sum_{\forall i}\left(\frac{{|t|}}{b_i}\right)^{a_i}\right)$$ with change of variable $$y(t)=\ln(f(t))$$ and forgetting the too low values $$|f(t|| in order to have $$f(t)>0$$ for the kept values : $$y(t)=\ln(C)-\sum_{\forall i}\left(\frac{{|t|}}{b_i}\right)^{a_i}$$ This is a sum of powers. In the above referenced paper this case is treated in full details only in the simplest case $$i=1$$. The method is generalisable for higher $$i$$ with successive numerical integrations. But this is not done in the paper and would require more preliminary studies.