Consider the family of distributions having the form $$f = \frac{1}{Z({\alpha,\beta,A,B})}\prod\_{j=1}^{n}\left(\frac{1}{1+e^{-x+A\_{j}}}\right)^{\alpha\_{j}}\prod\_{j=1}^{m}\left(\frac{1}{1+e^{x+B\_{j}}}\right)^{\beta\_{j}}$$


$Z$ is the normalizing constant. Typically $n$ and $m$ are both around 3. The $\alpha\_j$ and $\beta\_j$ are in $\mathbb{R}^+$ and the $A\_j$ and $B\_j$ in $\mathbb{R}$.


I believe a sufficient statistic for this family of distribution is the set of expectations
$$\left\{\mathbf{E}(\log(1+e^{-x+A\_j}))\right\}\_{j\leq n} \cup \left\{\mathbf{E}(\log(1+e^{x+B\_j}))\right\}\_{j \leq m}$$.


This motivates the following questions.


(1) Is there an analytic form for $\mathbf{E}(\log(1+e^{-x+A\_j}))$? if so (or if not), is there an efficient way to numerically approximate it?


(2) What of $Z$?


Calculating $Z$ would be a good first step it seems, as (if I'm not mistaken) $\frac{dZ}{d \alpha\_i} = -Z\mathbf{E}(\log(1+e^{-x+A\_j}))$



Computing a particular expection for a family of distribution
With the change of variable $y = 1/(1+e^{-x})$, the integral is a special case of the Lauricella hypergeometric series


<http://en.wikipedia.org/wiki/Lauricella_hypergeometric_series#Integral_representation_of_FD>


